Abstract
This paper considers the initial value problem for a class of fifth order dispersive models containing the fifth order KdV equation
The main results show that regularity or polynomial decay of the data on the positive half-line yields regularity in the solution for positive times.
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1 Introduction
In this work we study propagation of regularity and persistence of decay results for a class of fifth order dispersive models. For concreteness, the main theorems are stated for initial value problems of the form
where \(c_{j}\) are real constants, \(u:\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\) is an unknown function and \(u_0:\mathbb {R}\rightarrow \mathbb {R}\) is a given function. Eq. (1.1) contains the specific equation
which is the third equation in the sequence of nonlinear dispersive equations
known as the KdV hierarchy. Here the polynomials \(Q_j\) are chosen so that Eq. (1.3) has the Lax pair formulation
for \(L=\frac{d^2}{dx^2}-u(x)\) the Schrödinger operator [16]. The first two equations in the hierarchy are
and the KdV equation
With only slight modifications concerning the hypothesis on the initial data, the techniques in this paper apply to a large class of fifth order equations including the following models arising from mathematical physics:
modelling the water wave problem for long, small amplitude waves over shallow bottom [22], a model describing short and long wave interaction [1]
and Lisher’s model for motion of a lattice of anharmonic oscillators [18]
See also [24] and references therein.
Following Kato’s definition [9], the initial value problem (IVP) (1.1) is said to be locally well-posed in the Banach space X if for every \(u_0 \in X\) there exists \(T>0\) and a unique solution u(t) satisfying
where \(Y_T\) is an auxillary function space. Moreover, the solution map \(u_0 \mapsto u\) is continuous from X into the class (1.9). If T can be taken arbitrarily large, the IVP (1.1) is said to be globally well-posed. The persistence condition (1.9) states that the solution curve describes a dynamical system.
It is natural to study the IVP (1.1) in the Sobolev spaces
having norm
The homogeneous derivative D and its inhomogeneous counterpart J are defined via the Fourier multipliers
where \(\langle x \rangle = (1+x^2)^{1/2}\). The weighted spaces
also appear in our analysis. Additionally, we use the notation \(x_+=\max \{0,x\},x_{-}=\min \{0,x\}\) and write \(A \lesssim B\) to denote \(A \le cB\) when the value of the fixed constant c is immaterial. The floor and ceiling functions are denoted by \(\lfloor {x}\rfloor \) and \(\lceil {x}\rceil \), respectively.
The persistence property (1.9) doesn’t preclude all smoothing effects. For step-data, Murray [21] proved the existence of solutions to the initial value problem for the KdV Equation (1.5) in the class \(C^\infty (\{x,t : x\in \mathbb {R}, \; t>0)\})\) which weakly recover the initial data. Kato [9] described this quasiparabolic smoothing effect as stemming from the unidirectional dispersion inherent in the equation. He obtained a similar result for data having exponential decay on the positive half-line. The Kato estimates occur in the asymmetric spaces
where
in which the operator \(\partial _t+\partial _x^3\) is formally equivalent to \(\partial _t+(\partial _x-\beta )^3\). The use of asymmetric spaces leads to a result which is irreversible in time. Isaza et al. [8] extended the quasiparabolic smoothing effect to a large class of fifth order equations.
Theorem A
(Isaza et al. [8]) Let \(u \in C([0,T];H^6(\mathbb {R}))\) be a solution of the IVP associated to the equation
corresponding to initial data \(u_0 \in H^6(\mathbb {R}) \cap L^2(e^{\beta x} \; dx), \beta >0\), with
and
Then
and
Kato [9] demonstrated the existence of weak global solutions u to the KdV Equation (1.5) corresponding to initial data in \(L^2(\mathbb {R})\). A key step in his proof is the a priori estimate of \(\Vert u\Vert _{H^1(-R,R)}\) in terms of \(\Vert u_0\Vert _2\). In addition, his approach shows the following local smoothing effect.
Theorem B
(Kato [9]) Let \(s>3/2\) and \(0<T<\infty \). If \(u \in C([0,T]; H^s(\mathbb {R}))\) is the solution to (1.5), then
with the associated norm depending only on \(\Vert u_0\Vert _{H^s}\), R and T.
Roughly, the proof follows by observing that a smooth solution u to the IVP associated to the KdV Equation (1.5) satisfies the identity
for \(k\in \mathbb {Z}^+\). Selecting \(\psi =\psi (x)\) to be a sufficiently smooth, nonnegative, nondecreasing cutoff function, integration of the above identity in time yields local estimates of \(\partial _x^{k+1}u\) as each term on right-hand side can be controlled by \(\Vert u\Vert _{L_T^\infty H^k}\).
Isaza et al. applied Kato’s argument to study the propagation of regularity and persistence of decay of solutions to the k-generalized KdV and Benjamin–Ono equations in Refs. [7] and [6], respectively. Also working in asymmetric spaces, they observed that for a solution u to the KdV equation corresponding to data \(u_0 \in H^{s}(\mathbb {R})\) with \(s>3/4\), if \(\Vert x^{n/2}u_0\Vert _{L^2(0,\infty )}\) for some \(n\in \mathbb {Z}^+\), then for every \(x_0\in \mathbb {R}\), \(u(\cdot ,t) \in H^n(x_0,\infty )\) for positive times. More succinctly, one-sided decay on the initial data yields regularity in the solution. In this paper we extend their work to fifth order dispersive models. Before stating our results we review the local well-posedness theory for (1.1) and related models.
