Multi-dimensional Burgers Equation with Unbounded Initial Data: Well-Posedness and Dispersive Estimates

Abstract

The Cauchy problem for a scalar conservation law admits a unique entropy solution when the datum $u_0$ is a bounded measurable function (Kružhkov). The semi-group $(S_t)_{t\geqq 0}$ is contracting in the $L^1$-distance. For the multi-dimensional Burgers equation, we show that $(S_t)_{t\geqq 0}$ extends uniquely as a continuous semi-group over $L^p({\mathbb R}^n)$ whenever $1\leqq p<\infty$. We answer affirmatively a question raised by M. Crandall, by showing that $u(t):=S_tu_0$ is not only an abstract solution, but is actually an entropy solution to the Cauchy problem. When $p\leqq q\leqq \infty$ and $t>0$, we prove that $S_t$ maps $L^p({\mathbb R}^n)$ into $L^q({\mathbb R}^n)$. These results are based upon new dispersive estimates. The ingredients are on the one hand Compensated Integrability, and on the other hand a DeGiorgi-type iteration.

Notations. When $1\leqq p\leqq \infty$, the natural norm in $L^p({\mathbb R}^n)$ is denoted $\Vert \cdot \Vert _p$, and the conjugate exponent of p is $p'$. The total space-time dimension is $d=1+n$ and the coordinates are $x=(t,y)$. In the space of test functions, $\mathcal{D}^+({\mathbb R}^{1+n})$ is the cone of functions which take non-negative values. The partial derivative with respect to the coordinate $y_j$ is $\partial _j$, while the time derivative is $\partial _t$. The various finite positive constants that depend only the dimension, but not upon the solutions of our PDE, are denoted $c_d,c_{d,p}, c_{d,p,q}$ ; they usually differ from one inequality to another. We denote with $C_0(0,+\infty )$, the space of continuous functions over $(0,+\infty )$, which tend to zero at infinity. Mind that $C({\mathbb R}_+)$ is the space of bounded continuous functions over $[0,+\infty )$.

1 Introduction

Let us consider a scalar conservation law in $1+n$ dimensions:

$$\begin{aligned} \partial _tu+\sum _{i=1}^n\partial _if_i(u)=0,\qquad t>0,\,y\in {\mathbb R}^n. \end{aligned}$$

(1)

We complement this equation with an initial datum

$$\begin{aligned} u(0,y)=u_0(y),\qquad y\in {\mathbb R}^n. \end{aligned}$$

The flux $f(s)=(f_1(s),\ldots ,f_n(s))$ is a smooth vector-valued function of $s\in {\mathbb R}$. We recall the terminology that an entropy–entropy flux pair is a couple $(\eta ,q)$ where $s\mapsto \eta (s)$ is a numerical function, $s\mapsto q(s)$ a vector-valued function, such that $q'(s)\equiv \eta '(s)f'(s)$. Kružhkov’s entropies and their fluxes form a one-parameter family:

$$\begin{aligned} \eta _a(s)=|s-a|,\qquad q_a(s)=\mathrm{sgn}(u-a)\,(f(u)-f(a)). \end{aligned}$$

Together with the affine functions, the $\eta _a$’s span the cone of convex functions.

We recall that an entropy solution is a measurable function $u\in L^1_\mathrm{loc}([0,+\infty )\times {\mathbb R}^n)$ such that $f(u)\in L^1_\mathrm{loc}([0,+\infty )\times {\mathbb R}^n)$, which satisfies the Cauchy problem in the distributional sense that

$$\begin{aligned} \int _0^\infty \,\mathrm{d}t\int _{{\mathbb R}^n}(u\partial _t\phi +f(u)\cdot \nabla _y\phi )\,\mathrm{d}y+\int _{{\mathbb R}^n}u_0(y)\phi (0,y)\,\mathrm{d}y=0,\qquad \forall \phi \in \mathcal{D}({\mathbb R}^{1+n}), \end{aligned}$$

