On the regularity of solutions to a class of nonlinear dispersive equations

Abstract

We shall study special regularity properties of solutions to some nonlinear dispersive models. The goal is to show how regularity on the initial data is transferred to the solutions. This will depend on the spaces where regularity is measured.

1 Introduction

The aim of this work is to study special regularity properties of solutions to the initial value problem (IVP) associated to some nonlinear dispersive equations of Korteweg-de Vries (KdV) type and related models.

The starting point is a result found by Isaza, Linares and Ponce [15] concerning the solutions of the IVP associated to the k-generalized KdV equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu+\partial _x^3u +u^k\partial _xu=0, \,x, t\in \mathbb R, \, k\in \mathbb Z^{+},\\ u(x,0)=u_0(x). \end{array}\right. } \end{aligned}$$

(1.1)

To state our result we first recall the following local well-posedness for the IVP (1.1) established in [20]:

Theorem A1

[20] If $\,u_0\in H^{{3/4}^{+}}(\mathbb R)$, then there exist $T\!=\!T(\Vert u_0\Vert _{_{{\frac{3}{4}}^{+}\!, 2}};k)>0$ and a unique solution of the IVP (1.1) such that

• (i)
  $$\begin{aligned} u\in C([-T,T] : H^{{3/4}^{+}}(\mathbb R)), \end{aligned}$$
• (ii)
  $$\begin{aligned} \partial _x u\in L^4([-T,T]: L^{\infty }(\mathbb R)), \quad (\text {Strichartz}), \end{aligned}$$
• (iii)
  $$\begin{aligned} \underset{x}{\sup }\int _{-T}^{T} |J^r\partial _x u(x,t)|^2\,dt<\infty \quad \text {for } \ r\in [0,{3/4}^{+}], \end{aligned}$$

  (1.2)
• (iv)
  $$\begin{aligned} \int _{-\infty }^{\infty }\, \sup _{-T\le t\le T}|u(x,t)|^2\,dx < \infty , \end{aligned}$$

  with $\;J=(1-\partial _x^2)^{1/2}$. Moreover, the map data-solution, $\,u_0\rightarrow \,u(x,t)$ is locally continuous (smooth) from $\,H^{3/4+}(\mathbb R)$ into the class $\,X^{3/4+}_T\,$ defined in (1.2).

If $k\ge 2$, then the result holds in $H^{{3/4}}(\mathbb R)$.

We recall the definition of the Sobolev spaces $H^{{s}}(\mathbb R)$ for index $s\ge 0$:

$$\begin{aligned} H^{{s}}(\mathbb R)=\{f\in L^2(\mathbb R)\,:\,\Vert f\Vert _{s,2}=\Vert J^s f\Vert _2<\infty \} \end{aligned}$$

where

$$\begin{aligned} \Vert J^s f\Vert _2=\left( \int _{-\infty }^{\infty }(1+\xi ^2)^{s}|\widehat{f}(\xi )|^2d\xi \right) ^{1/2}, \end{aligned}$$

and $\,\widehat{f}$ denotes the Fourier transform of f. In particular, if $s>s'>0$, then

$$\begin{aligned} H^{s}(\mathbb R)\subset H^{s'}(\mathbb R) \subset L^2(\mathbb R). \end{aligned}$$

Thus, $H^{3/4}(\mathbb R)$ represents the Sobolev space of index 3 / 4 and

$$\begin{aligned} H^{{3/4}^{+}}(\mathbb R) =\bigcup _{s>3/4} H^{s}(\mathbb R). \end{aligned}$$

For a detailed discussion on the best available local and global well-posedness results of the IVP (1.1) we refer to [15, 22].

Now we enunciate the result obtained in [15] regarding propagation of regularities which motivates our study here:

Theorem A2

[15] If $u_0\in H^{{3/4}^{+}}(\mathbb R)$ and for some $\,l\in \mathbb Z^{+},\,\,l\ge 1$ and $x_0\in \mathbb R$

$$\begin{aligned} \Vert \,\partial _x^l u_0\Vert ^2_{L^2((x_0,\infty ))}=\int _{x_0}^{\infty }|\partial _x^l u_0(x)|^2dx<\infty , \end{aligned}$$

(1.3)

then the solution $u=u(x,t)$ of the IVP (1.1) provided by Theorem A1 satisfies that for any $v>0$ and $\epsilon >0$

$$\begin{aligned} \underset{0\le t\le T}{\sup }\,\int ^{\infty }_{x_0+\epsilon -vt } (\partial _x^j u)^2(x,t)\,dx<c, \end{aligned}$$

(1.4)

for $j=0,1, \dots , l$ with $c = c(l; \Vert u_0\Vert _{{3/4}^{+},2};\Vert \,\partial _x^l u_0\Vert _{L^2((x_0,\infty ))} ; v; \epsilon ; T)$.

In particular, for all $t\in (0,T]$, the restriction of $u(\cdot , t)$ to any interval of the form $(a, \infty )$ belongs to $H^l((a,\infty ))$.

Moreover, for any $v\ge 0$, $\epsilon >0$ and $R>0$

$$\begin{aligned} \int _0^T\int _{x_0+\epsilon -vt}^{x_0+R-vt} (\partial _x^{l+1} u)^2(x,t)\,dx dt< c, \end{aligned}$$

(1.5)

with $c = c(l; \Vert u_0\Vert _{_{{3/4}^{+},2}};\Vert \,\partial _x^l u_0\Vert _{L^2((x_0,\infty ))} ; v; \epsilon ; R; T)$.

This tells us that the $H^l$-regularity on the right hand side of the data travels forward in time with infinite speed. Notice that since the equation is reversible in time a gain of regularity in $H^s(\mathbb R)$ cannot occur so at $t>0$, $u(\cdot , t)$ fails to be in $H^j(\mathbb R)$ due to its decay at $-\infty$. In fact, it follows from the proof in [15] that for any $\delta >0$ and $t\in (0,T)$ and $j=1,\dots ,l$

$$\begin{aligned} \int _{-\infty }^{\infty } \frac{1}{\langle x_{-}\rangle ^{j+\delta }} (\partial _x^j u)^2(x,t)\,dx \le \frac{c}{t}, \end{aligned}$$

with $c= c(\Vert u_0\Vert _{{3/4}^{+},2}; \Vert \partial _x^j u_0\Vert _{L^2((x_0,\infty ))}; \,x_0; \,\delta )$.

The result in [15] (Theorem A2) has been extended to the generalized Benjamin–Ono (BO) equation [16] and to the Kadomtsev–Petviashvili II equation [17]. Hence, it is natural to ask if this propagation of regularity phenomenon is intrinsically related to the integrable character of the model or as in the KdV equation is due to the form of the solution of the associated linear problem. More precisely, to the structure of its fundamental solution, i.e. the Airy function (see (3.3) below).

Indeed, for the so called k-generalized dispersive BO equation,

$$\begin{aligned} \partial _tu+u^k\partial _xu- (-\partial _x^2)^{\alpha /2}\partial _xu=0, \quad k\in \mathbb Z^{+},\quad 1\le \alpha \le 2, \end{aligned}$$

which for $\alpha =1$ corresponds to the k-generalized BO equation and $\alpha =2$ to the k-generalized KdV equation, one has that the propagation of regularities (as that presented in Theorem A2) is only known in the cases $\alpha =1$ and $\alpha =2$.

Our first result shows that this fact seems to be more general. In particular, it is valid for solutions of the general quasilinear equation of KdV type, that is,

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu + a(u,\partial _xu,\partial _x^2u)\,\partial _x^3u + b(u,\partial _xu,\partial _x^2u)=0,\\ u(x,0)=u_0(x). \end{array}\right. } \end{aligned}$$

(1.6)

where the functions $a,b:\mathbb R^3\times [0,T]\rightarrow \mathbb R$ satisfy:

• (H1) $\,\,a(\cdot ,\cdot ,\cdot )$ and $b(\cdot ,\cdot ,\cdot )$ are $C^{\infty }$ with all derivatives bounded in $[-M, M]^3$, for any $M>0$,

• (H2) given $M>0$, there exists $\kappa >1$ such that

  $$\begin{aligned} 1/\kappa \le a(x, y, z)\le \kappa \quad \text {for any } \ (x,y,z) \in [-M,M]^3, \end{aligned}$$

  and

  $$\begin{aligned} \partial _z\,b(x,y,z)\le 0 \quad \text {for } \ (x,y,z) \in [-M,M]^3. \end{aligned}$$

To establish the propagation of regularity in solutions of (1.6) of the kind described in (1.4) we shall follow the arguments and results obtained by Craig, Kappeler and Strauss in [10].

Assuming the hypotheses (H1) and (H2), the local existence and uniqueness result established in [10] affirms:

Theorem A3

[10] Let $m\in \mathbb Z^{+}$, $m\ge 7$. For any $u_0\in H^m(\mathbb R)$, there exist $T=T(\Vert u_0\Vert _{7,2})>0$ and a unique solution $u=u(x,t)$ of the IVP (1.6) satisfying,

$$\begin{aligned} u\in L^{\infty }([0,T] ; H^m(\mathbb R)). \end{aligned}$$

Moreover, for any $R>0$

$$\begin{aligned} \int \limits _0^T\int \limits _{-R}^{R} (\partial _x^{m+1} u)^2(x,t)\,dxdt <\infty . \end{aligned}$$

For our purpose here we need some (weak) continuous dependence of the solutions upon the data. Hence, we shall first prove the following “refinement”  of Theorem A3.

Theorem 1.1

Let $m\in \mathbb Z^{+}$, $m\ge 7$. For any $u_0\in H^m(\mathbb R)$ there exist $T=T(\Vert u_0\Vert _{7,2})>0$ and a unique solution $u=u(x,t)$ of the IVP (1.6) such that

$$\begin{aligned} u\in C([0,T] : H^{m-\delta }(\mathbb R))\cap L^{\infty }([0,T] : H^m(\mathbb R)), \quad \text {for all } \ \delta >0, \end{aligned}$$

(1.7)

with

$$\begin{aligned} \partial _x^{m+1}u\in L^2([0,T] \times [-R,R]),\quad \text {for all } \ R>0. \end{aligned}$$

(1.8)

Moreover, the map data solution $u_0\mapsto u(\cdot , t)$ is locally continuous from $H^m(\mathbb R)$ into $C([0,T] : H^{m-\delta }(\mathbb R))$ for any $\delta >0$.

Since our objective here is to study propagation of regularities we shall not address the problem of (strong) persistence (i.e. $u_0\in X$, then the corresponding solution $u(\cdot ,t)$ describes a continuous curve on X, $u\in C([0, T] : X)$) and the (strong) continuous dependence $u_0\mapsto u(\cdot , t)$, (i.e. the map data $\rightarrow$ solution from X into C([0, T] : X) is continuous), so that the solutions of (1.6) generate a continuous flow in $H^m(\mathbb R)$, $m\ge 7$.

Our main result concerning the solution of the IVP (1.6) is the following:

Theorem 1.2

Let $n,m \in \mathbb Z^{+}$, $n>m\ge 7$. If $u_0\in H^m(\mathbb R)$ and for some $x_0\in \mathbb R$

$$\begin{aligned} \partial _x^j u_0\in L^2((x_0,\infty ))\quad \text {for}\quad j=m+1,\dots , n, \end{aligned}$$

then the solution of the IVP (1.6) provided by Theorem 1.1 satisfies that for any $\epsilon >0$, $v>0$, and $t\in [0,T)$

$$\begin{aligned} \int \limits _{x_0+\epsilon -vt}^{\infty }|\partial _x^ju(x,t)|^2\,dx\le c(\epsilon ; v; \Vert u_0\Vert _{m,2}; \Vert \partial _x^l u_0\Vert _{L^2((x_0, \infty ))}:l=m+1,\dots ,n),\nonumber \\ \end{aligned}$$

(1.9)

for $j=m+1, \dots , n$. Moreover, for any $\epsilon >0$, $v>0$, and $R>0$

$$\begin{aligned}&\int \limits _{0}^T\int \limits _{x_0+\epsilon -vt}^{x_0+R+vt} |\partial _x^{n+1} u(x,t)|^2 \,dxdt \nonumber \\&\quad \le c(\epsilon ; v; R;\Vert u_0\Vert _{m,2}; \Vert \partial _x^l u_0\Vert _{L^2((x_0, \infty ))}:l=m+1,\dots ,n). \end{aligned}$$

(1.10)

Several direct consequences can be deduced from Theorem 1.2 for instance (for further outcomes see [15])

Corollary 1.3

Let $u\in C([0,T] : H^m(\mathbb R))$, $m\ge 7$, be the solution of the IVP (1.6) provided by Theorem 1.1. If there exist $n>m$, $a\in \mathbb R$ and $\hat{t}\in (0,T)$ such that

$$\begin{aligned} \partial _x^nu(\cdot , \hat{t})\notin L^2((a,\infty )) \end{aligned}$$

then for any $t\in (0, \hat{t})$ and any $\beta \in \mathbb R$

$$\begin{aligned} \partial _x^nu(\cdot , t)\notin L^2((\beta ,\infty )). \end{aligned}$$

Theorem 1.2 tells us that the propagation phenomenon described in Theorem A2 still holds in solutions of the quasilinear problem (1.6). This result and those in KdV, BO, KPII equations seem to indicate that the propagation of regularity phenomena can be established in systems where Kato smoothing effect [18] can be proved by integration by parts directly in the differential equation.

Since our arguments follow closely those in [10] without lost of generality and for the sake of simplicity of exposition we shall restrict the proofs of Theorems 1.1 and 1.2 to the case of the model equation

$$\begin{aligned} \partial _tu + (1+(\partial _x^2u)^2) \partial _x^3u=0. \end{aligned}$$

(1.11)

However, we shall remark that combining the arguments below with those in [10] the results in Theorem 1.2 can be extended to solution of the IVP associated to the equation

$$\begin{aligned} \partial _tu+f(t,x, u, \partial _x u,\partial _x^2u,\partial _x^3u)=0, \end{aligned}$$

under appropriate assumptions on the structure, regularity and decay of the function $f(\cdot )$. These decay assumptions are quite strong, for example, one should assume roughly that $\,f(\cdot , t,0,0,0,0)\in \mathcal S(\mathbb R)$, (see [10]). This fact seems to rule out the possibility of having a propagation of regularity phenomenun of the kind described here in models involving periodic coefficients.

Next we consider the question of the propagation of other type of regularities besides those proved before i.e. for $u_0\in H^n((x_0, \infty ))$ for some $x_0\in \mathbb R$.

We recall that the next result can be obtained as a consequence of the argument given by Bona and Saut in [3].

Theorem 1.4

Let $k\in \mathbb Z^{+}$. There exists

$$\begin{aligned} u_0\in H^1(\mathbb R)\cap C^{\infty }(\mathbb R) \end{aligned}$$

with $\Vert u_0\Vert _{1,2}\ll 1$ so that the solution $u(\cdot ,t)$ of the IVP (1.1) is global in time if $k\ge 4$ with $u\in C(\mathbb R: H^1(\mathbb R))\cap L^1_{loc}(\mathbb R:W^{1,\infty }(\mathbb R))$ and satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} u(\cdot , t)\in C^1(\mathbb R), &{} t>0, \quad t\notin \mathbb Z^{+},\\ u(\cdot , t)\in C^1(\mathbb R\backslash \{0\})\backslash C^1(\mathbb R), &{} t\in \mathbb Z^{+}. \end{array}\right. \end{aligned}$$

(1.12)

Here, $W^{j,p}(\mathbb R)$ denotes the space of functions whose distributional derivatives up to order $\,j\,$ belong to $\,L^p(\mathbb R)$.