Utilizing the Lax pair formulation, initial value problems associated to equations in the KdV hierarchy (1.3) can be solved in a space of rapidly decaying functions using the inverse scattering method [4]. This method does not apply to dispersive equations of a more general form.
While studying the models (1.1), (1.6), (1.7) and (1.8), Ponce [24] remarked that the use of dispersive estimates appears essential to attain local well-posedness in Sobolev spaces. Using the energy method, sharp linear estimates and parabolic regularization, in Ref. [24] Ponce proved local well-posedness for the initial value problems associated to these equations in \(H^s(\mathbb {R})\), \(s\ge 4\).
Kenig et al. investigated the class
with \(j\in \mathbb {Z}^+\) and \(P : \mathbb {R}^{2j+1}\rightarrow \mathbb {R}\) (or \(\mathbb {C}^{2j+1}\rightarrow \mathbb {C}\)) a polynomial having no constant or linear terms. Using the contraction principle, they established in Refs. [12] and [11] that for a given equation in the class (1.14) there exists a positive real number \(s_0\) and nonnegative integer \(m_0\) depending only on the form of the polynomial P such that the corresponding IVP is locally well-posed in the weighted space \(X_{s,m}\) for all \(m\in \mathbb {Z}^+\), \(m \ge m_0\) and \(s \ge \max \{s_0,jm\}\). Thus equations of the form (1.14) preserve the Schwarz class. The use of weighted spaces stems from the observation that \([L;\varGamma ]=0\) for the vector fields
Given that each term of P has “enough” factors, it may be that the corresponding IVP is globally well-posed, that no weight is necessary or both. For further comments, see [17].
Following [19] and [20], Pilod [23] showed that certain initial value problems in the class (1.14) are in some sense ill-posed. In particular, if P contains the term \(u\partial _x^ku\) for \(k>j\), then the solution map \(H^s(\mathbb {R}) \ni u_0 \mapsto u \in C([0,T];H^s(\mathbb {R}))\) is not \(C^2\) at the origin for any \(s\in \mathbb {R}\). For equations of the form (1.1), Kwon demonstrated that the solution map is not even uniformly continuous by using the arguments of [13] and [14]. All of these facts result from uncontrollable interactions when both high and low frequencies are present in the initial data. Thus, in contrast to the KdV (1.5), equations of the form (1.1) cannot be solved using the contraction principle in \(H^s(\mathbb {R})\).
Differences between (1.1) and (1.5) also arise when applying the energy estimate method. Note that after integrating by parts, smooth solutions u to (1.1) satisfy
for \(k\in \mathbb {Z}^+\). After integrating in time, the right-hand side cannot be estimated in terms of \(\Vert u\Vert _{L_T^\infty H^k}\). Kwon [15] introduced a corrected energy and refined Strichartz estimate to overcome this loss of derivatives and obtained the following result.
Theorem C
(Kwon [15]) Let \(s>5/2\). For any \(u_0 \in H^s(\mathbb {R})\) there exists a time \(T \gtrsim \Vert u_0\Vert _{H^s}^{-10/3}\) and a unique real-valued solution u for the IVP (1.1) satisfying
Remark 1
A loss of derivatives can occur for equations for which LWP can be obtained in \(H^s(\mathbb {R})\) using the contraction principle (see Sect. 7).
Using an auxillary Bourgain space introduced in Refs. [2, 3], the local well-posedness of the IVP (1.1) in the energy space \(H^2(\mathbb {R})\) was established simultaneously by Kenig and Pilod [10] and Guoet al. [5]. Thus global well-posedness follows in the Hamiltonian case, i.e., when \(c_{2}=2c_{3}\).
Our main contribution is the incorporation of Kwon’s corrected energy and refined Strichartz estimate into the iterative argument used in Refs. [7] and [6]. We first describe the propagation of one-sided regularity exhibited by solutions to the IVP (1.1) provided by Theorem C.
Theorem 1
Let \(s>5/2\). Suppose \(u_0 \in H^{s}(\mathbb {R})\) and for some \(l \in \mathbb {Z}^+, x_0 \in \mathbb {R}\)
Then the solution u of IVP (1.1) provided by Theorem C satisfies
for any \(\nu \ge 0, \varepsilon >0\) and each \(m=0,1,\ldots ,l\) with
where T is given in Theorem C. In particular, for all \(t\in (0,T]\), the restriction of \(u(\cdot ,t)\) to any interval \((x_1,\infty )\) belongs to \(H^l(x_1,\infty )\).
Moreover, for any \(\nu \ge 0, \varepsilon >0\) and \(R>\varepsilon \)
with
Remark 2
Observe that (1.20) is a generalization of Kato’s local smoothing effect since we do not require \(u_0 \in H^l(\mathbb {R})\).
Remark 3
The constants appearing in Theorem 1 have the form of a polynomial in \(\nu \). For \(l\ge 6\), the degree of this dependence is \(d=8(l-5)\).
For fixed \(l\in \mathbb {Z}^+\), Theorem 1 is the base case for the situation where the derivatives of the initial data possess polynomial decay when restricted to the positive half-line. Our second result states that this decay persists.
Theorem 2
Let \(s>5/2\) and let \(n,l\in \mathbb {Z}^+\). Suppose \(u_0 \in H^{s}(\mathbb {R})\) and for each \(m=0,1,\ldots ,l\)
Then the solution u of IVP (1.1) provided by Theorem C satisfies
for any \(\varepsilon >0\) and each \(m=0,1,\ldots ,l\) with
for \(k=0,1,\ldots ,m\), where T is given in Theorem C. By local well-posedness, we may take \(\varepsilon =0\) for \(m \le s\).