(2)

together with the entropy inequalities

$$\begin{aligned}&\int _0^\infty \,\mathrm{d}t\int _{{\mathbb R}^n}(\eta _a(u)\partial _t\phi +q_a(u)\cdot \nabla _y\phi )\,\mathrm{d}y \nonumber \\&\quad +\int _{{\mathbb R}^n}\eta _a(u_0(y))\phi (0,y)\,\mathrm{d}y \geqq 0, \qquad \forall \,\phi \in \mathcal{D}^+({\mathbb R}^{1+n}),\,\forall \,a\in {\mathbb R}. \end{aligned}$$

(3)

The theory of this Cauchy problem dates back to 1970, when S. Kružhkov [10] proved that if $u_0\in L^\infty ({\mathbb R}^n)$, then there exists one and only one entropy solution in the class

$$\begin{aligned} L^\infty ({\mathbb R}_+\times {\mathbb R}^n)\cap C({\mathbb R}_+;L^1_\mathrm{loc}({\mathbb R}^n)). \end{aligned}$$

The parametrized family of operators $S_t:u_0\mapsto u(t,\cdot )$, which map $L^\infty ({\mathbb R}^n)$ into itself, form a semi-group. We warn the reader that $S_t:L^\infty \rightarrow L^\infty$ is not continuous, because of the onset of shock waves. Likewise, $t\mapsto u(t)$ is not continuous from ${\mathbb R}_+$ into $L^\infty ({\mathbb R}^n)$.

This semi-group nevertheless enjoys nice properties. On the one hand, a comparison principle says that if $u_0\leqq v_0$, then $S_tu_0\leqq S_tv_0$. For instance, the solution u associated with the datum $u_0$ is majorized by the solution $\bar{u}$ associated with the datum $(u_0)_+$, the positive part of $u_0$. On the other hand, if $v_0-u_0$ is integrable over ${\mathbb R}^n$, then $S_tv_0-S_tu_0$ is integrable too, and we have

$$\begin{aligned} \int _{{\mathbb R}^n}|S_tv_0-S_tu_0|(y)\,\mathrm{d}y\leqq & {} \int _{{\mathbb R}^n}|v_0-u_0|(y)\,\mathrm{d}y \end{aligned}$$

(4)

$$\begin{aligned} \int _{{\mathbb R}^n}(S_tv_0-S_tu_0)(y)\,\mathrm{d}y= & {} \int _{{\mathbb R}^n}(v_0-u_0)(y)\,\mathrm{d}y. \end{aligned}$$

(5)

Finally, $S_t$ maps $L^p\cap L^\infty ({\mathbb R}^n)$ into itself, and the function $t\mapsto \Vert S_tu_0\Vert _p$ is non-increasing.

Because of (4) and the density of $L^1\cap L^\infty ({\mathbb R}^n)$ in $L^1({\mathbb R}^n)$, the family $(S_t)_{t\geqq 0}$ extends in a unique way as a continuous semi-group of contractions over $L^1({\mathbb R}^n)$, still denoted $(S_t)_{t\geqq 0}$. When $u_0\in L^1({\mathbb R}^n)$ is unbounded, we are thus tempted to declare that $u(t,y):=(S_tu_0)(y)$ is the abstract solution of the Cauchy problem for (1) with initial datum $u_0$. At this stage, it is unclear whether $(S_t)_{t\geqq 0}$ can be defined as a semi-group over some $L^p$-space for $p\in (1,\infty )$, because the contraction property (4) occurs only in the $L^1$-distance, but in no other $L^p$-distance.

An alternate construction of $(S_t)_{t\geqq 0}$ over $L^1({\mathbb R}^n)$, based upon the Generation Theorem for nonlinear semigroups, was done by M. Crandall [2], who pointed out that it is unclear whether the abstract solution u is an entropy solution, because the local integrability of the flux f(u) is not guaranted.¹ The following problem is therefore an important one:

Identify the widest class of integrable initial data for which the abstract solution of (1) is actually an entropy solution.