The argument in [3] is based in a careful analysis of the asymptotic decay of the Airy function and the well-posedness of the IVP (1.1) with data $u_0(x)$ in appropriate weighted Sobolev spaces. This argument was simplified [for the case of two points in (1.12)] for the modified KdV equation $k=2$ in [23] without relying in weighted spaces. Here we shall give a direct proof of Theorem 1.4 which follows the approach in [23], i.e. it does not rely on the analysis of the decay of the Airy function and applies to all the nonlinearities.

Our method has the advantage that it can be extended to $W^{s,p}$-setting. More precisely, we shall show the following:

Theorem 1.5

• (a) Fix $k=2, 3, \dots$, let $p\in (2,\infty )$ and $j\ge 1$, $j\in \mathbb Z^{+}$. There exists

  $$\begin{aligned} u_0\in H^{3/4}(\mathbb R)\cap W^{j,p}(\mathbb R) \end{aligned}$$

  (1.13)

  such that the corresponding solution $u(\cdot ,\cdot )$ of the IVP (1.1)

  $$\begin{aligned} u\in C([-T,T]:H^{3/4}(\mathbb R)) \end{aligned}$$

  provided by Theorem A1 satisfies that there exists $t\in (0,T]$ such that

  $$\begin{aligned} u(\cdot , t)\notin W^{j,p}(\mathbb R^{+})\quad \text {and}\quad u(\cdot , -t)\notin W^{j,p}(\mathbb R^{+}). \end{aligned}$$

  (1.14)

• (b) For $k=1$, the same result holds for $j\ge 2$, $j\in \mathbb Z^{+}$.

Remark

It will follow from our proof that there exists $u_0$ as in (1.13) such that (1.14) holds in $\mathbb R^{-}$. Hence, one can conclude that regularities in $W^{j,p}(\mathbb R)$ for $p>2$ do not propagate forward or backward in time to the right or to the left.

Next we study the propagation of regularities in solutions to some related dispersive models. First, we consider the IVP associated to the Benjamin–Bona–Mahony BBM equation [2]

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu +\partial _xu +u\partial _xu-\partial _x^2\partial _tu=0, \quad x, t\in \mathbb R,\\ u(x,0)=u_0(x). \end{array}\right. } \end{aligned}$$

(1.15)

The BBM equation was proposed in [2] as a model for long surface gravity waves of small amplitude propagating in one dimension. It was introduced as a “regularized” version of the KdV equation. In most cases, the independent variable x characterizes position in the medium of propagation whilst t is proportional to elapsed time. The dependent variable u may be an amplitude, a pressure, a velocity or other measurable quantity, depending upon the physical system and the modeling stance taken.

We recall the local well-posedness for the IVP (1.15) obtained by Bona and Tzvetkov [5].

Theorem A4

[5] Let $s\ge 0$. For any $u_0\in H^s(\mathbb R)$ there exist $T=T(\Vert u_0\Vert _{s,2})>0$ and a unique solution u of the IVP (1.15)

$$\begin{aligned} u\in C([0,T] : H^s(\mathbb R)). \end{aligned}$$

Moreover, the map data-solution $u_0\mapsto u(\cdot , t)$ is locally continuous from $H^s(\mathbb R)$ into $C([0,T] : H^s(\mathbb R))$.

In [5] it was also shown that Theorem A4 is optimal in an appropriate sense (see [5] for details).

The following result describes the local propagation of regularities in solutions of the IVP (1.15).

Theorem 1.6

Let $u_0\in H^s(\mathbb R)$, $s\ge 0$. If for some $k\in \mathbb Z^{+}\cup \{0\}$, $\theta \in [0,1)$, and $\Omega \subseteq \mathbb R$ open

$$\begin{aligned} u_0\big |_{\Omega }\in C^{k+\theta }, \end{aligned}$$

then the corresponding solution $u\in C([0,T] : H^s(\mathbb R))$ of the IVP (1.15) provided by Theorem A4 satisfies that

$$\begin{aligned} u(\cdot , t)\big |_{\Omega } \in C^{k+\theta } \quad \text {for all } \ t\in [0,T]. \end{aligned}$$

Moreover,

$$\begin{aligned} u, \partial _tu \in C([0,T]: C^{k+\theta }(\Omega )). \end{aligned}$$

Remark

• (1) Theorem 1.6 tells us that in the time interval [0, T] in the $C^{k+\theta }$ setting no singularities can appear or disappear in the solution $u(\cdot ,t)$.

In particular, one has the following consequence of Theorem 1.6 and its proof:

Corollary 1.7

Let $u_0\in H^s(\mathbb R)$, $s\ge 0$. If for $a<x_0<b$, $k\in \mathbb Z^{+}\cup \{0\}$ and $\theta \in [0,1)$

$$\begin{aligned} u_0\big |_{(a,x_0)}, \;\;u_0\big |_{(x_0,b)}\in C^{k+\theta } \quad \text {and}\quad u_0\big |_{(a,b)}\notin C^{k+\theta }, \end{aligned}$$

then the corresponding solution $u\in X^s_T$ of the IVP (1.15) provided by Theorem A4 satisfies

$$\begin{aligned} u(\cdot ,t)\big |_{(a,x_0)}, \;\;u(\cdot , t)\big |_{(x_0,b)}\in C^{k+\theta } \quad \text {and}\quad u(\cdot , t)\big |_{(a,b)}\notin C^{k+\theta }. \end{aligned}$$

• (2) Theorems A2, 1.4, 1.6, and Corollary 1.7 show that solutions of the BBM equation and the KdV equation exhibit a quite different behavior regarding the propagation of regularities.

Next we consider the IVP associated to the Degasperis-Procesi (DP) equation (see [11]):

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu-\partial _x^2\partial _tu+ 4u\partial _{x}u=3\partial _xu\partial _x^2u+u\partial _{x}^3u,\,\, x,t\in \mathbb R,\\ u(x,0)=u_0(x). \end{array}\right. } \end{aligned}$$

(1.16)

The DP model was derived by Degasperis and Procesi as an example of an integrable system similar to the Camassa–Holm (CH) equation ([8])

$$\begin{aligned} \partial _tu+2\kappa \partial _xu-\partial _x^2\partial _tu+ 3u\partial _{x}u=2\partial _xu\partial _x^2u+u\partial _x^3u, \quad \kappa >0. \end{aligned}$$

(1.17)

Like the CH equation it possesses a Lax pair formulation and a bi-Hamiltonian structure leading to an infinite number of conservation laws.

Also, similar to the CH equation, the DP equation has been shown to exhibit multi-peakons solutions

$$\begin{aligned} u(x,t)=\underset{j=1}{\overset{n}{\sum }} \;\alpha _j(t) e^{-|x-x_j(t)|} \end{aligned}$$

where $(x_j(t))_{j=1}^n$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{dx_j}{dt}= \underset{k=1}{\overset{n}{\sum }} \alpha _j(t) e^{-|x-x_k(t)|}\\ \dfrac{d\alpha _j}{dt}= 2\alpha _j \,\underset{k=1}{\overset{n}{\sum }} \alpha _k(t) \,\mathrm{sgn}(x_j-x_k) \,e^{-|x-x_k(t)|}. \end{array}\right. } \end{aligned}$$

(1.18)

In [27] Yin shows that the IVP (1.16) is locally well-posed in $H^s(\mathbb R)$, $s>3/2$. It shall be remarked that in the Sobolev scale $H^s(\mathbb R)$ the result in [27] is the ”optimal” possible for the “strong” local well-posedness. More precisely, it was established by Himonas, Holliman and Grayshan [14] that for $s<3/2$ solutions of the IVP (1.16) exhibit an ill-posedness feature due to the so called norm inflection (see [14]).

One observes that the multi-peakons solutions barely fail to belong to these spaces since $e^{-|x|} \in H^s(\mathbb R)$ if and only if $s<3/2$. So we shall construct first a space where uniqueness and existence hold, where the “flow” (characteristics) is defined (see (1.21) below) and which contains the multi-peakons.

Theorem 1.8

Given $u_0\in H^{1+\delta }(\mathbb R)\cap W^{1,\infty }(\mathbb R)$, for some $\delta >0$. There exist $T\!\!=\!T(\Vert u_0\Vert _{1+\delta ,2}$, $\Vert u_0\Vert _{1,\infty })>0$ and a unique strong solution (limit of classical solution)

$$\begin{aligned} u\in C([0,T]:H^1(\mathbb R))\cap L^{\infty }([0,T]:H^{1+\delta }(\mathbb R))\cap L^{\infty }([0,T]:W^{1,\infty }(\mathbb R)). \end{aligned}$$

Moreover, if $u_{n_0}\rightarrow u_0$ in $H^{1+\delta }(\mathbb R)\cap W^{1,\infty }(\mathbb R)$, then the corresponding solutions $u_n, u$ satisfy that $u_n\rightarrow u$ in $C([0,T] : H^1(\mathbb R))$.

With Theorem 1.8 in hand we can show the following result regarding the propagation of regularities in solutions of the DP equation.

Theorem 1.9

Suppose $u_0\in H^{1+\delta }(\mathbb R)\cap W^{1,\infty }(\mathbb R)$, for some $\delta >0$, such that for some open subset $\Omega \subseteq \mathbb R$

$$\begin{aligned} u_0\big |_{\Omega }\in C^1. \end{aligned}$$

(1.19)

Then the local solution of the IVP (1.16) $u=u(x,t)$ provided by Theorem 1.8 satisfies

$$\begin{aligned} u(\cdot , t)\in C^1(\Omega _t), \end{aligned}$$

(1.20)

where $\Omega _t=\Phi _t(\Omega )$ and $\Phi _t(x_0)=X(t;x_0)$ is the map given by

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{dX}{dt}= u(X(t),t)\\ X(0)=x_0. \end{array}\right. } \end{aligned}$$

(1.21)

Remark

• (1) Notice that

  $$\begin{aligned} u\in C([0,T] : H^s(\mathbb R)) \hookrightarrow C_b(\mathbb R\times [0,T]) \end{aligned}$$

  and

  $$\begin{aligned} u\in L^{\infty }([0,T] : W^{1,\infty }(\mathbb R)) \end{aligned}$$

  which guarantees that the flow in (1.21) is well defined (see for instance [9]).

• (2) Theorem 1.9 describes the propagation of $C^1$ singularities. In particular one has the following consequence of Theorem 1.9.

Corollary 1.10

If $u_0\in H^{1+\delta }(\mathbb R)\cap W^{1,\infty }(\mathbb R)$, for some $\delta >0$, (and for some $\Omega \subseteq \mathbb R$ open) $u_0\big |_{\Omega }\in C^1(\Omega \backslash \{\tilde{x}_j\}_{j=1}^{N})$, then the corresponding solution u provides by Theorem 1.9 satisfies for $t\in [0, T]$

$$\begin{aligned} u(\cdot , t)\big |_{\Omega _t}\in C^1(\Omega _t\backslash \{x_j(t)\}_{j=1}^{N}) \end{aligned}$$

where $\Omega _t$ and $x_j(t)$ are defined by the flow $\Phi _t(x_0)=X(t; x_0)$ as in (1.21), $\Omega _t=\Phi (\Omega )$ and $x_j(t)=\Phi _t(\tilde{x}_j)$.

This tell us that all $C^1$-singularities for data in $H^{1+\delta }\cap \mathrm{Lip}$ propagate with the flow as in the case of multi-peakons see (1.18).

• (3) Since $m(x,t)= (1-\partial _x^2) u(x,t)$ satisfies

  $$\begin{aligned} \partial _tm+u\partial _xm+3\partial _xu\, m=0, \end{aligned}$$

  (1.22)

  the propagation at the $C^2$ level and beyond follows directly from the equation (1.22). Therefore, the result in Theorem 1.9 extends to $C^k$, $k\in \mathbb Z^{+}$, in (1.19) and (1.20).

• (4) A result like that described in Theorem 1.8 (when uniqueness, existence and the ”flow” is defined in a space containing the peakons) is unknown for the CH equation (see [6, 24, 26]). Similarly for Theorem 1.9.

Finally, we consider the 1D version of the Brinkman model [7]

$$\begin{aligned} {\left\{ \begin{array}{ll} \phi \partial _t\rho +\partial _x(\rho v)= F(t,\rho ),\\ (-\tilde{\mu }\partial _x^2+\frac{\mu }{\kappa }\big )v=-\partial _xP(\rho ). \end{array}\right. } \end{aligned}$$

(1.23)

This system models fluid flow in certain porous media. It has been useful to treat high porosity systems and a rigidly bounded porous medium. It also has been used to investigate different convective heat transfer problems in porous media (see [25] and references therein). Here $\rho$ is the fluid’s density, v its velocity, $P(\rho )$ is the pressure and F the external mass flow rate. The physical parameters $\mu$, $\kappa$, $\tilde{\mu }$, and $\phi$ represent the fluid viscosity, the porous media permeability, the pure fluid viscosity, and the porosity of the media, respectively. To simplify we shall assume (without loss of generality)

$$\begin{aligned} \phi =\mu =\kappa =\tilde{\mu }=1\quad \text {and}\quad F\equiv 0, \end{aligned}$$

and that $P(\rho )=\rho ^2$

It will be clear from our method of proof that these are not necessary restrictions. Thus the equation in (1.23) becomes

$$\begin{aligned} \partial _t\rho -\partial _x(\rho (1-\partial _x^2)^{-1}\partial _x(\rho ^2))=0 \quad x, t\in \mathbb R. \end{aligned}$$

(1.24)

In [1] Arbieto and Iorio established the local well-posedness of the associated IVP to (1.24) in $H^s(\mathbb R)$, $s>3/2$.

Our first outcome in this regard shows that the result in [1] can be improved to $H^s(\mathbb R)$, $s\ge 1$.

Theorem 1.11

Let $\rho _0\!\in \! H^s(\mathbb R)$, $s\ge 1$. There exist $T=T(\Vert u_0\Vert _{1,2})=c\,\Vert u_0\Vert _{1,2}^{-2}$ and a unique solution $\rho =\rho (x,t)$ such that

$$\begin{aligned} \rho \in C([0,T] : H^s(\mathbb R))\cap C^1((0,T) : H^{s-1}(\mathbb R)). \end{aligned}$$

Moreover, the map data solution is locally continuous from $H^s$ to $C([0,T] : H^s(\mathbb R))$.

With this local well-posedness theory we establish the following result concerning the propagation of regularities in solutions of the IVP associated to (1.24).