Moreover, for any \(\varepsilon >0\)
with \(\tilde{c}\) as in (1.24).
The hypothesis of Theorem 2 may seem unneccessarily strong, but a bootstrapping argument yields regularity of the solution for positive times by imposing decay on only the initial data and not its derivatives. Thus the next theorem can be seen as a weakening of the hypothesis of Theorem A in as much as exponential decay implies polynomial decay.
Theorem 3
Let \(s>5/2\). Suppose \(u_0 \in H^s(\mathbb {R})\) and for some \(n \in \mathbb {Z}^+\)
Then for every \(\delta >0\) and any pair \(m,k \in \mathbb {Z}^+\cup \{0\}\) satisfying
the solution u of IVP (1.1) provided by Theorem C satisfies, for \(k>0\)
for every \(\nu \ge 0, \varepsilon >0\), with
where T is given in Theorem C. For \(k=0\) and any \(R>\varepsilon \),
with \(\tilde{c}\) additionally depending on R.
The time reversible nature of Eq. (1.1) yields a number of consequences. Combining with the contrapositive of Theorems 1 and 3, we have the following.
Corollary 1
Assume that \(s>5/2\). Let \(u \in C([-T,T];H^{s}(\mathbb {R}))\) be a solution of (1.1) provided by Theorem C such that
Then for any \(t\in [-T,\hat{t})\) and any \(\beta \in \mathbb {R}\)
Suppose now that the initial data has regularity to the right but also contains a singularity, for instance \(u_0 \in H^{s}(\mathbb {R})\), \(u_0 \notin H^l(\mathbb {R})\) and
The persistence property (1.9) prohibits the solution from lying in \(H^l(\mathbb {R})\). However, as a consequence of Remark 3, we deduce that for positive times \(\partial _x^lu(\cdot ,t)\) has only polynomial growth to the left and thus lies in \(L^2_{\text {loc}}(\mathbb {R})\). That is, any singularities in \(\partial _x^lu(\cdot ,t)\) vanish for positive times. This is made precise by the next corollary to Theorem 1.
Corollary 2
Assume that \(s>5/2\). Let \(u \in C([-T,T];H^{s}(\mathbb {R}))\) be a solution of (1.1) provided by Theorem C. Suppose there exists \(l,m\in \mathbb {Z}^+\) with \(m \le l\) such that for some \(a,b\in \mathbb {R}\) with \(a<b\)
-
(i)
For any \(t\in (0,T]\) and any \(\varepsilon >0\)
$$\begin{aligned} \int _{-\infty }^\infty \frac{1}{\langle x_{-} \rangle ^{8(l-5)+\varepsilon }} \left( \partial _x^lu\right) ^2(x,t) \; dx \le c, \quad l\ge 6 \end{aligned}$$(1.32)with c depending on t and \(\varepsilon \).
-
(ii)
For any \(t\in [-T,0)\) and any \(\alpha \in \mathbb {R}\)
$$\begin{aligned} \int _\alpha ^\infty \left( \partial _x^mu\right) ^2(x,t) \; dx = \infty . \end{aligned}$$
Remark 4
The conclusion (1.32) holds for \(l=3,4,5\) with the appropriate modification to the weight.
As a consequence of Corollary 2 we see that, in general, regularity to the left does not propagate forward in time. Suppose in addition to (1.31) that
If this regularity persisted we could conclude from (1.32) that \(u(\cdot ,t) \in H^l(\mathbb {R})\) for positive times, contradicting the persistence property (1.9).
Beginning with Theorem 3 yields a similar corollary.
Corollary 3
Assume that \(s>5/2\). Let \(u \in C([-T,T];H^{s}(\mathbb {R}))\) be a solution of (1.1) provided by Theorem C. If for \(m,n\in \mathbb {Z}^+\), \(m<n\),
then for any \(t\in (0,T]\)
and for any \(t\in [-T,0)\)
Our proof technique does not rely on the particular values of the coefficients in (1.1), hence Theorems 1, 2 and 3 can be applied backwards in time. For instance, if u(x, t) is a solution of (1.1) with regularity to the right which propagates leftward, then \(u(-x,-t)\) has regularity to the left which propagates rightward. Therefore we can consider the situation when \(u(\cdot ,t_0)\) has decay or regularity to the right and \(u(\cdot ,t_1)\) has decay or regularity to the left, where \(t_0<t_1\).
Corollary 4
Assume that \(s>5/2\). Let \(u \in C([-T,T];H^{s}(\mathbb {R}))\) be a solution of (1.1) provided by Theorem C. If there exist \(n_j\in \mathbb {Z}^+\cup \{0\}\), \(j=1,2,3,4\), \(t_0,t_1\in [-T,T]\) with \(t_0<t_1\) and \(a,b\in \mathbb {R}\) such that
and
then
where
In Sect. 2 we construct cutoff functions which are needed to prove Theorems 1, 2 and 3. Theorems 1 and 2 are proved in Sects. 3 and 4, respectively. In Sect. 5 we prove Theorem 3. The proof of Corollary 2 is found in Sect. 6. We conclude in Sect. 7 with an extension to a more general class of fifth order models.