Our most complete results are about a special case: the so-called multi-dimensional Burgers equation

$$\begin{aligned} \partial _tu+\partial _1\frac{u^2}{2}\,+\cdots +\,\partial _n\frac{u^{n+1}}{n+1}=0. \end{aligned}$$

(6)

This equation was already considered by G. Crippa et al. [3], and more recently by L. Silvestre [18]. The particular flux in (6) is a prototype for genuinely nonlinear conservation laws, those which satisfy the assumption

$$\begin{aligned} \det (f'',\ldots ,f^{(n+1)})\ne 0. \end{aligned}$$

(7)

The latter condition is a variant of the non-degeneracy condition at work in the kinetic formulation of the equation (1) ; see [12] or [13].

Our first result deals with dispersive estimates.

Theorem 1.1

Let $1\leqq p\leqq q\leqq \infty$ be two exponents. Define two parameters $\alpha ,\beta$ depending on p and q by

$$\begin{aligned} \alpha (p,q)=\frac{h(q)}{h(p)}\,,\qquad h(p):=2+\frac{dn}{p}\, \end{aligned}$$

(8)

and

$$\begin{aligned} \beta (p,q)=h(q)(\delta (p)-\delta (q)),\qquad \delta (p):=\frac{n}{2p+dn}\,. \end{aligned}$$

(9)

Then there exists a finite constant $c_{d,p,q}$ such that for every initial datum $u_0\in L^1\cap L^\infty ({\mathbb R}^n)$, the entropy solution u(t) of the scalar conservation law (6) satisfies

$$\begin{aligned} \Vert u(t)\Vert _q\leqq c_{d,p,q}\,t^{-\beta (p,q)}\Vert u_0\Vert _p^{\alpha (p,q)},\qquad \forall \,t>0. \end{aligned}$$

(10)

Remarks

• The consistency of estimates (10) with the Hölder inequality is guaranted by the property that whenever $\theta \in (0,1)$,

  $$\begin{aligned} \left( \frac{1}{q}=\frac{1-\theta }{p}+\frac{\theta }{r}\right) \Longrightarrow \left\{ \begin{array}{l} \alpha (p,q)=1-\theta +\theta \alpha (p,r), \\ \\ \beta (p,q)=\theta \beta (p,r). \end{array}\right. \end{aligned}$$

  (11)

• The consistency under composition $(p,q)\wedge (q,r)\mapsto (p,r)$ is ensured by the rules

  $$\begin{aligned} \alpha (p,r)=\alpha (p,q)\alpha (q,r)\qquad \hbox {and}\qquad \beta (p,r)=\beta (q,r)+\beta (p,q)\alpha (q,r) \end{aligned}$$

  (12)

• In one space dimension, (10) gives back well-know results, such as Theorem 11.5.2 in [6].²

Theorem 1.1 has several important consequences. An obvious one is that the extension of $(S_t)_{t\geqq 0}$ as a semi-group over $L^1({\mathbb R}^n)$ satisfies the above estimates with $p=1$.

Corollary 1.1

If $u_0\in L^1({\mathbb R}^n)$ and $t>0$, then $S_tu_0\in \bigcap _{1\leqq q\leqq \infty } L^q({\mathbb R}^n)$ and we have

$$\begin{aligned} \Vert S_tu_0\Vert _q\leqq c_{d,q}\,t^{-\kappa /q'}\Vert u_0\Vert _1^{1-\nu /q'},\qquad \forall q\in [1,\infty ], \end{aligned}$$

where the exponents are given in terms of

$$\begin{aligned} \kappa =2\,\frac{d-1}{d^2-d+2}\,\quad \hbox {and}\quad \nu =\frac{d(d-1)}{d^2-d+2}\,. \end{aligned}$$

The next one is that the Cauchy problem is solvable for data taken in $L^p({\mathbb R}^n)$ for arbitrary exponent $p\in [1,\infty ]$. In particular, it solves Crandall’s concern.

Theorem 1.2

Let $p\in [1,\infty )$ be given. For every $t\geqq 0$, the operator $S_t:L^1\cap L^\infty ({\mathbb R}^n)\rightarrow L^1\cap L^\infty ({\mathbb R}^n)$ admits a unique continuous extension $S_t:L^p({\mathbb R}^n)\rightarrow L^p({\mathbb R}^n)$.