Theorem 1.12

Suppose $\rho _0\in H^s(\mathbb R)$, $s\ge 1$, such that

$$\begin{aligned} \rho _0\big |_{\Omega }\in C^k(\Omega ) \quad \text {for some } \ \Omega \subseteq \mathbb R\,\,\text {open}. \end{aligned}$$

Then the corresponding solution $\rho =\rho (x,t)$ of the IVP associated to (1.24) provided by Theorem 1.11 satisfies that

$$\begin{aligned} \rho (\cdot , t)\big |_{\Omega _t} \in C^k(\Omega _t), \end{aligned}$$

where $\Omega _t=\Phi _t(\Omega )$ and $\Phi _t(x_0)=X(t;x_0)$ is the map defined by

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{dX}{dt}= u(X(t),t),\\ X(0)=x_0. \end{array}\right. } \end{aligned}$$

The rest of this work is organized as follows: Theorems 1.1 and 1.2 will be proven in Sect. 2. Section 3 contains the proof of Theorems 1.4 and 1.5. Section 4 is concerned with Theorem 1.6, Sect. 5 with Theorems 1.8 and 1.9 and Sect. 6 with Theorems 1.11 and 1.12.

2 Quasilinear KdV type equations

For simplicity in the presentation and without loss of generality we will consider the following IVP associated to a quasilinear KdV type satisfying the hypotheses (H1)-(H2) in (1.6)

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _tu +\big (1+(\partial _x^2u)^2\big )\partial _x^3u=0, &{} x\in \mathbb R, \ t>0,\\ u(x,0)=u_0(x)\in H^m(\mathbb R), &{} m\ge 7. \end{array}\right. \end{aligned}$$

(2.1)

Proof of Theorem 1.1 For $\epsilon \in (0,1)$ consider the parabolic problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu +\big (1+(\partial _x^2u)^2\big )\partial _x^3u=-\epsilon \partial _x^4u,\\ u(x,0)=u_0(x)\in H^7(\mathbb R). \end{array}\right. } \end{aligned}$$

(2.2)

Denoting by

$$\begin{aligned} K_t*f= e^{-\epsilon t\partial _x^4}f=\big (e^{-\epsilon t(2\pi \xi )^4}\widehat{f} \,\big )^{\vee } \end{aligned}$$

using that

$$\begin{aligned} \Vert \partial _x^l(K_t*f)\Vert _2\le \frac{c_l}{(\epsilon t)^{l/4}}\,\Vert f\Vert _2, \end{aligned}$$

and writing the solution $u^{\epsilon }$ of (2.2) in its equivalent integral equation version

$$\begin{aligned} u^{\epsilon }(t)=K_t*u_0+\int \limits _0^t K_{t-t'}*\big (1+(\partial _x^2u^{\epsilon })^2\big )\partial _x^3u^{\epsilon }(t')\,dt', \end{aligned}$$

(2.3)

one sees that there exist

$$\begin{aligned} T_{\epsilon }=T_{\epsilon }(\epsilon ; \Vert u_0\Vert _{m,2})>0 \text { with } T_{\epsilon }\sim \mathrm{O}(\epsilon ^{3/4}) \end{aligned}$$

and a unique $u^{\epsilon }$ solution of (2.2) such that

$$\begin{aligned} u^{\epsilon }\in C([0, T_{\epsilon }] : H^m(\mathbb R))\cap C^{\infty }((0, T_{\epsilon }) : H^{\infty }(\mathbb R)). \end{aligned}$$

(2.4)

Next we apply the argument in [10] to the IVP (2.2) to obtain a priori estimate which allows us to extend the local solution $u^{\epsilon }(\cdot )$ in a time interval [0, T] with $T=T(\Vert u_0\Vert _{m,2})>0$ independent of $\epsilon$ in the class described in (2.4). The only difference with the argument provided in [10] is the term on the RHS of the equation in (2.2). This can be easily handled using that

$$\begin{aligned} -\epsilon \int \partial _x^{4+l}u^{\epsilon }\partial _x^l u^{\epsilon }\chi _l(x,t)\,dx= & {} -\epsilon \int \partial _x^{2+l}u^{\epsilon }\partial _x^{2+l}u^{\epsilon }\chi _l(x,t)\,dx\nonumber \\&-2\epsilon \int \partial _x^{2+l}u^{\epsilon }\partial _x^{1+l}u^{\epsilon }\partial _x\chi _l(x,t)\,dx\nonumber \\&-\epsilon \int \partial _x^{2+l}u^{\epsilon }\partial _x^{l}u^{\epsilon }\partial _x^2\chi _l(x,t)\,dx\nonumber \\= & {} -\epsilon \int (\partial _x^{2+l}u^{\epsilon })^2 \chi _l(x,t)\,dx\nonumber \\&-2\epsilon \int (\partial _x^{2+l}u^{\epsilon })(\partial _x^lu^{\epsilon })\partial _x^2\chi _l(x,t)\,dt\nonumber \\&+\frac{\epsilon }{2}\int (\partial _x^lu^{\epsilon })(\partial _x^lu^{\epsilon })\partial _x^4\chi _l(x,t)\,dt\nonumber \\= & {} E_1+E_2+E_3, \end{aligned}$$

(2.5)

with $\chi _l(u)$ such that

$$\begin{aligned} \frac{3}{2}\partial _x [(1+(\partial _x^2u)^2) \chi _l]-(l+1)\partial _x(1+(\partial _x^2u)^2)\chi _l=0,\quad l\in \mathbb Z^{+}, \end{aligned}$$

(2.6)

i.e.

$$\begin{aligned} \chi _l= (1+(\partial _x^2u^{\epsilon })^2)^{-c_l} , \quad c_l=\frac{2}{3}(l+1)-1. \end{aligned}$$

Thus one sees that $E_1$ has the appropriate sign,

$$\begin{aligned} E_2\le & {} \frac{\epsilon }{2} \int (\partial _x^{2+l}u^{\epsilon })^2 \chi _l(x,t)\,dx + 2\epsilon \int (\partial _x^lu^{\epsilon })^2 \frac{(\partial _x^2\chi _{l})^2}{\chi _l}(x,t)\,dx\\= & {} E_{2,1} + E_{2,2}, \end{aligned}$$

where $E_{2,1}$ is absorbed by $E_1$. Terms $E_{2,2}$ and $E_3$ are at the level of the estimate involving derivatives of order l.

Hence, we can conclude that there exist $T=T(\Vert u_0\Vert _{7,2})>0$ and a unique solution

$$\begin{aligned} u^{\epsilon }\in C([0,T]:H^7(\mathbb R))\cap C((0,T):H^{\infty }(\mathbb R)) \end{aligned}$$

of (2.2) with

$$\begin{aligned} \underset{0<\epsilon <1}{\sup }\; \underset{0\le t\le T}{\sup } \Vert u^{\epsilon }(t)\Vert _{7,2}\le M_7= M_7(\Vert u_0\Vert _{7,2}). \end{aligned}$$

(2.7)

Moreover, if $u_0\in H^m(\mathbb R)$ with $m\ge 7$, $m\in \mathbb Z^{+}$, then

$$\begin{aligned} u^{\epsilon }\in C([0,T]:H^m(\mathbb R))\cap C((0,T):H^{\infty }(\mathbb R)) \end{aligned}$$

with

$$\begin{aligned} \underset{0<\epsilon <1}{\sup }\; \underset{0\le t\le T}{\sup } \Vert u^{\epsilon }(t)\Vert _{m,2}\le M_m= M_m(\Vert u_0\Vert _{m,2}). \end{aligned}$$

Next we want to show that $(u^{\epsilon })_{\epsilon >0}$ converges in $L^{\infty }([0,T] :H^3(\mathbb R))$.

Thus we consider

$$\begin{aligned} \partial _tu^{\epsilon }+(1+(\partial _x^2u^{\epsilon })^2)\partial _x^3u^{\epsilon }= -\epsilon \partial _x^4u^{\epsilon }\end{aligned}$$

and

$$\begin{aligned}&\partial _t\partial _x^3u^{\epsilon }+(1+(\partial _x^2u^{\epsilon })^2)\partial _x^6u^{\epsilon }+4\partial _x(1+(\partial _x^2u^{\epsilon })^2)\partial _x^5u^{\epsilon }\\&\quad +6\partial _x^2u^{\epsilon }\partial _x^4u^{\epsilon }\partial _x^4u^{\epsilon }+12\partial _x^3u^{\epsilon }\partial _x^3u^{\epsilon }\partial _x^4u^{\epsilon }=-\epsilon \partial _x^7u^{\epsilon }\end{aligned}$$

Similarly for $u^{\epsilon '}$ with $0<\epsilon '<\epsilon <1$. Now we write the equations for $w=w^{\epsilon ,\epsilon '}=u^{\epsilon }-u^{\epsilon '}$ and $\partial _x^3 w$, i.e.

$$\begin{aligned}&\partial _tw+(1+(\partial _x^2u^{\epsilon })^2)\partial _x^3w+\partial _x^3u^{\epsilon '}(\partial _x^2u^{\epsilon }+\partial _x^2u^{\epsilon '})\partial _x^2w\nonumber \\&\quad =-\epsilon '\partial _x^3w-(\epsilon -\epsilon ')\partial _x^4u^{\epsilon '}\end{aligned}$$

(2.8)

and

$$\begin{aligned}&\partial _t\partial _x^3w+(1+(\partial _x^2u^{\epsilon })^2)\partial _x^6w+(\partial _x^6u^{\epsilon '})(\partial _x^2u^{\epsilon }+\partial _x^2u^{\epsilon '})\,\partial _x^2w\nonumber \\&\qquad +8\partial _x^2u^{\epsilon }\partial _x^3u^{\epsilon }\partial _x^5w+4\partial _x^4u^{\epsilon '}\,\partial _x\big [ (\partial _x^2u^{\epsilon }+\partial _x^2u^{\epsilon '})\partial _x^2w\big ]\nonumber \\&\qquad + 6\partial _x^2u^{\epsilon }(\partial _x^4u^{\epsilon }+\partial _x^4u^{\epsilon '})\partial _x^4w +6(\partial _x^4u^{\epsilon '})^2\partial _x^2w\nonumber \\&\qquad +12(\partial _x^3u^{\epsilon })^2\partial _x^4w+12(\partial _x^4u^{\epsilon '})(\partial _x^3u^{\epsilon }+\partial _x^3u^{\epsilon '})\partial _x^3w\nonumber \\&\quad =-\epsilon ' \partial _x^7w-(\epsilon -\epsilon ')\partial _x^7u^{\epsilon '}. \end{aligned}$$

(2.9)

We multiply (2.8) by w and integrate the result to get

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int w^2+\frac{3}{2}\int \partial _x((\partial _x^2u^{\epsilon })^2)(\partial _x w)^2-\frac{1}{2}\int \partial _x^3((\partial _x^2u^{\epsilon })^2)w^2\\&\qquad -\int (\partial _x^3u^{\epsilon })(\partial _x^2u^{\epsilon }+\partial _x^2u^{\epsilon '})(\partial _xw)^2+\frac{1}{2}\int \partial _x^2\big [\partial _x^3u^{\epsilon '}(\partial _x^2u^{\epsilon }+\partial _x^2u^{\epsilon '})\big ]w^2\\&\quad =-(\epsilon -\epsilon ')\int \partial _xu^{\epsilon '}w \end{aligned}$$

and multiply (2.9) by $\partial _x^3w\,\chi$ and integrate the result to get (after some integration by parts)

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int (\partial _x^3w)^2\chi -\int (\partial _x^3w)^2\partial _x\chi \\&\qquad +\int \Big \{\frac{3}{2} \partial _x\big [(1+(\partial _x^2u^{\epsilon })^2)\chi \big ]-(3+1)\partial _x(1+(\partial _x^2u^{\epsilon })^2)\,\chi \Big \} (\partial _x^4w)^2\\&\qquad -\frac{1}{2}\int \partial _x^3(1+(\partial _x^2u^{\epsilon })^2\chi )(\partial _x^3w)^2+4\int \partial _x^2(\partial _x^2u^{\epsilon }\partial _x^3u^{\epsilon }\chi )(\partial _x^3w)^2\\&\qquad -3\int \partial _x[\partial _xu^{\epsilon }(\partial _x^4u^{\epsilon }+\partial _x^4u^{\epsilon '})\chi ] (\partial _x^3w)^2 -6\int \partial _x[(\partial _x^3u^{\epsilon })^2\chi ] (\partial _x^3w)^2\\&\qquad +\int \partial _x^6u^{\epsilon '}(\partial _x^2u^{\epsilon }+\partial _x^2u^{\epsilon '})\partial _x^2w\partial _x^3w+ 4\int \partial _x^5u^{\epsilon '}\partial _x\big ((\partial _x^2u^{\epsilon }+\partial _x^2u^{\epsilon '})\partial _x^2w\big )\partial _x^3w\chi \\&\qquad +6\int (\partial _x^4u^{\epsilon '})^2\partial _x^2w\partial _x^3w \chi + 12\int (\partial _x^3u^{\epsilon }+\partial _x^3u^{\epsilon '})\partial _x^4u^{\epsilon '}(\partial _x^3w)^2\chi \\&\quad =-\epsilon ' \int \partial _x^7w\partial _x^3w \chi -(\epsilon -\epsilon ') \int \partial _x^7u^{\epsilon '}\partial _x^3w \chi . \end{aligned}$$

As in [10] one chooses $\chi =\chi _{3}$ (see (2.6)) such that

$$\begin{aligned} \frac{3}{2} \partial _x[(1+(\partial _x^2u^{\epsilon })^2)\chi ]-(3+1)\partial _x\big (1+(\partial _x^2u^{\epsilon })^2\big )\chi \equiv 0, \end{aligned}$$

i.e.

$$\begin{aligned} c_l=\frac{2}{3}(l+1)-1>0 \end{aligned}$$

and

$$\begin{aligned} \chi _3=(1+(\partial _x^2u^{\epsilon })^2)^{-c_3}, \end{aligned}$$

which by hypothesis and previous results in (2.7) it follows that

$$\begin{aligned} \underset{[0,T]}{\sup } \Vert (u^{\epsilon }-u^{\epsilon '})(t)\Vert _{3,2}\le c\big (\Vert u_0\Vert _{7,2}\big )\Big [ \Vert u_0^{\epsilon }-u_0^{\epsilon '}\Vert _{3,2}+(\epsilon +\epsilon ')\Big ]. \end{aligned}$$

Hence

$$\begin{aligned} (u^{\epsilon })_{\epsilon }\subseteq C([0,T] : H^7(\mathbb R)) \end{aligned}$$

(uniformly bounded) converges as $\epsilon \downarrow 0$ in $L^{\infty }([0,T] : H^3(\mathbb R))$ to $u\in C([0,T]: H^3(\mathbb R))$.

Moreover, from (2.7) it follows that for any $\delta >0$

$$\begin{aligned} u^{\epsilon }\rightarrow u \,\,\text {in}C([0,T] : H^{7-\delta }(\mathbb R)) \end{aligned}$$

and

$$\begin{aligned} u\in L^{\infty }([0,T] : H^7(\mathbb R)) \end{aligned}$$

is a solution of the IVP (2.1). But by the uniqueness established in [10] this solution is unique so it agrees with that provided in Theorem A3.