2 Construction of Cutoff Function
In this section we construct cutoff functions which are needed to prove Theorems 1, 2 and 3. Define the polynomial
which satisfies
with \(0<\rho ,\rho '\) for \(0<x<1\). Much of the complexity of our construction airses when handling the ratio which appears in (3.2), see Sect. 3 below. Thus we note that the expression
is continuous for \(x\in [0,1]\) and vanishes at the endpoints. For \(\varepsilon ,b>0\), define \(\chi \in C^5(\mathbb {R})\) by
By construction \(\chi \) is positive for \(x\in (\varepsilon ,\infty )\) and all derivatives are supported in \([\varepsilon ,b+\varepsilon ]\). A scaling argument and (2.1) provides
and for \(j=1,2,3,4,5\)
A computation produces
and for \(j=1,2,3,4,5\)
where \(q_0,\ldots ,q_5\) are polynomials. In each of the previous two cases, the right-hand side is continuous on the interval \(x\in [\varepsilon ,b+\varepsilon ]\), hence bounded. These computations lead to the following estimates, which will be used in a later inductive argument:
and for \(j=1,2,3,4,5\)
Additionally, we define \(\chi _n \in C^5(\mathbb {R})\) via the formula
It is helpful to make the auxillary definition
whose only real root occurs at \(y\approx 1.29727\). Note that for \(n\in \mathbb {Z}^+\)
which is positive for \(\varepsilon < x \le b+\varepsilon \). Hence the expression
is continuous in this interval. To prove that it is bounded in \([\varepsilon ,b+\varepsilon ]\), we must only analyze the limit \(x \rightarrow \varepsilon ^+\). First observe
so that
Each term of \(\chi _n'''\) has a factor of \((x-\varepsilon )^3\) implying the above limit vanishes. Hence
and so
Each term of (2.6) is nonnegative and \(\chi '\) is supported in \([\varepsilon ,b+\varepsilon ]\), hence
Using the Leibniz rule, it similarly follows for \(j=1,2,3,4,5\) that
Assuming \(n\ge 3\), notice that (2.7) and
imply
A similar argument holds for \(n=1,2\). Next we prove for \(j=1,2,3,4,5\)
This follows by definition when \(b+\varepsilon \le x\); thus it suffices to prove
We demonstrate the details for \(j=1\), the remaining cases being similar. In this case
Assuming \(\varepsilon \le x \le b+\varepsilon \),
Note that \(\frac{x-\frac{\varepsilon }{3}}{b+\varepsilon }<1\) so that p does not vanish in \([\varepsilon ,b+\varepsilon ]\). Hence this above expression is continuous and bounded on this interval. Similarly for the second term
This proves (2.11) in the case \(j=1\).
3 Proof of Theorem 1
In this section, we prove Theorem 1. We show several lemmas which are needed to prove Theorems 1, 2 and 3. The first lemma is an analogue of (1.13) to implement Kato’s energy estimate argument which is proved by Isaza et al. [8].
Lemma 1
Let \(u \in C^{\infty }([0,T];H^\infty (\mathbb {R}))\) be a solution to IVP
and let \(\psi \in C^5(\mathbb {R}^2)\) satisfy \(\partial _x\psi \ge 0\). Then we have
By interpolation we have the following lemma, which is required to apply the inductive hypothesis.
Lemma 2
Suppose \(u_0\in L^2(\mathbb {R})\) and for some \(l\in \mathbb {Z}^+\), \(l\ge 2\), \(x_0\in \mathbb {R}\)
For any \(k=1,2,\ldots ,l-1\) and \(\delta >0\)
We reproduce for convenience a lemma of Isaza et al. [7].
Lemma 3
Let \(j_1,j_2,j_3\in \mathbb {Z}^+\) and \(\varepsilon ,b>0\). Suppose \(\psi (x;\varepsilon ,b)\) has support in \([\varepsilon ,\infty )\), \(\psi \ge 0\) and \(\psi (x;\varepsilon ,b)\ge 1\) whenever \(x\ge b+\varepsilon \). Then
In particular, we may choose \(\psi =\chi , \chi ', \chi _n\) or \(\chi _n'\).
Proof
Using Cauchy–Schwarz and Young’s inequality, followed by the Sobolev embedding, we have
since \(\psi (x;\varepsilon ,b)\) is nonnegative, supported on \([\varepsilon ,\infty )\) and \(\psi (x;\varepsilon ,b)\ge 1\) when \(x \ge b+\varepsilon \). Furthermore, Young’s inequality yields
This completes the proof of Lemma 3. \(\square \)
We now turn to the proof of Theorem 1. As the argument is translation invariant, we consider only \(x_0=0\). Additionally, the estimates are performed for nonlinearity \(u\partial _x^3u\); a later remark explains how to control other terms. We invoke constants \(c_0,c_1,c_2,\ldots ,\) depending only on the parameters
whose value may change from line to line. We explicitly record dependence on the parameter \(\nu \) using the notation \(c(\nu ;d)\), which indicates a constant taking the form of a degree-d polynomial in \(\nu \):
We first describe the formal calculations and later provide justification using a limiting argument. Let u be a smooth solution of IVP (1.1), differentiate the equation l-times and apply (3.2) with \(\phi (x,t)=\chi (x+\nu t;\varepsilon ,b)\). Using properties (2.4) and (2.5) to expand the region of integration in the first term, we arrive at
where
We have used the convention that when \(\varepsilon \) and b are suppressed, \(\chi (x)=\chi (x;\varepsilon ,b)\). The argument proceeds via induction on l where, for fixed l, we integrate (3.7) in time, integrate B by parts and apply a correction to account for the loss of derivatives.