The family $(S_t)_{t\geqq 0}$ is a continuous semi-group over $L^p({\mathbb R}^n)$. If $u_0\in L^p({\mathbb R}^n)$, the function u(t, y) defined by $u(t)=S_tu_0$ is actually an entropy solution of the Cauchy problem for (6) with initial datum $u_0$.

Finally, $S_t(L^p({\mathbb R}^n))$ is contained in $\bigcap _{p\leqq q\leqq \infty }L^q({\mathbb R}^n)$ and the estimates (10) are valid for every data $u_0$ in $L^p({\mathbb R}^n)$.

The proof of Theorem 1.1 will be done in two steps. The first one consists in establishing the estimate (10) when $q=p^*$ is given by the formula

$$\begin{aligned} p^*=d\left( 1+\frac{p}{n}\right) . \end{aligned}$$

To this end, we apply Compensated Integrability to a suitable symmetric tensor, whose row-wise divergence is a bounded measure with controlled mass. This argument involves the theory recently developped by the first author in [14, 15]. The second step is an iteration in DeGiorgi’s style, based on the preliminary work [18] by the second author; see also the original paper by E. DeGiorgi [7] or the review paper by A. Vasseur [21]. This technique allows us to establish an $L^\infty$-estimate, which extends the dispersive estimate to $q=+\infty$. Then using the Hölder inequality, we may interpolate between this result and the decay of $t\mapsto \Vert u(t)\Vert _p$, and treat every exponent $q>p$.

We notice that the symmetric tensor mentionned above extends to a multi-dimensional context the one already used when $n=1$ by L. Tartar [20] to prove the compactness of the semi-group, and by F. Golse [8] (see also [9]) to prove some kind of regularity.

Previous dispersive estimates ($n=1$). In one space dimension, (6) reduces to the original Burgers equation. Its Kružhkov solution satisfies the Oleinik inequality $\partial _yu\leqq \frac{1}{t}$ , which does not involve the initial datum at all. Ph. Bénilan & M. Crandall [1] proved

$$\begin{aligned} TV\left( \frac{u(t)^2}{2}\right) \leqq \frac{2\Vert u_0\Vert _1}{t} \end{aligned}$$

(13)

by exploiting the homogeneity of the flux $f(s)=\frac{s^2}{2}$. Inequality (13) implies an estimate

$$\begin{aligned} \Vert u(t)\Vert _\infty \leqq 2\,\sqrt{\frac{2\Vert u_0\Vert _1}{t}\,}, \end{aligned}$$

(14)

which is a particular case of Corollary 1.1.

C. Dafermos [5] proved a general form of (13) in situations where the flux f may have one inflexion point and the datum $u_0$ has bounded variation, by a clever use of the generalized backward characteristics. His argument involves the order structure of the real line. Backward characteristics are not unique in general. Given a base point in the upper half-plane, one has to define and analyse the minimal and the maximal ones. The description of backward characteristics seems to be much more complicated in higher space dimensions, and Dafermos’ strategy has not been applied successfully beyond the 1-D case.

Enhanced decay. Because of a scaling property which will be described in the next section, the dispersion (10) is optimal, as long as we involve only the $L^p$-norms, and we exclude any extra information about the initial datum. It is however easy to obtain a better decay as time t goes to infinity. Let us give one example, by taking an initial datum $u_0$ such that

$$\begin{aligned} 0\leqq u_0(y)\leqq v_0(y_1),\qquad v_0\in L^1({\mathbb R}). \end{aligned}$$

By the maximum principle, we have $u(t,y)\leqq v(t,y_1)$, where v is the solution of the 1-dimensional Burgers equation associated with the initial datum $v_0$. We have, therefore, that

$$\begin{aligned} \Vert u(t)\Vert _\infty \leqq 2\,\sqrt{\frac{2\Vert v_0\Vert _1}{t}\,}\,, \end{aligned}$$

where the decay rate $t^{-\frac{1}{2}}$ is independent of the space dimension. In particular this decay is faster than that given by Corollary 1.1 when $n\geqq 3$.