In particular, for any $R>0$

$$\begin{aligned} \int \limits _0^T\int \limits _{-R}^{R} |\partial _x^8 u(x,t)|^2\,dxdt \le c = c(T; R; \Vert u_0\Vert _{7,2}). \end{aligned}$$

(2.10)

Moreover, the previous argument applied to two solutions

$$\begin{aligned} u, v\in C([0,T] : H^{7-\delta }(\mathbb R))\cap L^{\infty }([0,T] : H^7(\mathbb R)) \end{aligned}$$

with data $u_0, v_0$ respectively, shows that

$$\begin{aligned} \underset{[0,T]}{\sup } \Vert (u-v)(t)\Vert _{3,2} \le c\big ( M(\Vert u_0\Vert _{7,2},\Vert v_0\Vert _{7,2})\big ) \Vert u_0-v_0\Vert _{3,2}. \end{aligned}$$

Hence, the map data solution from $H^7(\mathbb R)$ into $C([0,T] : H^{7-\delta }(\mathbb R))$ is locally continuous for any $\delta >0$.

Proof of Theorem 1.2 As in the previous section we will consider the following IVP

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu +\big (1+(\partial _x^2u)^2\big )\partial _x^3u=0, x\in \mathbb R,\;t>0,\\ u(x,0)=u_0(x)\in H^m(\mathbb R), \;\;\; m\ge 7. \end{array}\right. } \end{aligned}$$

(2.11)

Without loss of generality we shall assume that $m=7$ and that $x_0=0$. Thus we have that the solution $u(\cdot )$ of the IVP (2.11) provided by Theorem 1.1 satisfies

$$\begin{aligned} u\in C([0,T] : H^{7-\delta }(\mathbb R))\cap L^{\infty }([0,T] : H^7(\mathbb R)), \text {for all}\;\; \delta >0 \end{aligned}$$

(2.12)

and (2.10).

As in [15] we define the family of cut off functions: for $\epsilon >0$ and $b\ge 5\epsilon$, let $\chi _{_{\epsilon , b}}\in C^{\infty }(\mathbb R)$ such that

$$\begin{aligned} \chi _{_{\epsilon , b}}= {\left\{ \begin{array}{ll} 0, x\le \epsilon ,\\ 1, x\ge b, \end{array}\right. } \end{aligned}$$

(2.13)

$$\begin{aligned} \mathrm{supp}\,\chi _{_{\epsilon , b}} \subseteq [\epsilon , \infty ), \quad \mathrm{supp}\,\chi _{_{\epsilon , b}}' \subseteq [\epsilon , b], \end{aligned}$$

(2.14)

$$\begin{aligned} \chi _{_{\epsilon , b}}'(x)\ge 0 \quad \text {with} \quad \chi _{_{\epsilon , b}}'(x)\ge \frac{1}{b-3\epsilon } 1_{[3\epsilon , \,b-2\epsilon ]}(x). \end{aligned}$$

(2.15)

Thus,

$$\begin{aligned} \chi _{_{\epsilon /3, b+\epsilon }}'(x)\ge c_j\,|\chi _{_{\epsilon , b}}^{(j)}(x)|, \quad x\in \mathbb R, \quad j=1, 2, 3. \end{aligned}$$

(2.16)

Next we follow the argument in [10]. Thus we formally apply $\partial _x^j$, $j=8, \dots , m$ to the equation in (2.11) and multiply the result by

$$\begin{aligned} \partial _x^ju \, \partial _x^j\psi _j=\partial _x^ju \,\psi _{j,v,\epsilon ,b}, \end{aligned}$$

(2.17)

with $\psi _{j,v,\epsilon ,b}(x,t)$ to be determine below, to get (after some integration by parts)

$$\begin{aligned}&\frac{1}{2} \frac{d}{dt} \int (\partial _x^ju)^2 \psi _j(x,t)\,dx -\frac{1}{2}\int (\partial _x^ju)^2 \partial _t\psi _j(x,t)\,dx\nonumber \\&\qquad +\int (\partial _x^{j+1}u)^2\Big \{ \frac{3}{2} \partial _x\big ((1+(\partial _x^2u)^2\big )\psi _j\big ) -j\big [ \partial _x(1+(\partial _x^2u)^2)\psi _j\big ]\nonumber \\&\qquad -\big [\partial _x(1+(\partial _x^2u)^2)\psi _j\big ]\Big \}\,dx\nonumber \\&\qquad +\int Q_j\big ( (\partial _x^lu)_{|l|\le j} ; (\partial _x^r\psi _j)_{|r|\le 3}\big )(x,t)\,dx\nonumber \\&\quad \equiv \frac{1}{2} \frac{d}{dt} \int (\partial _x^ju)^2 \psi _j(x,t)\,dx +E_{1,j}+E_{2,j}+E_{3,j}. \end{aligned}$$

(2.18)

where $Q_j(\cdot , \cdot )$ is a polynomial in its variables, linear in the components $(\psi _j, \partial _x\psi _j, \partial _x^2\psi _j, \partial _x^3\psi _j)$, and at most quadratic in the highest derivatives of u, i.e. $(\partial _x^ju)$, involving at most $7+2j$ derivatives of u.

First we consider $E_{2,j}$ which determines the choices of $\psi _j=\psi _{j,v,\epsilon ,b}$. As in [10] we choose $\psi _j=\psi _{j,v,\epsilon ,b}$ such that for $v>0$

$$\begin{aligned} \frac{3}{2} \partial _x\big ((1+(\partial _x^2u)^2\big )\psi _j\big )-(j+1) \partial _x\big (1+(\partial _x^2u)^2\big )\psi _j = \chi _{_{\epsilon , b}}' (x+vt) \end{aligned}$$

(2.19)

with

$$\begin{aligned} \psi _j(x,t) \rightarrow 0 \quad \text {as } \ x\downarrow -\infty , \end{aligned}$$

(2.20)

i.e.

$$\begin{aligned} (1+(\partial _x^2u)^2)\partial _x\psi _j -\Big (1-\frac{2}{3}(j+1)\Big ) \partial _x\big (1+(\partial _x^2u)^2\big )\psi _j = \frac{2}{3}\chi _{_{\epsilon , b}}' (x+vt).\qquad \quad \end{aligned}$$

(2.21)

Hence if

$$\begin{aligned} d_j \equiv 1-\frac{2}{3}(j+1), \end{aligned}$$

(2.22)

then

$$\begin{aligned} \psi _j=\psi _{j,v,\epsilon ,b} \equiv \frac{2}{3} (1+(\partial _x^2u)^2)^{-d_j}(x,t)\! \int \limits _{-\infty }^x \!(1+(\partial _x^2u)^2)^{d_j-1}(s,t) \chi _{_{\epsilon ,b}}'(s+vt)\,ds. \end{aligned}$$

We observe that

$$\begin{aligned} \psi _j(x,t)=\psi _{j,v,\epsilon ,b}(x,t)\ge 0 \end{aligned}$$

(2.23)

with

$$\begin{aligned} \mathrm{supp}\, \psi _{j,v,\epsilon ,b}(\cdot , t)\subseteq [\epsilon -vt,\infty ), \quad t\in [0,T] \end{aligned}$$

(2.24)

and

$$\begin{aligned} \psi _{j,v,\epsilon ,b}(\cdot , t)\in L^{\infty }(\mathbb R) \end{aligned}$$

(2.25)

with

$$\begin{aligned} \Vert \psi _{j,v,\epsilon ,b}(\cdot , t)\Vert _{\infty } \le c=c(v,\epsilon , b; \Vert u_0\Vert _{7,2}). \end{aligned}$$

(2.26)

With this choice of $\psi _j(\cdot )=\psi _{j,v,\epsilon ,b}(\cdot )$, $E_{2,j}$ becomes

$$\begin{aligned} E_{2,j}=\int (\partial _x^{j+1}u)^2(x,t) \chi _{_{\epsilon ,b}}'(x+vt)\, dx. \end{aligned}$$

(2.27)

Also one sees [using (2.10)–(2.12) and (H2)] that for any $T>0$, $v>0$, $j\in \mathbb Z^{+}$, $j\ge 8$, there exists $c= c(T; v; j; k)\in (1,\infty )$ such that for any $(x,t)\in \mathbb R\times [0,T]$

$$\begin{aligned} c^{-1} \chi _{_{\epsilon ,b}}(x+vt) \le \psi _{j,v,\epsilon ,b}(x,t) \le c\, \chi _{_{\epsilon ,b}}(x+vt). \end{aligned}$$

(2.28)

Moreover (with c as above)

$$\begin{aligned} |\partial _x\psi _{j,v,\epsilon , b}(x,t)|\le & {} \big (\chi _{_{\epsilon ,b}}(x+vt) + \chi _{_{\epsilon ,b}}'(x+vt)\big )\nonumber \\\le & {} c \big ( \psi _{j,v,\epsilon , b}(x,t)+\chi _{_{\epsilon ,b}}'(x+vt)\big ), \end{aligned}$$

(2.29)

$$\begin{aligned} |\partial _x^2\psi _{j,v,\epsilon , b}(x,t)|\le & {} \big ( (\chi _{_{\epsilon ,b}}+\chi _{_{\epsilon ,b}}')(x+vt)\big )\nonumber \\&+ |\chi _{_{\epsilon ,b}}''(x+vt)|\big ), \end{aligned}$$

(2.30)

and

$$\begin{aligned} |\partial _x^3\psi _{j,v,\epsilon , b}(x,t)|\le & {} \big (\big (\chi _{_{\epsilon ,b}}+\chi _{_{\epsilon ,b}}')(x+vt)\nonumber \\&+ |\chi _{_{\epsilon ,b}}''(x+vt)|+ |\chi _{_{\epsilon ,b}}^{(3)}(x+vt)|\big ), \end{aligned}$$

(2.31)

Therefore, combining (2.28) and (2.16) one has that

$$\begin{aligned} |\partial _x^r \psi _{j,v,\epsilon ,b}(x,t)|\le c\big (\psi _{j,v,\epsilon ,b}(x,t)+\chi _{_{\epsilon ,b}}'(x+vt)+\chi _{_{\epsilon /3,b+\epsilon }}(x+vt), \end{aligned}$$

(2.32)

$r=2, 3$, for any $(x,t)\in \mathbb R\times [0,T]$.

Also using the Eqs. (2.12), and (2.28) it follows that

$$\begin{aligned} |\partial _t\psi _{j,v,\epsilon ,b}(x,t)|\le & {} c \big (\chi _{_{\epsilon ,b}}(x+vt)+\chi _{_{\epsilon ,b}}'(x+vt)\big )\nonumber \\\le & {} c \big ( \psi _{j,v,\epsilon ,b}(x,t)+ \chi _{_{\epsilon ,b}}'(x+vt)\big ). \end{aligned}$$

(2.33)

We now turn to the estimate of $E_{1,j}$ in (2.18). First, we consider the case $j=8$. In this case using (2.33) we have that

$$\begin{aligned} |E_{1,8}|\le & {} \frac{c}{2} \int (\partial _x^8u)^2 \psi _{j,v,\epsilon ,b} (x,t)\, dx\nonumber \\&+ c\,\big |\int (\partial _x^8u)^2 \chi _{_{\epsilon ,b}}'(x+vt)\, dx\big | = E_{1,8,1}+E_{1,8,2}. \end{aligned}$$

(2.34)

We notice that $E_{1,8,1}$ is a multiple of the term we are estimating, so it will be part of the Gronwall’s inequality. For the term $E_{1,8,2}$ we observe that given $T>0$ and $v>0$, there exists $R>0$ such that

$$\begin{aligned} \chi _{_{\epsilon ,b}}'(x+vt)\le 1_{(-R,R)} (x) \end{aligned}$$

(2.35)

for all $(x,t)\in \mathbb R\times [0,T]$. Therefore, the bound of $E_{1,8,2}$ follows from (2.10) after integrating in time in the estimate. So at the level $j=8$ it only remains to consider $E_{3,8}$ in (2.18).

Using the structure of $E_{3,8}$ commented after (2.18) and the bounds in (2.28)–(2.32) it follows that in this case $j=8$, the term $E_{3,8}$ can be written as a sum of terms of the form

$$\begin{aligned} \int P_{3,8,1} \big ((\partial _x^lu)_{| l |\le 4} \big )\, (\partial _x^8u)^2\, \psi _{8,v,\epsilon ,b}(x,t) \,dx \quad (\equiv E_{3,8,1}), \end{aligned}$$

(2.36)

$$\begin{aligned} \int P_{3,8,2,r}\big ((\partial _x^l u)_{|l|\le 4-r}\big ) \,(\partial _x^8u)^2\, \chi _{_{\epsilon ,b}}^{(r)}(x+vt) \,dx \quad (\equiv E_{3,8,2}) \end{aligned}$$

(2.37)

with $r=1, 2, 3$, or terms involving lower order derivatives $l=0,\dots ,7$,

$$\begin{aligned}&\int P_{3,8,3}\big ((\partial _x^l u)_{|l|\le 7}\big )\, \chi _{_{\epsilon ,b}}^{(r)}(x+vt) \,dx \quad (\equiv E_{3,8,3}),\end{aligned}$$

(2.38)

$$\begin{aligned}&\int P_{3,8,4}\big ((\partial _x^l u)_{|l|\le 7}\big )\, \psi _{8,v,\epsilon ,b}(x, t) \,dx \quad (\equiv E_{3,8,4}), \end{aligned}$$

(2.39)

where

• $P_{3,8,1} (\cdot )$ and $P_{3,8,2,r} (\cdot )$ are polynomials of degree at most 2,

and

• $P_{3,8,3} (\cdot )$ and $P_{3,8,4} (\cdot )$ are polynomials of degree at most 4.

In this case, $j=8$, the terms in $E_{3,8,3}$ and $E_{3,8,4}$ are bounded since one has (2.12) and the fact that

$$\begin{aligned} \underset{r=3}{\overset{3}{\sum }}\; \Vert \chi _{_{\epsilon ,b}}^{(r)}\Vert _{\infty } +\Vert \psi _{8,v,\epsilon ,b}\Vert _{L^{\infty }(\mathbb R\times [0,T])}\le c, \end{aligned}$$

with $c = c(v,\epsilon , b, \sup _{[0,T]} \Vert u(t)\Vert _{7,2})$.

The term $E_{3,8,2}$ can be estimated using (2.10) as in (2.35).

Finally, the term in (2.36) is the one we are estimating and will be handled by Gronwall.

Hence, gathering the above information we conclude in the case $j=8$ that

$$\begin{aligned}&\underset{[0,T]}{\sup } \int (\partial _x^8u)^2(x,t)\,\psi _{8,v, \epsilon , b}(x,t)\,dx +\int \limits _0^T\int (\partial _x^9u)^2(x,t)\, \chi _{_{\epsilon , b}}'(x+vt)\,dxdt\nonumber \\&\quad \le c = c(\Vert u_0\Vert _{7,2}; \Vert \partial _x^8u_0\Vert _{L^2((0,\infty ))}; v; \epsilon ; b). \end{aligned}$$

(2.40)

We notice that by (2.28) that (2.40) implies that for $\epsilon >0$, $b>5\epsilon$, $v>0$,

$$\begin{aligned}&\underset{[0,T]}{\sup } \int (\partial _x^8u)^2(x,t)\,\chi _{_{ \epsilon , b}}(x+vt)\,dx +\int \limits _0^T\int (\partial _x^9u)^2(x,t)\, \chi _{_{\epsilon , b}}'(x+vt)\,dxdt\nonumber \\&\quad \le c = c(\Vert u_0\Vert _{7,2}; \Vert \partial _x^8u_0\Vert _{L^2((0,\infty ))}; v; \epsilon ; b). \end{aligned}$$

(2.41)

Once we have established the desired result (2.41) for the case $j=8$ we sketch the iterative argument for the general case $j=8, \dots , m$.