\( {{Case}\,l=1}\) Integrating in the time interval [0, t] and applying (2.3), we obtain
where \(0\le t\le T\). After integrating by parts, we find
The inequality (2.3) and the Sobolev embedding imply
Integrating the inequality (3.7) and combining (3.8) and (3.10), we obtain
As the right-hand side is independent of t, the result follows.
\({{Case}\,l=2}\) Similar to the previous case, integrating in the time interval [0, t], we find
where \(0\le t\le T\). After integrating by parts, we see
This expression exhibits a loss of derivatives in that the term
can be controlled neither by the well-posedness theory nor by the \(l=1\) case (without the technique introduced in Sect. 7). In [15], Kwon introduced a modified energy to overcome a similar issue. In particular, a smooth solution u to the IVP (1.1) satisfies the following identity:
where \(\chi ^{(j)}\) denotes \(\chi ^{(j)}(x+\nu t)\). We use this identity to eliminate (3.13) from (3.12), yielding
where the notation \(\widetilde{\partial _x^{j_1}u}\) indicates this factor may be omitted. That is, since \(0\le j_1,j_2 \le 2\),
Integrating in the time interval [0, t], applying (2.3) and the Sobolev embedding, we obtain
since \(\max \{j_1,j_2,j_3\}\le 2\). The fundamental theorem of calculus and Sobolev embedding yield
The first term on the right-hand side is controlled by the Sobolev embedding, the hypothesis on the initial data and Lemma 2. The second and third term illustrate the iterative nature of the argument, as they can be bounded by the \(l=1\) result. The two remaining integrals are finite by property (2.3). Therefore
Integrating inequality (3.7), using (3.11), (3.18) and the hypothesis on the initial data, we have
As the right-hand side is independent of t, the result follows.
\(\underline{\hbox {Case}\,l=3}\) Integrating in the time interval [0, t] and applying the \(l=1\) result, we obtain
where \(0\le t\le T\). After integrating by parts, we find
This expression exhibits a loss of derivatives in the term
A smooth solution u to the IVP (1.1) satisfies the following identity:
where \(\chi ^{(j)}\) denotes \(\chi ^{(j)}(x+\nu t)\), which we use to eliminate (3.21) from (3.20). Thus, ignoring coefficients, we may write
where the notation \(\widetilde{\partial _x^{j_1}u}\) indicates this factor may be omitted. Integrating in the time interval [0, t], applying (2.5), the Sobolev embedding and the \(l=1\) result yields
Similarly, integrating in the time interval [0, t], applying (2.3) and the Sobolev embedding, we find
Hence the fundamental theorem of calculus and Sobolev embedding yield
Similar to the \(l=2\) case, the first term on the right-hand side is controlled by the hypothesis on the initial data. The second and third terms are finite by the \(l=2\) case. Therefore
Integrating inequality (3.7), using (3.19), (3.27) and the hypothesis on the initial data,
Applying Gronwall’s inequality produces
This proves the desired result with \(l=3\).
\({{Cases}\,l=4,5,6}\) Due to the structure of the IVP, the cases \(l=4,5,6\) must be handled individually. The analysis is omitted as it is similar to the cases \(l=3\) and \(l\ge 7\). It can be proved that
where the values of d are summarized in Table 1.
\({{Case}\,l\ge 7}\) In the course of this case, we will prove that for \(l\ge 7\), the final constant obtained after integrating both sides of (3.7) takes the form of a polynomial in \(\nu \) with degree \(8(l-5)\).
Integrating in the time interval [0, t] and applying the \(l-2\) result (assuming \(l>7\)) we have
where \(0\le t\le T\). For \(l=7\), this expression has degree 5 in \(\nu \). We write
where
and \(3+k\le l-k<l\) for \(1\le k\le \lceil {l/2}\rceil -2\). Integrating by parts, we have
where
and, in \(B_{12}\), we have omitted coefficients depending only on l using the expression (3.30). Then integrating in the time interval [0, t], where \(0 \le t \le T\), we obtain
by the Sobolev embedding and (2.5). Applying the result for cases \(l-1\) and \(l-2\),
Observe that term \(B_2\) only occurs when \(l\ge 5\). For \(l>5\), note that \(4+k<l\). The inequality (3.5) produces
after suppressing constants depending on l. Integrating in the time interval [0, t],
The strongest \(\nu \)-dependence for \(B_2\) arises from analyzing terms of the form:
Each factor in (3.33) is finite by the result for cases \(l-k\) and \(4+k\). The inductive hypothesis further implies that the \(\nu \)-dependence has the form of a polynomial in \(\nu \) having degree
Hence
Integrating the inequality (3.7) in the time interval [0, t], where \(0 \le t \le T\), we have
using the hypothesis on the initial data, (3.28), (3.31) and (3.34). Thus it only remains to estimate the integral involving
which exhibits a loss of derivatives. Assuming that u satisfies the IVP (1.1), we rewrite this term by considering the correction factor
Observe that integrating \(\widetilde{C_3}\) by parts reveals
where
Here \(\chi ^{(j)}\) denotes \(\chi ^{(j)}(x+\nu t;\varepsilon ,b)\). The fundamental theorem of calculus leads to
We now concern ourselves with estimating the right-hand side of this expression. By the Sobolev embedding, hypothesis on the initial data, Lemma 2 and the result for case \(l-1\), we have
which is uniformly bounded by the inductive hypothesis. Applying (3.5), we obtain
Integrating in the time interval [0, t] and following the argument applied to term \(B_2\), we see that the strongest \(\nu \)-dependence for \(C_1\) arises from analyzing the term
Each factor in (3.41) is finite by the result for cases 6 and \(l-1\). Hence for the base case \(l=7\), the right-hand side is bounded by \(c(\nu ;16)\). For \(l>7\), the inductive hypothesis further yields that the \(\nu \)-dependence has the form of a polynomial in \(\nu \) with degree determined by
Thus
It will be clear from the remainder of the argument that (3.41) produces the overall strongest \(\nu \)-dependence, hence justifying this inductive calculation.