The way this faster decay is compatible with the optimality of (10) is well explained by a study of the growth of the support of the solution. In the most favorable case where the datum $u_0$ is bounded with compact support, the argument above yields $\Vert u(t)\Vert _\infty =O((1+t)^{-1/2})$. It is easy to infer that the width of $\mathrm{Supp}(u(t))$ in the $y_1$-direction expands as $O(\sqrt{t}\,)$ (one might have used the comparison with the solution v above). Likewise, the width in the $y_2$-direction is an $O(\log t)$ and that in the other $y_k$-directions remains bounded because

$$\begin{aligned} \int _0^\infty (1+t)^{-\frac{k}{2}}\,\mathrm{d}t<\infty . \end{aligned}$$

On the contrary, if $u_0\in L^1({\mathbb R}^n)$ has compact support but is not bounded by an integrable fonction $v_0(y_1)$ as above, Corollary 1.1 gives only $\Vert u(t)\Vert _\infty =O(t^{-\kappa })$. It turns out that $n\kappa \geqq 1$ when $n\geqq 2$, and therefore

$$\begin{aligned} \int _0t^{-n\kappa }\,\mathrm{d}t=+\infty . \end{aligned}$$

This suggests that the width of the support in the $y_n$-direction is immediately infinite; the support of u(t) is unbounded for every $t>0$. The solution has a tail in the last direction, and this tail is responsible for a slow $L^\infty$-decay, at rate $t^{-\kappa }$ instead of $t^{-\frac{1}{2}}$ .

This analysis suggests in particular that the fundamental solution$U_m$, if it exists, should have an unbounded support in the space variable when $n\geqq 2$. The terminology denotes an entropy solution of (6), say a non-negative one, with the property that

$$\begin{aligned} U_m(t){\mathop {\longrightarrow }\limits ^{t\rightarrow 0+}}m\,\delta _{y=0} \end{aligned}$$

in the vague sense of bounded measures. In particular,

$$\begin{aligned} \int _{{\mathbb R}^n}U_m(t,y)\,\mathrm{d}y\equiv m. \end{aligned}$$

This behaviour is in strong constrast with the one-dimensional situation, where

$$\begin{aligned} U_m(t,y)=\frac{y}{t}\,\mathbf{1}_{(0,\sqrt{2m t}\,)} \end{aligned}$$

is compactly supported at every time.

The existence of a fundamental solution is left as an open problem. It should play an important role in the time-asymptotic analysis of entropy solutions of finite mass. This asymptotics has been known in one-space dimension since the seminal works by P. Lax [11] and C. Dafermos [4].

Preliminary works. The authors posted, separately, recent preprints on this subject in ArXiv database, see [16, 19]. The present paper supersedes both of them.

Outline of the article. Since Compensated Integrability plays a important role in this work and is a rather new tool, we end this introductory section with a short presentation of the theory. In Section 2, we prove a special case of the dispersive estimate (10), that for the pairs $(p,p^*)$. We treat the case $(p,\infty )$ in Section 3. This allows us to extend the equation (10) to every pair (p, q) with $p\leqq q$. The construction of the semi-group over every $L^p$-space is done in Section 4. We show in Section 5 how these ideas adapt to a scalar equation when the fluxes $f_j$ are monomials. The last section describes how the first argument, which involves Compensated Integrability, can be adapted to conservation laws with arbitrary flux.

1.1 Compensated Integrability for Symmetric Positive Tensors

We keep the notations $d=1+n$, $n\geqq 1$ above. The cone of positive semi-definite symmetric real matrices is $\mathbf{Sym}_d^+$. If $\Omega \subset {\mathbb R}^d$ and $x\in \Omega \rightarrow A(x)\in \mathbf{Sym}_d$ is a symmetric tensor, we denote $\mathrm{Div}\,A$ its row-wise divergence, understood in the distributional sense that

$$\begin{aligned} (\mathrm{Div}\,A)_i=\sum _{j=0}^n\frac{\partial a_{ij}}{\partial x_j}=\frac{\partial a_{i0}}{\partial t}+\sum _{j=1}^n\frac{\partial a_{ij}}{\partial y_j}\,. \end{aligned}$$