Assuming the step $j=m_0\in \{ 8, \dots , m\}$, i.e. for $j=8, \dots ,m_0$,

$$\begin{aligned}&\underset{[0,T]}{\sup } \int (\partial _x^j u)^2(x,t)\,\chi _{_{ \epsilon , b}}(x+vt)\,dx +\int \limits _0^T\int (\partial _x^{j+1}u)^2(x,t)\, \chi _{_{\epsilon , b}}'(x+vt)\,dxdt\nonumber \\&\quad \le c = c(\Vert u_0\Vert _{7,2}; \Vert \partial _x^{m_0}u_0\Vert _{L^2((0,\infty ))}; v; \epsilon ; b), \end{aligned}$$

(2.42)

we shall prove it for $j=m_0+1$.

We repeat the argument in (2.17) and (2.18) to get $E_{1,m_0+1}$, $E_{2, m_0+1}$, and $E_{3, m_0+1}$.

The estimate for $E_{2, m_0+1}$ is similar to that given in (2.19)–(2.27).

To handle the term $E_{1,m_0+1}$ we observe that from (2.33) (with $j=m_0+1$) one has an estimate as in (2.33), (2.34) with $m_0+1$ instead of 8, i.e. the terms $E_{1, m_0+1,1}$ and $E_{1,m_0+1,2}$. As in the previous case $E_{1, m_0+1,1}$ is a multiple of the term we are estimating and $E_{1, m_0+1,2}$ can be bounded, after time integration, by the second term in the right hand side of (2.42) by taking there an appropriate value of $\epsilon$ and b. So it only remains to consider the terms in $E_{3, m_0+1}$, where (see (2.18))

$$\begin{aligned} E_{3, m_0+1}= \int Q_{m_0+1} \big ( (\partial _x^lu)^2_{|l|\le m_0+1} ; (\partial _x^r\psi _{m_0+1})_{|r|\le 3}\big ) \,dx. \end{aligned}$$

where $Q_{m_0+1}(\cdot ,\cdot )$ is a polynomial in its variables, linear in the ($\psi _{m_0+1}$, $\partial _x\psi _{m_0+1}$, $\partial _x^2\psi _{m_0+1}$, $\partial _x^3\psi _{m_0+1}$) components and at most quadratic in the highest derivatives of u, i.e. $(\partial _x^{m_0+1}u)$ involving at most $7+2(m_0+1)$ derivatives of u.

To handle $E_{3,m_0+1}$ one combines the global (in space) estimate in (2.12) with that in (2.10) and those obtained in the previous estimates $j=8,\dots ,m_0$, i.e. (2.42) with $j=8,\dots , m_0$ to obtain the desired estimate. The proof follows the argument provided in details in [15] Sect. 3. Therefore it will be omitted.

Finally, to justify the previous formal computation we recall that the argument in the proof of Theorem 1.1 shows that u in (2.12)–(2.10) is the limit in the $C([0,T] : H^{7-\delta }(\mathbb R))$-norm (for any $\delta >0$) of smooth solutions (weak form of the continuous dependence upon the data). In particular, we have that u is the uniform limit of smooth solutions in $\mathbb R\times [0,T]$. Hence, by performing the above (formal) argument in the smooth solutions one obtains a uniform bounded sequence in the norms described in (1.7) and (1.8). Hence, considering the uniform boundedness, the weak convergence and passing to the limit we obtain the desired result.

3 Dispersive Blow-up

Consider the IVP associated to the k-generalized Korteweg–de Vries equation,

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu+\partial _x^3u +u^k\partial _xu=0, \quad k=1, 2, 3, \dots , \;x\in \mathbb R, \quad t>0,\\ u(x,0)=u_0(x). \end{array}\right. } \end{aligned}$$

(3.1)

Proof of Theorem 1.4 Let $\phi (x)=e^{-2|x|}$ and consider the linear IVP,

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t v+\partial _x^3v=0, x\in \mathbb R, \;t>0,\\ v(x,0)=v_0 \end{array}\right. } \end{aligned}$$

(3.2)

whose solution is given by

$$\begin{aligned} v(x,t)= V(t)v_0(x)= e^{-t\partial _x^3}v_0= S_t*v_0(x) \end{aligned}$$

(3.3)

where

$$\begin{aligned} S_t(x)=\frac{1}{\sqrt{3}{3t}} A_i\left( \frac{x}{\sqrt{3}{3t}}\right) \end{aligned}$$

and $A_i(\cdot )$ denotes the Airy function.

Define

$$\begin{aligned} u_0(x)=\underset{j=1}{\overset{\infty }{\sum }} \alpha _j\,V(-j)\phi (x), \quad \alpha _j>0. \end{aligned}$$

(3.4)

If

$$\begin{aligned} \underset{j=1}{\overset{\infty }{\sum }} \alpha _j \ll 1, \end{aligned}$$

we have that $u_0\in H^1(\mathbb R)$, in fact in $u_0\in H^{3/2-}(\mathbb R)$. This in particular guarantees the global existence of solutions in $H^1(\mathbb R)$ for the IVP (3.1).

Step 1. Reduction to linear case.

Case $k=2, 3, \dots$

Since the nonlinear part of the solution u of (3.1), i.e.

$$\begin{aligned} z(t)=\int \limits _0^t V(t-t') \,u^k\partial _xu(t')\,dt' \end{aligned}$$

is in $C([0,\infty ) : H^2(\mathbb R))$ (see [23] for the proof in the case $k=2$, since our solution is $C([0,T]: H^{1}(\mathbb R))$, the argument works for all $k\ge 2$). It suffices to consider the linear part, $V(t)u_0$.

Case $k=1$

We observe that from (3.3), (3.4) it follows that $u_0\in H^{{3/2}^{-}}(\mathbb R)$.

Next we recall the identity deduced in [12]: for $\beta \in (0,1)$ and $t\in \mathbb R$

$$\begin{aligned} |x|^{\beta } V(t)f = V(t)(|x|^{\beta }f)+ V(t)\{\Phi _{t,\beta }(\widehat{f\,})\}^{\vee } \end{aligned}$$

(3.5)

with

$$\begin{aligned} \Vert \Phi _{t, \beta }(\widehat{f\,})\Vert _2\le c(1+|t|) \Vert f\Vert _{2\beta ,2}. \end{aligned}$$

(3.6)

Hence, for $\beta \in (0,3/4)$

$$\begin{aligned} |x|^{\beta }u_0\in L^2(\mathbb R)\qquad \text {if}\qquad \underset{j=1}{\overset{\infty }{\sum }} \,\alpha _j \,j^{\beta }<\infty . \end{aligned}$$

Assuming the last inequality we have that for $u_0$ as above the corresponding solutions of the IVP for the KdV equation satisfies for any $T>0$

$$\begin{aligned} u\in C([0,T] : H^{{3/2}^{-}}(\mathbb R)\cap L^2(|x|^{{3/4}-\epsilon }\,dx)), \end{aligned}$$

(3.7)

$$\begin{aligned} J^{3/2^{-}}\partial _xu\in L^{\infty }_x(\mathbb R: L^2([0,T])), \quad \text {(smoothing effect)} \end{aligned}$$

(3.8)

and

$$\begin{aligned}&\int \limits _0^T \Vert D^{\alpha \theta /2} J^s u(\cdot , t)\Vert _p^q\,dt<\infty \quad \text {for } \ 0<s<3/2, \quad \text {(Strichartz)}\nonumber \\&\quad (q,p)=(6/\theta (\alpha +1), 2/(1-\theta )), \quad \theta \in (0,1),\quad 0\le \alpha \le 1/2. \end{aligned}$$

(3.9)

As in the previous case $k=2,3,\dots$ we shall show that for any $t\in [0,T]$

$$\begin{aligned} z(t)=\int \limits _0^t V(t-t') u\partial _x u (t')\,dt' \in C^1(\mathbb R) \end{aligned}$$

(3.10)

by proving that

$$\begin{aligned} z(t)\in C([0,T] : H^{3/2^{-}+1/6}(\mathbb R)). \end{aligned}$$

(3.11)

Using the inequality (see [21])

$$\begin{aligned} \underset{0\le t\le T}{\sup }\Vert \partial _x \int \limits _0^t V(t-t') F(\cdot , t')\,dt'\Vert _2 \le c\Vert F\Vert _{L^1_xL^2_T} \end{aligned}$$

(3.12)

one has that

$$\begin{aligned}&\underset{0\le t\le T}{\sup }\Vert D^{3/2^{-}+1/6}\int \limits _0^t V(t-t') u\partial _xu (t')\, dt'\Vert _2 \le \Vert D^{1/2^{-}+1/6}(u\partial _x u)\Vert _{L^1_xL^2_T}\\&\quad \le \big ( \Vert u\Vert _{L^{6/5}_xL^3_T}\Vert D^{3/2^{-}+1/6}u\Vert _{L^6_xL^6_T} +E_1\big ) \end{aligned}$$

where the terms in $E_1$ are easy to control by considering the commutator estimates in the Appendix of [21] and interpolated norms of the previous terms to be considered below, so we omit this proof.

Now

$$\begin{aligned} \Vert D^{3/2^{-}+1/6}u\Vert _{L^6_xL^6_T}<\infty \end{aligned}$$

from (3.9) with $p=q=6$, $\theta =2/3$, $\alpha =1/2$ and using the inequality (2.11) in [13],

$$\begin{aligned} \Vert J^{\gamma \,a}(\langle x\rangle ^{(1-\gamma ) b}f)\Vert _2 \le c\Vert \langle x\rangle ^b f\Vert _2^{1-\gamma }\Vert J^af\Vert _2^{\gamma },\;\;a,b>0, \;\gamma \in (0,1), \end{aligned}$$

we deduce

$$\begin{aligned} \Vert u\Vert _{L^{6/5}_xL^3_T}\le & {} c\Vert \langle x\rangle ^{1/2+} u\Vert _{L^3_TL^3_x}\le c T^{1/3}\Vert \langle x\rangle ^{1/2+} u\Vert _{L^{\infty }_TL^3_x}\\\le & {} c T^{1/3}\Vert J^{1/6} (\langle x\rangle ^{1/2+} u)\Vert _{L^{\infty }_TL^2_x}\\\le & {} cT^{1/3}\Vert J^{3/2-}u\Vert _{L^{\infty }_TL^2_x}^{1-\gamma }\Vert \langle x\rangle ^{3/4-} u\Vert _{L^{\infty }_TL^2_x}^{\gamma } \end{aligned}$$

with $\gamma$ such that $\gamma \,{3/4}^{-}= {1/2}^{+}$ (i.e. $\gamma >2/3$) and $(1-\gamma ){3/2}^{-}>1/6$.

As in the case $k=2, 3, \dots$ we have reduced ourselves to consider the linear associated problem so the nonlinearity after Step 1 is not relevant for our purposes.

Step 2. Estimate for $V(t)\phi$, $(t>0)$.

Assume that $v_0\in L^2(\mathbb R)$ and $e^{x}v_0\in L^2(\mathbb R)$

Now consider $w(x,t)=e^x v(x,t)$. Following Kato [18] we set $v(x,t)= e^{-x} w(x,t)$ where w is solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t w +(\partial _x-1)^3 w=0,\\ w(x,0)= e^x v_0(x). \end{array}\right. } \end{aligned}$$

(3.13)

Since

$$\begin{aligned} (\partial _x - 1)^3= \partial _x^3 -3\partial _x^2+3\partial _x-1 \end{aligned}$$

one has that

$$\begin{aligned} w(x,t)= & {} V(t) e^{3t\partial _x^2} e^{-3t\partial _x} e^t \big ( e^x v_0(x)\big )\\= & {} V(t) e^{3t\partial _x^2} e^{-3t\partial _x} \big (e^{x+t} v_0(x)\big ) \end{aligned}$$

and

$$\begin{aligned} V(t)v_0= v(x,t)= e^{-x} V(t)e^{3t\partial _x^2} \big (e^{x-2t} v_0(x-3t)\big ). \end{aligned}$$

We notice using the heat kernel properties that

$$\begin{aligned} \partial _x^m V(t)v_0 \sim e^{-x} V(t)\Big (\partial _x^m e^{3t\partial _x^2}\Big )\Big ( e^{x-2t} v_0(x-3t)\Big ). \end{aligned}$$

It follows that

$$\begin{aligned} \Vert \,e^{x} \,\partial _x^m V(t)v_0\Vert _2\sim \frac{c_m}{(3t)^{m/2}}\Vert e^{x-2t} v_0(x-3t)\Vert _2\sim \frac{c_m}{(3t)^{m/2}} e^t, \end{aligned}$$

since

$$\begin{aligned} \Vert e^{x-2t} v_0(x-3t)\Vert _2= e^t\Vert e^{x-3t} v_0(x-3t)\Vert _2= c\,e^t. \end{aligned}$$

Similarly, if $t<0$, we have an IVP analogous to the one in (3.13) the operator $-(\partial _x+1)^3$ instead of $(\partial _x-1)^3$. Thus

$$\begin{aligned} V(t)v_0= & {} e^x V(t) e^{-3t\partial _x^2} e^{-3t\partial _x} (e^{-t} e^{-x} v_0)\\= & {} e^x V(t) e^{-3t\partial _x^2} e^{-3t\partial _x} \Big (e^{-x-t} v_0(x)\Big )\\= & {} e^{x} V(t) e^{-3t\partial _x^2} \Big ( e^{-x-4t} v_0(x-3t)\Big ). \end{aligned}$$

and so we have

$$\begin{aligned} \partial _x^m V(t)v_0 \sim e^{x} V(t)\Big (\partial _x^m e^{-3t\partial _x^2}\Big )\Big ( e^{x-4t} v_0(x+3t)\Big ) \end{aligned}$$

and

$$\begin{aligned} \Vert \,e^{-x}\,\partial _x^m V(t)v_0\Vert _2\sim \,\frac{c_m}{(3t)^{m/2}} e^{-t}. \end{aligned}$$

(3.14)

Step 3. Next we prove that

$$\begin{aligned} \underset{j=1}{\overset{\infty }{\sum }} \alpha _j V(-j) \phi \in C^{\infty }(\mathbb R) \end{aligned}$$

or equivalently

$$\begin{aligned} \underset{j=1}{\overset{\infty }{\sum }} \alpha _j e^{-x} V(-j) \phi \in C^{\infty }(\mathbb R). \end{aligned}$$

To do this, it suffices to show that

$$\begin{aligned} \underset{j=1}{\overset{\infty }{\sum }} \alpha _j e^{-x}\Big (\partial _x^m V(-j) \phi \Big ) \in L^2(\mathbb R) \;\; \text { for all} \;\;m \end{aligned}$$

or equivalently

$$\begin{aligned} \underset{j=1}{\overset{\infty }{\sum }} \alpha _j \frac{c_m}{(3j)^{m/2}}\, e^j <\infty . \end{aligned}$$

Step 4. For each $t>0$, $t\notin \mathbb Z^{+}$, we claim that

$$\begin{aligned} V(t)u_0= \underset{j=1}{\overset{\infty }{\sum }} \alpha _j V(t-j)\phi \in C^1(\mathbb R). \end{aligned}$$

Combining (3.14) and the assumption

$$\begin{aligned} \underset{j=1}{\overset{\infty }{\sum }}\, \alpha _j \frac{1}{3|t-j|} e^{|t-j|} <\infty \end{aligned}$$

one has $V(t)u_0 \in H^2_\mathrm{loc} (\mathbb R)\subseteq C^1(\mathbb R)$.