Integrating in time, using the Sobolev embedding and inductive hypothesis, we find
Integrating in time and using (2.5), (3.5), the Sobolev embedding and the inductive hypothesis, we have
Expanding but ignoring binomial coeffiecients, we write \(C_4=C_{41}+C_{42}\) with
and
Similar to \(C_2\) and \(C_3\),
Similar to \(B_2\), ignoring constants we have
after applying (3.5). Finally, assuming \(l>7\), we obtain
(or \(c(\nu ;3)\) when \(l=7\)) using the Sobolev embedding and inductive case \(l-3\).
Inserting the above into (3.39) and (3.35), then using nonnegativity of \(\chi ,\chi '\), we find
Hence Gronwall’s inequality yields
This concludes the proof for the case of smooth data.
Now we use a limiting argument to justify the previous computations for arbitrary \(u_0 \in H^{s}(\mathbb {R})\) with \(s>5/2\). Fix \(\rho \in C_0^\infty (\mathbb {R})\) with \(\text {supp}\;\rho \subseteq (-1,1)\), \(\rho \ge 0\), \(\int \rho (x)\;dx=1\) and
The solution \(u^\mu \) of IVP (1.1) corresponding to smoothed data \(u_0^\mu = \rho _\mu *u_0\), \(\mu \ge 0\), satisfies
Hence we may conclude
where
To see that this bound is independent of \(\mu >0\), first note
As \(\chi \equiv 0\) for \(x<\varepsilon \), restricting \(0<\mu <\varepsilon \) it follows
Thus by Young’s inequality
From Kwon’s local well-posedness result [15] we have
and so we may replace the bound \(c=c(\mu )\) with \(\tilde{c}\) as in (1.19).
As the solution depends continuously on the initial data,
Combining this fact with the \(\mu \)-uniform bound \(\tilde{c}\), weak compactness and Fatou’s lemma, the theorem holds for all \(u_0 \in H^{s}(\mathbb {R})\) with \(s>5/2\). This completes the proof of Theorem 1 for nonlinearity \(u\partial _x^3u\).
Including nonlinearity \(\partial _xu\partial _x^2u\), term B in (3.7) will contain a term
As this nonlinearity has a total of three derivatives, integrating by parts produces a form very similar to (3.29). The nonlinearity \(u^2\partial _xu\), containing only a single derivative, shows no loss of derivatives (see Sect. 7 for a more thorough treatment). This completes the proof of Theorem 1.
4 Proof of Theorem 2
In this section we prove Theorem 2. Let u be a smooth solution of IVP (1.1), differentiate the equation l-times and apply (3.2) with \(\phi (x,t)=\chi _n(x+\nu t;\varepsilon ,b)\) to arrive at
where
The proof proceeds by induction on l, however, for fixed l we induct on n. The base case \(n=0\) coincides with the propagation of regularity result. We invoke constants \(c_0,c_1,c_2,\ldots ,\) depending only on the parameters
as well as the decay assumptions on the initial data (1.22).
\({{Case}\,l=0}\) Using properties (2.8) and (2.9), we see
and so integrating in the time interval [0, t], we have
where \(0\le t\le T\). Additionally,
Integrating (4.1) in the time interval [0, t], combining (4.3) and (4.4), we have
using the hypothesis on the initial data. Gronwall’s inequality yields
Note that induction in n was not required in this case.
\({{Case}\,l=1}\) Using properties (2.8) and (2.9), we have
and so integrating in the time interval [0, t], we find
where \(0\le t\le T\). After integrating by parts, we find
This expression exhibits a loss of derivatives requiring a correction. A smooth solution u to the IVP (1.1) satisfies the following identity
after integrating by parts, where \(\chi _n^{(j)}\) denotes \(\chi _n^{(j)}(x+\nu t)\). Substituting (4.7), we can write (4.6) as a linear combination of the following terms
The fundamental theorem of calculus and the Sobolev embedding yield
where \(0 \le t \le T\). This term is finite by hypothesis (1.22) and the case \(l=0\). Next,
which is finite by case \(l=0\). Using (2.11) and the Sobolev embedding, we obtain
The first term is finite by induction on n in the current case \(l=1\), whereas the second term is finite by the case \(l=0\). The Sobolev embedding implies
Finally the inequality (2.11) and the Sobolev embedding yield
which is finite by case \(l=0\). Integrating (4.1) in the time interval [0, t] and combining the above, we have
The result follows by Gronwall’s inequality.
\({{Cases}\,l=2,3,4,5}\) Due to the structure of the IVP, the cases \(l=2,3,4,5\) must be handled individually. The analysis is omitted, however, as it is similar to the cases presented.