Step 5. For $t=n\in \mathbb Z^{+}$ we affirm that

$$\begin{aligned} V(n)u_0 = \alpha _n\phi + \underset{j\ne n}{\underset{j=1}{\overset{\infty }{\sum }}}\, \alpha _j V(n-j) \phi \equiv \alpha _n\phi +\Phi _n \end{aligned}$$

(3.15)

with $\Phi _n\in C^1$.

As before using (3.14) and taking

$$\begin{aligned} \underset{j\ne n}{ \underset{j=1}{\overset{\infty }{\sum }}}\, \alpha _j \frac{1}{3|n-j|} e^{|n-j|} <\infty \end{aligned}$$

it follows that $\Phi _n\in H^2_\mathrm{loc} (\mathbb R)$ which yields (3.15).

By setting $\alpha _j= c\,e^{-j^2}$ with c small enough we obtain the desired result.

Proof of Theorem 1.5

• (a) First we consider the case $k=2, 3, \dots$. We recall the Strichartz estimates for solutions of the linear IVP (3.2) established in [20]

  $$\begin{aligned} \left( \int \limits _{-\infty }^{\infty } \Vert D^{\alpha \theta /2} V(t)f\Vert _p^q\,dt\right) ^{1/q} \le c\Vert f\Vert _2, \end{aligned}$$

  (3.16)

with $(q, p)= (6/{\theta (\alpha +1)}, 2/(1-\theta ))$, $0\le \theta \le 1\;$ and $\;0\le \alpha \le 1/2$.

In particular for $p\in (2,\infty )$ and $\alpha =1/2$, the estimate (3.16) becomes

$$\begin{aligned} \left( \int \limits _{-\infty }^{\infty } \Vert D^{(p-2)/4p} V(t)f\Vert _p^{4p/(p-2)}\,dt\right) ^{(p-2)/4p} \le c\Vert f\Vert _2. \end{aligned}$$

(3.17)

We take $\tilde{u}_0\in H^s(\mathbb R)$ with $s=j-\frac{p-2}{4p}=j-\hat{p}>3/4$ with $\tilde{u}_0\notin W^{j,p}(\mathbb R^{+})$.

From (3.17) it follows that there exists $\hat{t}\in (0,T/2)$ such that

$$\begin{aligned} V(\pm \hat{t}\,)\tilde{u}_0,\; V(\pm 2\hat{t}\,)\tilde{u}_0 \in W^{r,p}\quad \text {with } \ r=s+\frac{p-2}{4p}=j. \end{aligned}$$

Thus we consider the initial data

$$\begin{aligned} u_0=V(\hat{t}\,)\tilde{u}_0+ V(-\hat{t}\,)\tilde{u}_0. \end{aligned}$$

(3.18)

Observe that $u_0\in H^s(\mathbb R)$, so since

$$\begin{aligned} u(t)=V(t)u_0-\int \limits _0^t V(t-t') u^k \partial _x u (t')\,dt'= V(t)u_0 + z(t), \end{aligned}$$

from the argument in [23] one has that

$$\begin{aligned} z\in C([-T,T]:H^{s+1}(\mathbb R)) \hookrightarrow C([-T,T] : W^{j,p}(\mathbb R)). \end{aligned}$$

Also one sees that

$$\begin{aligned} V(\hat{t}\,)u_0= V(2\hat{t}\,)\tilde{u}_0+\tilde{u}_0 \notin W^{j,p}(\mathbb R^{+}). \end{aligned}$$

Similarly for $V(-\hat{t}\,)u_0$. Gathering this information we obtain the desired result.

• (b) Now we turn to the proof of the case $k=1$. We observed that the argument of proof in Theorem 1.4 (Step 1) shows that in the case $k=1$ if $\tilde{u}_0\in H^{\hat{\!j}}(\mathbb R)\cap L^2(|x|^{j/2}\,dx)$ with $\hat{\!j}=j+1/2-1/p-1/{12}$ (thus $H^{\hat{\!j}}\hookrightarrow W^{j,p}$) with $\tilde{u}_0\notin W^{j,p}(\mathbb R)$ one has that

$$\begin{aligned} z(t)=\int \limits _0^t V(t-t') u\partial _xu(t')\,dt'\in C([-T,T] : H^{j+\frac{1}{2}-\frac{1}{p}-\frac{1}{12}}(\mathbb R))\hookrightarrow W^{j,p}(\mathbb R). \end{aligned}$$

Once this has been established the rest of the proof follows the argument provided for the case $k=2, 3, \dots$.

4 BBM equation

Proof of Theorem 1.6 We shall restrict ourselves to consider the most general case $s=0$, i.e. $u_0\in L^2(\mathbb R)$. Thus, from the local well-posedness theory in Theorem A4 ([5]) there exist $T=T(\Vert u_0\Vert _{2})>0$ and a unique solution $u=u(x,t)$ of the IVP (1.15) such that

$$\begin{aligned} u\in C([0,T] : L^2(\mathbb R)). \end{aligned}$$

(4.1)

We rewrite the BBM equation in (1.15)

$$\begin{aligned} \partial _tu+\partial _xu+u\partial _xu-\partial _x^2\partial _tu=0 \end{aligned}$$

as the integro-differential equation

$$\begin{aligned} \partial _t u=-\partial _x J^{-2}(u+u^2/2) \end{aligned}$$

(4.2)

where

$$\begin{aligned} J^{-2}f=(1-\partial _x^2)^{-1}f=\frac{1}{2} e^{-|x|}*f. \end{aligned}$$

We observe that

$$\begin{aligned} \partial _x^2J^{-2}=J^{-2}-I. \end{aligned}$$

(4.3)

Since $u\in C([0,T] : L^2(\mathbb R))$ Sobolev embedding theorem guarantees that

$$\begin{aligned} \partial _xJ^{-2} u \in C([0,T] : H^1(\mathbb R)) \hookrightarrow C([0,T] : C_{\infty }(\mathbb R)) , \end{aligned}$$

(4.4)

where

$$\begin{aligned} \,C_{\infty }(\mathbb R)=\{ f :\mathbb R\rightarrow \mathbb R\,:\,f\;\text {continuous with}\;\;\lim _{|x|\rightarrow \infty }f(x)=0\}. \end{aligned}$$

Also, since

$$\begin{aligned} u^2\in C([0,T] : L^{1}(\mathbb R)), \end{aligned}$$

one has that

$$\begin{aligned} \partial _xJ^{-2} (u^2)=- \frac{1}{2} \text {sgn}(x) e^{-|x|}*u^2\in C([0,T]: C_b(\mathbb R)), \end{aligned}$$

(4.5)

where $\,C_{b}(\mathbb R)=C(\mathbb R)\cap L^{\infty }(\mathbb R)$. Hence, combining (4.2), (4.4) and (4.5) it follows that

$$\begin{aligned} u(x,t)= & {} u_0(x)-\int \limits _0^t \partial _xJ^{-2}(u+u^2/2)(x,\tau )\,d\tau \nonumber \\= & {} u_0(x) + z(x,t), \end{aligned}$$

(4.6)

with

$$\begin{aligned} z(x,t)=z\in C([0,T]: C_b(\mathbb R)). \end{aligned}$$

Thus, if for some open $\Omega \subset \mathbb R$, $u_0\big |_{\Omega }\in C(\Omega )$, then

$$\begin{aligned} u\big |_{\Omega \times [0,T]} \in C([0,T]:C(\Omega )). \end{aligned}$$

(4.7)

Moreover, using (4.3) one has that

$$\begin{aligned} \partial _x z(x,t)= & {} -\int \limits _0^t \partial ^2_xJ^{-2}(u+u^2/2)(x,\tau )\,d\tau \nonumber \\= & {} \int \limits _0^t ((u+u^2/2)-J^{-2}(u+u^2/2))(x,\tau )\,d\tau . \end{aligned}$$

(4.8)

Thus, from (4.7)

$$\begin{aligned} (u+u^2/2)\big |_{\Omega \times [0,T]} \in C([0,T]:C(\Omega )), \end{aligned}$$

and by (4.1) and an argument similar to that in (4.4)–(4.5)

$$\begin{aligned} J^{-2}(u+u^2/2)\in C([0,T]:C_b(\mathbb R)), \end{aligned}$$

therefore we conclude that if $u_0\big |_{\Omega }\in C(\Omega )$, then

$$\begin{aligned} z\big |_{\Omega \times [0,T]}\in C([0,T]: C^1(\Omega )). \end{aligned}$$

Hence, from (4.6) : if for some $\Omega \subset \mathbb R$ open, $u_0\big |_{\Omega }\in C^{\theta }(\Omega )$ with $\theta \in (0,1]$, then

$$\begin{aligned} u\big |_{\Omega \times [0,T]} \in C([0,T]:C^{\theta }(\Omega )). \end{aligned}$$

(4.9)

Now using the previous step (4.9) with $\theta =1$, i.e. if for $\Omega \subset \mathbb R$ open, $u_0\big |_{\Omega }\in C^{1}(\Omega )$, then

$$\begin{aligned} u\big |_{\Omega \times [0,T]} \in C([0,T]:C^1(\Omega )). \end{aligned}$$

(4.10)

This combined with (4.8) implies that

$$\begin{aligned} z(x,t)=z\in C([0,T]: C^2(\Omega )). \end{aligned}$$

Therefore, if $u_0\big |_{\Omega }\in C^{1+\theta }(\Omega )$ for some $\theta \in (0,1]$, then from (4.6) it follows that

$$\begin{aligned} u\big |_{\Omega \times [0,T]} \in C^1([0,T]:C^{1+\theta }(\Omega )). \end{aligned}$$

(4.11)

It is clear that by reapplying this argument one gets the desired result.

5 Degasperis–Procesi equation

In this section we shall consider the IVP associated to the Degasperis–Procesi (DP) equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu-\partial _t\partial _x^2u+4u\partial _xu= 3\partial _xu\partial _x^2u+u\partial _x^3u, \quad x\in \mathbb R, \quad t>0,\\ u(x,0)=u_0(x). \end{array}\right. } \end{aligned}$$

(5.1)

The equation in (5.1) can be rewritten in the integro-differential form

$$\begin{aligned} \partial _tu+u\partial _x u+\frac{3}{2} \,(1-\partial _x^2)^{-1}\partial _x (u^2)= 0, \end{aligned}$$

(5.2)

where

$$\begin{aligned} (1-\partial _x^2)^{-2}f=J^{-2}f= \frac{1}{2} e^{-|x|}*f. \end{aligned}$$

Notice that

$$\begin{aligned} \partial _x^2 J^{-2} = J^{-2} - I. \end{aligned}$$

Proof of Theorem 1.8 In [27] it was shown that the IVP associated to the Eq. (5.2) is locally well-posed in $H^s(\mathbb R)$ for $s>3/2$.

Let $u^{\epsilon }$ be the solution corresponding to the initial data $\rho _{\epsilon }*u_0 =u_0^{\epsilon }$, with $u_0\in H^{1+\delta }(\mathbb R)\cap W^{1,\infty }(\mathbb R),\;\delta >0,$ and $\rho _{\epsilon }$ denoting the usual mollifiers. Thus,

$$\begin{aligned} u^{\epsilon }\in C([0,T_{\epsilon }]:H^{\infty }(\mathbb R))\cap ... \end{aligned}$$

(5.3)

To estimate $T_{\epsilon }$ we recall that using the commutator estimates in [19], see (5.8)–(5.9), and simpler inequalities as those below one obtains the formal energy estimate: for any $v_0\in H^s(\mathbb R)$ with $s> 3/2$ the corresponding solution to the DP equation in (5.2) $v\in C([0,T] : H^s(\mathbb R))$ with $T=T(\Vert v_0\Vert _{s,2})>0$ obtained in [27] satisfies:

$$\begin{aligned} \frac{d}{dt}\Vert v (t)\Vert _{s,2}\le c_s (\Vert v(t)\Vert _{\infty } +\Vert \partial _x v(t)\Vert _{\infty })\,\Vert v(t)\Vert _{s,2}. \end{aligned}$$

(5.4)

The estimate (5.4) implies the following continuation principle : given $v_0\in H^s(\mathbb R),\,s>3/2$, then the corresponding solution $v\in C([0,T] : H^s(\mathbb R))$ of the IVP associated to (5.2) can be extended in the time interval $[0,T^*]$ with $T^*>T$ satisfying that $v\in C([0,T^*]:H^s(\mathbb R))$ whenever

$$\begin{aligned} \int _0^{T^*}(\Vert v(t)\Vert _{\infty } +\Vert \partial _x v(t)\Vert _{\infty })\,dt <\infty . \end{aligned}$$

(5.5)

A priori estimate We shall show that if $u_0\in H^{1+\delta }(\mathbb R)\cap W^{1,\infty }(\mathbb R),\,\delta >0$, then $T_{\epsilon }$ defined above can be estimated independently of $\epsilon$, i.e. $T_{\epsilon }=T,\;\forall \epsilon >0$, with

$$\begin{aligned} T=O( (\Vert u_0\Vert _{1,2}+\Vert u_0\Vert _{1,\infty })^{-1})\,\,\,\text {as}\,\,\,(\Vert u_0\Vert _{1,2}+\Vert u_0\Vert _{1,\infty })\downarrow 0. \end{aligned}$$

Applying energy estimates we have that

$$\begin{aligned} \frac{d}{dt} \Vert u^{\epsilon }(t)\Vert _2\le c\,\Vert u^{\epsilon }(t)\Vert _{\infty }\Vert u^{\epsilon }(t)\Vert _2 \end{aligned}$$

(5.6)

after using that

$$\begin{aligned} \Vert \partial _x J^{-2}(u^{\epsilon })^2\Vert _2=c \Vert \,\text {sgn}(x)e^{-|x|}*(u^{\epsilon })^2\Vert _2 \le \Vert (u^{\epsilon })^2\Vert _2\le \Vert u^{\epsilon }\Vert _{\infty }\Vert u^{\epsilon }\Vert _2. \end{aligned}$$

Also, as far as the characteristics flow is defined, i.e. $X^{\epsilon }(t; x_0)$,

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{dX^{\epsilon }}{dt}(t) = u^{\epsilon } ( X^{\epsilon }(t), t),\\ \\ X^{\epsilon }(0)=x_0, \end{array}\right. } \end{aligned}$$

one has that

$$\begin{aligned} \frac{d}{dt} u^{\epsilon }(X^{\epsilon }(t; x_0), t)= -\frac{3}{2} \partial _x J^{-2} ( (u^{\epsilon })^2). \end{aligned}$$

Since

$$\begin{aligned} \Vert \partial _x J^{-2}(u^{\epsilon })^2\Vert _{\infty } \le \Vert (u^{\epsilon })^2\Vert _1\le \Vert u^{\epsilon }\Vert _2^2, \end{aligned}$$

it follows that

$$\begin{aligned} \frac{d}{dt} \Vert u^{\epsilon }(t)\Vert _{\infty } \le c\,\Vert u^{\epsilon }(t)\Vert _2^2. \end{aligned}$$

(5.7)

Next, we recall the following estimates deduced in [19]: for any $r>0$

$$\begin{aligned} \Vert [J^r;f]g\Vert _2\le c_r(\Vert \partial _xf\Vert _{\infty }\Vert J^{r-1}g\Vert _2+\Vert g\Vert _{\infty }\Vert J^rf\Vert _2), \end{aligned}$$

(5.8)

and

$$\begin{aligned} \Vert J^r(fg)\Vert _2\le c_r(\Vert f\Vert _{\infty }\Vert J^{r}g\Vert _2+\Vert g\Vert _{\infty }\Vert J^rf\Vert _2). \end{aligned}$$

(5.9)

Combining (5.8), (5.9) one gets that

$$\begin{aligned}&\frac{d}{dt}\Vert u^{\epsilon }(t)\Vert _{1+\delta ,2}\nonumber \\&\quad \le c(\Vert \partial _x u^{\epsilon }(t)\Vert _{\infty }\Vert u^{\epsilon }(t)\Vert _{\delta ,2}+ c\Vert u^{\epsilon }(t)\Vert _{\infty }\Vert u^{\epsilon }(t)\Vert _{1+\delta ,2}). \end{aligned}$$

(5.10)

Finally, since

$$\begin{aligned} \partial _t(\partial _x u^{\epsilon })+u^{\epsilon }\partial _x(\partial _x u^{\epsilon })+(\partial _x u^{\epsilon })(\partial _x u^{\epsilon })+ \frac{3}{2}\Big (J^{-1} (u^{\epsilon })^2-(u^{\epsilon })^2\Big )=0, \end{aligned}$$

(5.11)

then

$$\begin{aligned}&\frac{d}{dt}(\partial _xu^{\epsilon }(X^{\epsilon }(t;x_0),t))\\&\quad +\left( (\partial _xu^{\epsilon })^2 +\frac{3}{2} (J^{-1}(u^{\epsilon })^2-(u^{\epsilon })^2)\right) (X^{\epsilon }(t;x_0),t)=0. \end{aligned}$$

Thus, using that

$$\begin{aligned} \Vert J^{-2} (u^{\epsilon })^2-(u^{\epsilon })^2\Vert _{\infty }\le c\,\Vert u^{\epsilon }\Vert _{\infty }^2, \end{aligned}$$

we conclude that if $u_0\in H^{1+\delta }(\mathbb R)\cap W^{1,\infty }(\mathbb R)$ (actually $H^s(\mathbb R)$ with $s>1/2$ instead of $1+\delta$ will suffice for this step), there exists $T=T(\Vert u_0\Vert _{1+\delta ,2}; \Vert u_0\Vert _{1,\infty })>0$ such that

$$\begin{aligned} \underset{[0,T]}{\sup } \Big (\Vert u^{\epsilon }(t)\Vert _{1+\delta ,2}+\Vert u^{\epsilon }(t)\Vert _{1,\infty }\Big ) \le M\Big (\Vert u_0\Vert _{1+\delta ,2}+\Vert u_0\Vert _{1,\infty }\Big ). \end{aligned}$$

(5.12)

We recall that if $v\in C(\mathbb R\times [0,T])\cap L^{\infty }([0,T]:W^{1,\infty }(\mathbb R))$, then the characteristic flow $X_v(t; x_0)=X(t; x_0)$ solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{dX}{dt}(t) = v ( X(t), t),\\ X(0)=x_0, \end{array}\right. } \end{aligned}$$

(5.13)

is well-defined. Thus, combining these facts with the continuation principle in (5.4)–(5.5) we conclude that

$$\begin{aligned} (u^{\epsilon })_{\epsilon >0}\subset C([0,T]:H^{\infty }(\mathbb R)), \end{aligned}$$

with T as in (5.12). As it was remarked above for this step one just needs $u_0\in H^s(\mathbb R)\cap W^{1,\infty }(\mathbb R)$, $s>1/2$.

Convergence as $\epsilon \downarrow 0$ Defining $w=u^{\epsilon }-u^{\epsilon '}$ one gets the equation

$$\begin{aligned} \partial _t w +u^{\epsilon }\partial _x w+w\partial _x u^{\epsilon }+\frac{3}{2} \partial _x J^{-2}\big ((u^{\epsilon }+u^{\epsilon '})w\big )=0. \end{aligned}$$

(5.14)

Thus

$$\begin{aligned} \frac{d}{dt}\Vert w(t)\Vert _2\le & {} c\Big ( \Vert \partial _x u^{\epsilon }\Vert _{\infty }+\Vert \partial _x u^{\epsilon '}\Vert _{\infty }\Big )\Vert w(t)\Vert _2\nonumber \\&+ c\Big ( \Vert u^{\epsilon }\Vert _{\infty }+\Vert u^{\epsilon '}\Vert _{\infty }\Big )\Vert w(t)\Vert _2. \end{aligned}$$

(5.15)

Hence

$$\begin{aligned} u^{\epsilon }\rightarrow u \quad \text {in } \ C([0,T] :L^2(\mathbb R))\quad \text {as } \ \epsilon \downarrow 0, \end{aligned}$$

and consequently from (5.12)

$$\begin{aligned} u^{\epsilon }\rightarrow u \quad \text {in } \ C([0,T] : H^{1}(\mathbb R)). \end{aligned}$$

(5.16)

Moreover,

$$\begin{aligned} u\in C([0,T] : H^{1}(\mathbb R))\cap L^{\infty }([0,T]: H^{1+\delta }(\mathbb R)), \end{aligned}$$

(5.17)

with

$$\begin{aligned} \partial _t u^{\epsilon } \rightarrow \partial _t u\quad \text {in } \ C([0,T] :L^2(\mathbb R))\quad \text {as } \ \epsilon \downarrow 0. \end{aligned}$$

This tells us that $u=u(x,t)$ is the solution of the DP equation (5.2) with data $u(x,0)=u_0(x)\in H^{1+\delta }(\mathbb R)\cap W^{1,\infty }(\mathbb R)$, where the equation is realized in $C([0,T]:L^2(\mathbb R))$. Furthermore, since $u\in C([0,T] : C_b(\mathbb R))$ (5.12) and (5.16) imply that $u\in L^{\infty }([0,T] : W^{1,\infty } (\mathbb R))$ with norm bounded by M as in (5.12), thus the characteristic flow $X_u(t;x_0)$, see (5.13), is defined.

Notice that (5.15) implies uniqueness and a weak continuous dependence of the solutions upon the data, i.e. if $u_{0_n} \rightarrow u_0$ in $H^1$ with $u_{0_n}, u_0\in H^{1+\delta }(\mathbb R)$ uniformly bounded, then the corresponding solutions $u_n$ converges to u in the $C([0,T]:H^1(\mathbb R))$-norm.

It is clear form our proof above that a weaker version of the Theorem 1.8 holds for $u_0\in H^{1+\delta }(\mathbb R)\cap W^{1,\infty }(\mathbb R)$, with $s>1/2$. However we fixed $s=1+\delta$, $\delta >0$, such that the equation is realized in $C([0,T]: L^2(\mathbb R))$.

Proof of Theorem 1.9 Here we establish the propagation of regularity at the $C^1$ level since at the $C^2$ level and beyond it follows by writing $m=(1-\partial _x^2)u$ and considering the equation

$$\begin{aligned} \partial _t m+u\partial _xm+3\partial _x u\, m=0. \end{aligned}$$

Notice first that if $f=\frac{3}{2} \partial _x J^{-2}(u^2)$ with u as in (5.17) one has that

$$\begin{aligned} f \in C([0,T] : H^{2}(\mathbb R)). \end{aligned}$$

(5.18)

Therefore, if $u_0\Big |_{\Omega }\in C^1$ with $\Omega \subseteq \mathbb R$ open, then

$$\begin{aligned} u(\cdot ,t)\Big |_{\Omega _t}\in C^1, \quad \Omega _t=\Phi _t(\Omega ) \end{aligned}$$

where $\Phi _t(x_0)=X_u(t; x_0)=X(t;x_0)$ defined as the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{dX}{dt}=u(X(t), t),\\ X(0)=x_0. \end{array}\right. } \end{aligned}$$

Since the solution $u(\cdot , \cdot )$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u+u\partial _x u+f=0\\ u(x,0)=u_0, \end{array}\right. } \end{aligned}$$

(5.19)

with $f=f(x,t)$ as in (5.18) one sees that

$$\begin{aligned} \frac{d}{dt} u(X(t;x_0), t) = f(X(t; x_0), t) \end{aligned}$$

and

$$\begin{aligned} u(X(t;x_0), t)= u_0(x) +\int \limits _0^t f(X(\tau ; x_0), \tau )\,d\tau . \end{aligned}$$

This yields the desired result.

6 Brinkman model 1-D case

This section is concerned with the IVP associated to the Brinkman model,

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \rho =\partial _x \big (\rho (1-\partial _x^2)^{-1}\partial _x (\rho ^2)\big ), \;\;\;x\in \mathbb R, \;t>0,\\ \rho (x,0)=\rho _0(x). \end{array}\right. } \end{aligned}$$

(6.1)

We shall use that

$$\begin{aligned} J^{-2}f= (1-\partial _x^2)^{-1}f =\frac{1}{2} e^{-|x|}*f \end{aligned}$$

(6.2)

and

$$\begin{aligned} \partial _x^2 J^{-2} =J^{-2} - I. \end{aligned}$$

(6.3)

Proof of Theorem 1.11 Let $\rho ^{\epsilon }$ be the solution corresponding to initial data $\rho _0^{\epsilon }(x)=G_{\epsilon }*\rho _0(x)$, with $G_{\epsilon }(x)=\epsilon ^{-1}G\big (x/\epsilon \big )$, $G\in C^{\infty }_0(\mathbb R)$, $G(x)\ge 0$, $\int G(x)dx =1$, and $\int x\,G(x)dx=0$. We recall that in [1] the local well-posedness of the IVP (6.1) in $H^s(\mathbb R)$, $s>3/2$ was established.

A priori estimate in $H^1(\mathbb R)$ To simplify the notation we shall use $\rho$ instead of $\rho ^{\epsilon }$ in (5.3), (5.4) and (6.8).

Energy estimates show that

$$\begin{aligned} \frac{d}{dt}\Vert \rho (t)\Vert _2^2= & {} \int \partial _x\rho (1-\partial _x^2)^{-1}\partial _x(\rho ^2)\rho \,dx\nonumber \\&+ \int \rho (1-\partial _x^2)^{-1}\partial _x^2(\rho ^2)\rho \,dx\nonumber \\= & {} \frac{1}{2}\,\int \rho (1-\partial _x^2)^{-1}\partial _x^2(\rho ^2)\rho \,dx\nonumber \\\le & {} \frac{1}{2} \Vert (J^{-2} - I)(\rho ^2)\Vert _{\infty }\Vert \rho (t)\Vert _2^2\nonumber \\\le & {} c \Vert \rho \Vert _{\infty }^2\Vert \rho (t)\Vert _2^2. \end{aligned}$$

(6.4)

since $\Vert J^{-2}(\rho ^2)\Vert _{\infty }\le c \Vert \rho ^2\Vert _{\infty }$, and

$$\begin{aligned} \frac{d}{dt}\Vert \partial _x\rho (t)\Vert _2^2= & {} \int \partial _x^2\big (\rho (1-\partial _x^2)^{-1}\partial _x(\rho ^2))\partial _x\rho \,dx\nonumber \\= & {} \int \partial _x^2\rho (1-\partial _x^2)^{-1}\partial _x(\rho ^2)\partial _x\rho \,dx\nonumber \\&+2\int \partial _x \rho (1-\partial _x^2)^{-1}\partial _x^2(\rho ^2)\partial _x\rho \,dx\nonumber \\&+\int \rho (1-\partial _x^2)^{-1}\partial _x^3(\rho ^2)\partial _x\rho \,dx\nonumber \\\le & {} c\Vert \rho \Vert _{\infty }^2\Vert \partial _x\rho (t)\Vert _2^2. \end{aligned}$$

(6.5)

Notice that from Sobolev embedding theorem and (6.4)–(6.5) one gets that

$$\begin{aligned} \frac{d}{dt}\Vert \rho ^{\epsilon }(t)\Vert _{1,2} \le c \Vert \rho ^{\epsilon }(t)\Vert _{1,2}^3. \end{aligned}$$

Therefore, there exists

$$\begin{aligned} T=T(\Vert \rho _0^{\epsilon }\Vert _{1,2})=T(\Vert \rho _0\Vert _{1,2})=c \,\Vert \rho _0\Vert _{1,2}^{-2}, \end{aligned}$$

(6.6)

such that

$$\begin{aligned} \sup _{[0,T]}\Vert \rho ^{\epsilon }(t)\Vert _{1,2} \le 2 \Vert \rho _0\Vert _{1,2}. \end{aligned}$$

(6.7)

Similarly,

$$\begin{aligned} \frac{d}{dt}\Vert \partial _x^2\rho (t)\Vert _2 \le c\Vert \rho \Vert ^2_{\infty }\Vert \partial _x^2\rho (t)\Vert _2. \end{aligned}$$

(6.8)

Combining (6.7), (6.8) it follows that if $\rho _0\in H^1(\mathbb R)$, then

$$\begin{aligned} \underset{[0,T]}{\sup } \Vert \partial ^2_x\rho ^{\epsilon }(t)\Vert _2= \mathrm{O}(\epsilon ^{-1}). \end{aligned}$$

(6.9)

Convergence as $\epsilon \downarrow 0$. Let $\rho$ and $\tilde{\rho }$ be solutions of (6.1). Thus, $w=\rho -\tilde{\rho }$ satisfies the equation

$$\begin{aligned} \partial _t w-\partial _x(w(1-\partial _x^2)^{-1}\partial _x(\rho ^2))-\partial _x(\tilde{\rho }(1-\partial _x^2)^{-1}\partial _x(\rho ^2-{\tilde{\rho }}^2))=0. \end{aligned}$$

(6.10)

Multiplying (6.10) by w and integrating in x we have

$$\begin{aligned} \frac{d}{dt}\Vert w(t)\Vert _2^2= & {} \int \partial _x(w(1-\partial _x^2)^{-1}\partial _x(\rho ^2))w\,dx\nonumber \\&+\int \partial _x(\tilde{\rho }(1-\partial _x^2)^{-1}\partial _x(\rho ^2-{\tilde{\rho }}^2))w\,dx\nonumber \\= & {} W_1 + W_2. \end{aligned}$$

(6.11)

By integration by parts

$$\begin{aligned} W_1= & {} \int \partial _xw(1-\partial _x^2)^{-1}\partial _x(\rho ^2) w\,dx\nonumber \\&+ \int w(1-\partial _x^2)^{-1}\partial _x^2(\rho ^2)w\,dx\nonumber \\\le & {} c\,\Vert \rho ^2\Vert _{\infty }\Vert w(t)\Vert _2^2 \end{aligned}$$