\({{Case}\,l\ge 6}\) Integrating in the time interval [0, t] and using properties (2.10) and (2.11), we have
which is finite by induction on n. Recall (3.29) and (3.30), wherein we wrote
with the term \(B_{11}\) exhibiting a loss of derivatives. Integrating in the time interval [0, t], we see
where we have used (2.11). The first term is finite by the case \(l-1\) and the third is finite by induction on n, hence
Observe that term \(B_2\) only occurs when \(l\ge 5\). For \(l>5\), note that \(4+k<l\). The inequality (3.5) yields
where we have suppressed constants depending on l. Integrating in the time interval [0, t], we see
as factors in the summation are estimated via (2.11) and the inductive hypothesis.
Assuming that u satisfies the IVP (1.1), we rewrite this term by considering the correction factor
where
Integrating \(\widetilde{C_1}\) by parts, we have
where
Here \(\chi _n^{(j)}\) denotes \(\chi _n^{(j)}(x+\nu t;\varepsilon ,b)\). The fundamental theorem of calculus yields
We now concern ourselves with estimating the right-hand side of this expression. First note
is bounded by the hypothesis (1.22) and the case \(l-1\). Similarly to \(B_2\) and \(B_{12}\), integrating in the time interval [0, t], using (3.5) and property (2.11), we obtain
where the term containing \((\partial _x^{l+1}u)^2\chi _n'\) is controlled using the induction case \(l-1\), as in (4.15).
Using (3.5) and the inductive hypothesis, we see
similar to \(B_2\). The same technique applies to \(C_3\) and \(C_4\).
Integrating (4.1) in the time interval [0, t] and combining the above, we find that there exists constants as in (4.2) such that
The result follows by Gronwall’s inequality. To handle the case of arbitrary data \(u_0 \in H^{s}(\mathbb {R})\) with \(s>5/2\), a limiting argument similar to the proof of Theorem 1 is used. This completes the proof of Theorem 2.
5 Proof of Theorem 3
In this section we prove Theorem 3. Integration by parts yields the next lemma.
Lemma 4
Suppose for some \(l\in \mathbb {Z}^+\)
Then for every \(0<\delta <T\), there exists \(\hat{t}\in (0,\delta )\) such that
To prove Theorem 3, it suffices to consider an example; fix \(n=9\) in the hypothesis of the theorem. Then we may apply Theorem 2 with \((l,n)=(0,9)\). Thus, after applying Lemma 4, there exists \(t_0\in (0,\delta /2)\) such that
Hence we may apply Theorem 2 with \((l,n)=(2,8)\) and find \(t_1 \in (t_0,\delta /2)\) such that
Continuing in this manner, applying Theorem 2 with \((l,n)=(4,7),(6,6),\ldots ,(18,0)\) provides the existince of \(\hat{t}\in (\delta /2,\delta )\) such that
Finally, we can apply Theorem 1 with \(l=19\), completing the proof.
6 Proof of Corollary 2
The proof of Corollary 2 relies on the following lemma, which follows by considering a dyadic decomposition of the interval \([0,\infty )\). Observe that the lemma also applies when integrating a nonnegative function on the interval \([-(a+\varepsilon ),-\varepsilon ]\), implying decay on the left half-line.
Lemma 5
Let \(f:[0,\infty ) \rightarrow [0,\infty )\) be continuous. If for \(a>0\)
then for every \(\varepsilon >0\)
Now we prove Corollary 2.
Proof
Recall that for \(l\ge 6\), Theorem 1 with \(x_0=0\) states
For fixed \(t\in (0,T)\)
Theorem 1 with \(\nu =0\) yields control of II, so we focus on I. For \(\nu ^*\) large enough, \(\nu >\nu ^*\) implies
Applying Lemma 5 with \(a=\nu t\) and \(\alpha =8(l-5)\), we find
for \(\varepsilon >0\). This completes the proof of Corollary 2. \(\square \)
7 Extensions to Other Models
In this section we prove the following extension of Theorem 1, which applies to those equations described by Theorem A.
Theorem 4
Consider the class of initial value problems
where \(Q:\mathbb {R}^4\rightarrow \mathbb {R}\) is a polynomial having no constant or linear terms. Let u be a solution to IVP (7.1) satisfying
such that \(m \ge m_0\) and \(s \ge \max \{s_0,2m\}\) for a nonnegative integer \(m_0\) and positive real number \(s_0\) determined by the form of the nonlinearity Q. If \(u_0 \in X_{s,m}\) additionally satisfies
for some \(l \in \mathbb {Z}^+, x_0 \in \mathbb {R}\), then u satisfies
for any \(\nu \ge 0, \varepsilon >0\) and each \(k=0,1,\ldots ,l\) with
Moreover, for any \(\nu \ge 0, \varepsilon >0\) and \(R>\varepsilon \)
with
Remark 5
Due to the similarities in the proof technique, the comments in this section can be modified to prove extensions of Theorems 2 and 3 to the class (7.1).
Remark 6
Establishing local well-posedness of the IVP (7.1) in the weighted Sobolev spaces \(X_{s,m}\) imposes minimum values on m and s, see for instance the contraction principle technique used by Kenig et al. in [12] and [11]. Thus the values of \(m_0\) and \(s_0\) are determined by considering both the local well-posedness as well as our proof of the propagation of regularity. As we see below, these considerations may differ.
Remark 7
A slight modification to the energy inequality (3.2) allows one to loosen the restriction that Q not contain any linear terms. In particular, the theorem applies to the model (1.6) when coupled with an appropriate local well-posedness theorem. Provided suitable cutoff functions exist, modifications to (3.2) also extend the technique to a class of higher order equations containing the KdV heirarchy.
Proof
Though not strictly necessary, we break the proof into cases based on the form of the nonlinearity Q(u). We treat the case \(x_0=0\) as the argument is translation invariant. Following the proof of Theorem 1, let u be a smooth solution of the IVP (7.1). Differentiating the equation l-times, applying (3.2) and using properties of \(\chi \), we arrive at
The proof proceeds by induction on \(l\in \mathbb {Z}^+\). For a given nonlinearity Q(u), there exists \(l_0\in \mathbb {Z}^+\) such that the cases \(l=0,1,\ldots ,l_0\) can be proved by choosing \(s_0\) large enough. Thus it suffices to prove only the inductive step. We describe the formal calculations, omitting the limiting argument.
Integrating in the time interval [0, t] and applying the \(l-2\) result we have
where \(0\le t\le T\) and \(c_0\) as in (7.4). We now turn to term B.
Case 1 Suppose Q is independent of both \(\partial _x^2u\) and \(\partial _x^3u\). Then there exists \(N\in \mathbb {Z}^+\) such that, after integrating by parts, B is a linear combination of terms of the form
and
where \(1 \le j_3 \le 5\) and \(3 \le k \le l+1\). Hence no loss of derivatives occurs. Integrating in the time interval [0, t], applying the induction hypothesis and the Sobolev embedding
provided \(s_0>7/2\), with \(c_0\) and \(c_1\) as in (7.4). Combining with (7.8), after integrating (7.7) in time and using the hypothesis on the initial data we have
The result follows by an application of Gronwall’s inequality. The value of \(m_0\) is determined by the LWP theory.
Case 2 Suppose Q is a linear combination of quadratic terms (with the exception of \(u\partial _x^2u\)). After integrating by parts B is a linear combination of terms of the form
as well as lower order terms. The correction technique of Theorem 1 can be modified to account for this loss of derivatives. For example, if \(Q(u)=\partial _x^2u\partial _x^3u\), then integrating by parts and supressing coefficients
where \(\tilde{B}\) is controlled by induction. For the second term, we impose \(s_0>9/2\) to control \(\Vert \partial _x^4u\Vert _{L_x^\infty }\). For the first term, consider the correction
In general, more than one correction may be necessary. The remainder of the proof is similar to Theorem 1, thus the value of \(m_0\) is determined by the LWP theory. Note that if Q additionally contained higher degree terms independent of \(\partial _x^2u\) and \(\partial _x^3u\), the above argument applies. Equations in the class (1.1) are of this form.
Case 3 The remaining nonlinearities in the class (7.1) exhibit a loss of derivatives which, in general, cannot be controlled by the correction technique. We illustrate the argument in this case by focusing on the example equation
The IVP associated to this equation is locally well-posed in \(H^s(\mathbb {R}), s\ge 2\), using the contraction mapping principle. However, our modification to the proof of Theorem 1 will require the use of weighted Sobolev spaces.
After integrating by parts and supressing coefficients
where \(\tilde{B}\) is controlled by induction. Combining with (7.8), after integrating (7.7) in time and using the hypothesis on the initial data we have
Focusing on the last term in the above line,
We check three cases to show the inductive case \(l-1\) bounds the second factor. First, the integral vanishes for \(j+1<\varepsilon -\nu T\). For \(\varepsilon <j\) we apply the inductive hypothesis with \(\nu =0\). Otherwise we utilize a pointwise bound on \(\chi \)
The technique for bounding the first factor is described in the next theorem. In general, there exists a nonnegative integer n depending on the form of the polynomial Q such that the following quantities must be estimated:
assuming u is a Schwarz solution of IVP (7.1). With such an estimate in hand, the result follows by an application of Gronwall’s inequality. \(\square \)
Theorem 5
Let \(k\in \mathbb {Z}^+\cup \{0\}\) and u be a Schwartz solution of the IVP (7.1) corresponding to initial data \(u_0\in \mathscr {S}(\mathbb {R})\). Then there exists a nonnegative integer \(m_0\) (depending on Q and k) and positive real number \(s_0\ge 2m_0\) such that
The idea is to apply a Sobolev type inequality in the t-variable and show that the resulting summation converges by imposing enough spatial decay on the solution. Acheiving this goal requires the following lemma.
Lemma 6
If \(f \in C^2(\mathbb {R}^2)\), then
for any \(L,T>0\).
We now turn to the proof of Theorem 5.
Proof
For concreteness, we show details for \(k=0\). Applying Lemma 6,
Focusing on the worst term \(\Vert \partial _{xt}u\Vert _{L_T^1L_x^1}\) and applying
we arrive at
Looking at the second term and using the differential equation we have
Then
and so we impose \(s_0\ge 12, m_0\ge 4\) (compared to the \(H^2(\mathbb {R})\) local well-posedness). The estimates for the remaining terms are similar, completing the case \(k=0\). \(\square \)
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Acknowledgments
A portion of this work was completed while J.S was visiting the Department of Mathematics at the University of California, Santa Barbara whose hospitality he gratefully acknowledges. The authors thank Professor Gustavo Ponce for giving us valuable comments. J.S is partially supported by JSPS, Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation and by MEXT, Grant-in-Aid for Young Scientists (A) 25707004.
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Segata, JI., Smith, D.L. Propagation of Regularity and Persistence of Decay for Fifth Order Dispersive Models. J Dyn Diff Equat 29, 701–736 (2017). https://doi.org/10.1007/s10884-015-9499-x
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DOI: https://doi.org/10.1007/s10884-015-9499-x