(6.12)

and

$$\begin{aligned} W_2= & {} \int \partial _x{\tilde{\rho }} (1-\partial _x^2)^{-1}\partial _x((\rho +{\tilde{\rho }})w)w\,dx\nonumber \\&+\int \tilde{\rho }(1-\partial _x^2)^{-1}\partial _x^2((\rho +{\tilde{\rho }}) w))w\,dx \nonumber \\\le & {} \Vert \partial _x{\tilde{\rho }}\Vert _2\Vert (1-\partial _x^2)^{-1}\partial _x((\rho +{\tilde{\rho }})w)\Vert _{\infty }\Vert w\Vert _2\nonumber \\&+\Vert \tilde{\rho }\Vert _{\infty }\Vert (\rho +\tilde{\rho })w\Vert _2\Vert w\Vert _2. \end{aligned}$$

(6.13)

Since

$$\begin{aligned} \Vert (1-\partial _x^2)^{-1}\partial _x((\rho +{\tilde{\rho }})w)\Vert _{\infty }\le \Vert (\rho +{\tilde{\rho }})w)\Vert _{1}\le \Vert \rho +{\tilde{\rho }}\Vert _2\Vert w\Vert _2, \end{aligned}$$

(6.14)

combining (6.11)–(6.14) we obtain that

$$\begin{aligned} \frac{d}{dt}\Vert w(t)\Vert _2^2 \le c (\underset{[0,T]}{\sup } \Vert \rho (t)\Vert _{1,2};\underset{[0,T]}{\sup } \Vert \tilde{\rho }(t)\Vert _{1,2})\Vert w(t)\Vert _2^2. \end{aligned}$$

Thus, if $0<\epsilon '<\epsilon$, then

$$\begin{aligned} \underset{[0,T]}{\sup } \Vert (\rho ^{\epsilon }-\rho ^{\epsilon '})(t)\Vert _2 \le c(\Vert \rho _0\Vert _{1,2}) \Vert \rho ^{\epsilon }_0-\rho ^{\epsilon '}_0\Vert _2 =\mathrm{o}(\epsilon ),\quad \text {as } \ \epsilon \;\downarrow 0. \end{aligned}$$

(6.15)

Similarly, for any two strong solutions $\rho$, $\tilde{\rho }\in C([0,T]:H^1(\mathbb R))$ one has

$$\begin{aligned} \underset{[0,T]}{\sup } \Vert (\rho -\tilde{\rho })(t)\Vert _2 \le c(\Vert \rho _0\Vert _{1,2};\Vert \tilde{\rho }_0\Vert _{1,2}) \,\Vert \rho _0-\tilde{\rho }_0\Vert _2. \end{aligned}$$

Next, we shall estimate for $\Vert \partial _x(\rho ^{\epsilon }-\rho ^{\epsilon '})(t)\Vert _2$. Let $w=\rho ^{\epsilon }-\rho ^{\epsilon '},$ so w satisfies the equation

$$\begin{aligned}&\partial _t w-\partial _x(w(1-\partial _x^2)^{-1}\partial _x((\rho ^{\epsilon })^2))\\&\quad -\partial _x(\rho ^{\epsilon '} (1-\partial _x^2)^{-1}\partial _x((\rho ^{\epsilon }+\rho ^{\epsilon '})w))=0. \end{aligned}$$

Thus

$$\begin{aligned} \frac{d}{dt} \Vert \partial _xw(t)\Vert _2^2= & {} \int \partial _x^2(w(1-\partial _x^2)^{-1}\partial _x((\rho ^{\epsilon })^2))\partial _xw\,dx\nonumber \\&+\int \partial _x^2(\rho ^{\epsilon '} (1-\partial _x^2)^{-1}\partial _x((\rho ^{\epsilon }+\rho ^{\epsilon '})w))\partial _x w\,dx\nonumber \\= & {} \widetilde{W}_1+\widetilde{W}_2. \end{aligned}$$

(6.16)

But

$$\begin{aligned} \widetilde{W}_1= & {} \int \partial _x^2w(1-\partial _x^2)^{-1}\partial _x((\rho ^{\epsilon })^2))\partial _xw\,dx\\&+2\int \partial _x w(1-\partial _x^2)^{-1}\partial _x^2((\rho ^{\epsilon })^2))\partial _xw\,dx\\&+\int w(1-\partial _x^2)^{-1}\partial _x^3((\rho ^{\epsilon })^2))\partial _xw\,dx \end{aligned}$$

and so

$$\begin{aligned} |\widetilde{W}_1|\le c\Vert \rho ^{\epsilon }\Vert _{\infty }^2 \Vert \partial w(t)\Vert _2^2+ c\Vert \rho ^{\epsilon }\Vert _{1,2}^2 \Vert w(t)\Vert _2^{1/2}\Vert \partial _x w(t)\Vert _2^{3/2}, \end{aligned}$$

after integrating by parts and using that

$$\begin{aligned} \int w(1-\partial _x^2)^{-1}\partial _x^3((\rho ^{\epsilon })^2)\partial _xw\,dx= - \int w(\partial _x-\partial _xJ^{-2})((\rho ^{\epsilon })^2)\partial _xw\,dx. \end{aligned}$$

Also,

$$\begin{aligned} \widetilde{W}_2= & {} \int \partial _x^2\rho ^{\epsilon '} (1-\partial _x^2)^{-1}\partial _x((\rho ^{\epsilon }+\rho ^{\epsilon '})w)\partial _x w\,dx\\&+2\int \partial _x\rho ^{\epsilon '} (1-\partial _x^2)^{-1}\partial _x^2((\rho ^{\epsilon }+\rho ^{\epsilon '})w)\partial _x w\,dx\\&+\int \rho ^{\epsilon '} (1-\partial _x^2)^{-1}\partial _x^3((\rho ^{\epsilon }+\rho ^{\epsilon '})w)\partial _x w\,dx\\= & {} \widetilde{W}_{2,1}+\widetilde{W}_{2,2}+\widetilde{W}_{2,3}. \end{aligned}$$

Then

$$\begin{aligned} |\widetilde{W}_{2,1}|\le & {} \Vert \partial _x^2\rho ^{\epsilon '}\Vert _2 \Vert (\rho ^{\epsilon }+\rho ^{\epsilon '})w\Vert _1\Vert \partial _x w\Vert _2\\\le & {} \Vert \partial _x^2\rho ^{\epsilon '}\Vert _2 \Vert \rho ^{\epsilon }+\rho ^{\epsilon '}\Vert _2\Vert w\Vert _2\Vert \partial _x w\Vert _2,\\ |\widetilde{W}_{2,2}|\le & {} \Vert \partial _x\rho ^{\epsilon '}\Vert _2 \Vert (\rho ^{\epsilon }+\rho ^{\epsilon '})w\Vert _{\infty }\Vert \partial _x w\Vert \\\le & {} \Vert \partial _x\rho ^{\epsilon '}\Vert _2\Vert \rho ^{\epsilon }+\rho ^{\epsilon '}\Vert _{\infty }\Vert w\Vert _{\infty }\Vert \partial _x w\Vert , \end{aligned}$$

and

$$\begin{aligned} |\widetilde{W}_{2,3}|\le & {} \Vert \rho ^{\epsilon '}\Vert _{\infty } \Vert \partial _x((\rho ^{\epsilon }+\rho ^{\epsilon '})w)\Vert _2\Vert \partial _x w\Vert _2\\\le & {} \Vert \rho ^{\epsilon '}\Vert _{\infty }(\Vert \rho ^{\epsilon }+\rho ^{\epsilon '}\Vert _{\infty }\Vert \partial _xw\Vert _2+ \Vert \partial _x(\rho ^{\epsilon }+\rho ^{\epsilon '})\Vert _2\Vert w\Vert _{\infty })\Vert \partial _x w\Vert _2. \end{aligned}$$

We observe that considering (6.7), (6.9), and (6.15) one sees that

$$\begin{aligned} |\widetilde{W}_{2,1}| \le c\, \mathrm{o}(1)\,\Vert \partial _xw\Vert _2 \,\,\text {as}\,\,\epsilon \downarrow 0. \end{aligned}$$

Inserting the above estimates in (6.16), and then using (6.9)–(6.15) it follows that $\rho ^{\epsilon }\rightarrow \rho$ with $\rho \in C([0,T] : H^1(\mathbb R))\cap C^1((0,T):L^2(\mathbb R))$ and T as in (6.6). The continuous dependence of the solution upon the data can be proved in a similar manner (see [4] or [22] Chapter 9). This basically completes the proof of Theorem 1.11.

Proof of Theorem 1.12 Using (6.3) we write the equation in (6.1) as

$$\begin{aligned} \partial _t\rho -(1-\partial _x^2)^{-1}\partial _x(\rho ^2)\partial _x\rho =(J^{-2}-I)(\rho ^2)\rho . \end{aligned}$$

Hence, formally one has that

$$\begin{aligned}&\partial _t\partial _x\rho -(1-\partial _x^2)^{-1}\partial _x(\rho ^2)\partial _x\partial _x \rho \nonumber \\&\quad =2(J^{-2}-I)(\rho ^2)\partial _x\rho -2\rho ^2\partial _x\rho +\partial _xJ^{-2}(\rho ^2)\rho , \end{aligned}$$

(6.17)

$$\begin{aligned}&\partial _t\partial ^2_x\rho -(1-\partial _x^2)^{-1}\partial _x(\rho ^2)\partial _x\partial ^2_x \rho \nonumber \\&\quad =3(J^{-2}-I)(\rho ^2)\partial ^2_x\rho -2\rho ^2\partial _x^2\rho +3\partial _x(J^{-2}-I)(\rho ^2)\partial _x\rho \nonumber \\&\qquad + \partial ^2_xJ^{-2}(\rho ^2)\rho -\big [\partial _x^2(\rho ^2)\rho -2\rho ^2\partial ^2_x\rho ], \end{aligned}$$

(6.18)

and for the general case $k\in \mathbb Z^+$

$$\begin{aligned}&\partial _t\partial ^k_x\rho -(1-\partial _x^2)^{-1}\partial _x(\rho ^2)\partial _x\partial ^k_x \rho \nonumber \\&\quad =(k+1)(J^{-2}-I)(\rho ^2)\partial ^k_x\rho -2\rho ^2\partial _x^k\rho \nonumber \\&\qquad +a^k_{k-1}\partial _x(J^{-2}-I)(\rho ^2)\partial _x^{k-1}\rho \nonumber \\&\qquad +a^k_{k-2}\partial ^2_x(J^{-2}-I)(\rho ^2)\partial _x^{k-2}\rho \nonumber \\&\qquad +\dots + a^k_{1}\partial ^{k-1}_x(J^{-2}-I)(\rho ^2)\partial _x\rho \nonumber \\&\qquad +\partial ^k_xJ^{-2}(\rho ^2)\rho -\big [\partial _x^k(\rho ^2)\rho -2\rho ^2\partial ^k_x\rho ]. \end{aligned}$$

(6.19)

From Theorem 1.11 given $\rho _0\in H^1(\mathbb R)$, there exist $T>0$ (as in (6.6)) and a unique strong solution $\rho \in C([0,T]: H^1(\mathbb R))$ of the IVP (6.1). We introduce the notation

$$\begin{aligned} a(x,t)\equiv J^{-2}\partial _x(\rho ^2)\in C([0,T]: H^1(\mathbb R)) \hookrightarrow C([0,T] :C^1_b(\mathbb R)), \end{aligned}$$

(6.20)

and

$$\begin{aligned} (J^{-2}-I)\rho ^2,\;\rho ^2 \in C([0,T]: H^1(\mathbb R))\hookrightarrow C([0,T] :C^1_b(\mathbb R)). \end{aligned}$$

(6.21)

In particular, from (6.20) it follows that the flow $\Phi _t(x_0)=X(t;x_0)$ given by the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d\;}{dt}X=a(X,t),\\ X(0)=x_0, \end{array}\right. } \end{aligned}$$

(6.22)

is well defined for $t\in [0,T]$. For $\Omega \subset \mathbb R$ we define $A^T_{\Omega }$ as

$$\begin{aligned} A^T_{\Omega }=\{(x,t)\,:\,x\in \Phi _t(\Omega ),\quad t\in [0,T]\}. \end{aligned}$$

Setting $\mu _k= \partial _x^k\rho$, $k=1, \dots , m$, the equations in (6.17)–(6.19) can be written as

$$\begin{aligned} \partial _t \mu _k+a(x,t)\partial _x \mu _k = b_k(x,t)\mu _k+c_k(x,t), \end{aligned}$$

(6.23)

with a(x, t) as (6.20),

$$\begin{aligned} b_k(x,t)\equiv (k+1)(J^{-2}-I)(\rho ^2)-2\rho ^2\in C([0,T] :C^1_b(\mathbb R)), \end{aligned}$$

(see (6.21)) and

$$\begin{aligned} c_k(x,t)\equiv & {} a^k_{k-1}\partial _x(J^{-2}-I)(\rho ^2)\partial _x^{k-1}\rho \nonumber \\&+a^k_{k-2}\partial ^2_x(J^{-2}-I)(\rho ^2)\partial _x^{k-2}\rho \nonumber \\&+\dots + a^k_{1}\partial ^{k-1}_x(J^{-2}-I)(\rho ^2)\partial _x\rho \nonumber \\&+\partial ^k_xJ^{-2}(\rho ^2)\rho -\big [\partial _x^k(\rho ^2)\rho -2\rho ^2\partial ^k_x\rho ]. \end{aligned}$$

(6.24)

Thus, for $k=1$, if $\rho _0\big |_{\Omega }\in C^1$ for some open set $\Omega \subseteq \mathbb R$, since

$$\begin{aligned} c_1(x,t)=\partial _xJ^{-2}(\rho ^2)\rho \in C([0,T] :C^1_b(\mathbb R)), \end{aligned}$$

then using the Eq. (6.23) with $k=1$, it follows that

$$\begin{aligned} \mu _1\big |_{A^T_{\Omega }}=\partial _x\rho (\cdot ,\cdot )\big |_{A^T_{\Omega }}\in C. \end{aligned}$$

If $\rho _0\big |_{\Omega }\in C^2$, by combining the previous case $k=1$ and the fact that $\rho \in C([0,T] :H^1(\mathbb R))$ it follows that

$$\begin{aligned} c_2(x,t)= & {} 3\partial _x(J^{-2}-I)(\rho ^2)\partial _x\rho +\partial ^2_xJ^{-2}(\rho ^2)\rho \nonumber \\&-\left[ \partial _x^2(\rho ^2)\rho -2\rho ^2\partial ^2_x\rho \right] \end{aligned}$$

(6.25)

satisfies that

$$\begin{aligned} c_2(\cdot ,\cdot )\big |_{A^T_{\Omega }}\in C, \end{aligned}$$

then using the equation (6.23) with $k=2$, one concludes that

$$\begin{aligned} \mu _2\big |_{A^T_{\Omega }}= \partial _x^2\rho (\cdot ,\cdot )\big |_{A^T_{\Omega }}\in C. \end{aligned}$$

For the general case $k\in \mathbb Z^+$, this iterative argument will yield the result if assuming that

$$\begin{aligned} \partial _x^j\rho (\cdot ,\cdot )\big |_{A^T_{\Omega }}\in C,\quad j=1,2,\ldots ,k, \end{aligned}$$

one can show that

$$\begin{aligned} c_{k+1}(\cdot ,\cdot )\big |_{A^T_{\Omega }}\in C. \end{aligned}$$

But this follows directly by the explicit form of $c_{k+1}(\cdot ,\cdot )$ in (6.24).