If $\mu$ is a bounded measure over either $\Omega$ or $\partial \Omega$, its total mass is denoted $\Vert \mu \Vert _\mathcal{M}$; in practice, $\mu$ is often vector-valued and this norm is the total mass of $|\mu |$

The following result is taken from Theorems 2.2 and 2.3 of [15]. We denote $Q_\tau =(0,\tau )\times {\mathbb R}^n$ a slab in time-space coordinates:

Theorem 1.3

Let $A:Q_\tau \rightarrow \mathbf{Sym}_d^+$ be integrable. Let us assume that $\mathrm{Div}\,A$ is component-wise a bounded measure over $Q_\tau$, and that the normal traces $A_{0\bullet }:=A\mathbf {e}_t$ at initial and final times ($t=0$ and $t=\tau$) are bounded measures over ${\mathbb R}^n$. Then the function $(t,y)\mapsto (\det A)^{\frac{1}{n}}$ is integrable over $Q_\tau$, and there holds and inequality

$$\begin{aligned} \int _0^\tau \,\mathrm{d}t\int _{{\mathbb R}^n}(\det A)^{\frac{1}{n}}\mathrm{d}y\leqq & {} c_d\left( \Vert A_{0\bullet }(0,\cdot )\Vert _{\mathcal{M}({\mathbb R}^n)} \right. \nonumber \\&\left. +\Vert A_{0\bullet }(\tau ,\cdot )\Vert _{\mathcal{M}({\mathbb R}^n)}+\Vert \mathrm{Div}A\Vert _{\mathcal{M}(Q_\tau )}\right) ^{1+\frac{1}{n}}, \end{aligned}$$

(15)

where $c_d$ is a universal constant. In particular, $c_d$ does not depend on $\tau$.

For the sake of completeness, we point out that Theorem 1.3 is none of Theorems 2.2 and 2.3 of [15]. On the one hand, Theorem 2.2 is for a bounded domain $\Omega$, instead of a slab. On the other hand, Theorem 2.3 deals with a divergence-free tensor, thus ignores the contribution of $\mathrm{Div}\,A$ in the right-hand side of (15). However, the proof of the theorem above goes exactly the same way as that of Theorem 2.3: multiply A by a cut-off function $\phi _R(y)=\phi (\frac{y}{R})$ with $0\leqq \phi \leqq 1$ and $\phi \equiv 1$ in the unit ball, apply Theorem 2.2 in $(0,\tau )\times B$ where the ball B contains the support of $\phi _R$, then pass to the limit as $R\rightarrow +\infty$ with the help of Fatou’s Lemma.

Notice that the assumption that the normal trace be a (vector-valued) bounded measure amounts to saying that the row-wise divergence of the extension of A by $0_d$ away from $Q_\tau$ is still a (vector-valued) bounded measure. Therefore the theorem can be viewed as a statement about such tensors over the whole space ${\mathbb R}^d$, whose support is contained in $Q_\tau$.

Compensated Integrability is a far-reaching generalization of Gagliardo’s Inequality: Given d functions $f_j({\hat{x}_j})$, where the j-th one does not depend upon the j-th coordinate, the product

$$\begin{aligned} f(x)=\prod _{j=0}^nf_j({\hat{x}_j}) \end{aligned}$$

obeys the estimate

$$\begin{aligned} \Vert f\Vert _{L^1({\mathbb R}^d)}\leqq \prod _{j=0}^n\Vert f_j\Vert _{L^n({\mathbb R}^n)}. \end{aligned}$$

Since Gagliardo’s Inequality is at stake in the proof of the embedding $W^{1,1}\subset L^{\frac{d}{d-1}}$, it is not surprising that (15), applied to the tensor $A(x)=u(x)I_n$ gives

$$\begin{aligned} \Vert u\Vert _{L^{\frac{d}{d-1}}({\mathbb R}^d)}\leqq c_d\Vert \nabla u\Vert _{L^1({\mathbb R}^d)}, \end{aligned}$$

at least for compactly supported non-negative functions.

Compensated Integrability is also related to several topics in geometry, for instance the Isoperimetric inequality [14, 15] and Minkowski’s problem [15].

One important application of Compensated Integrability concerns gas dynamics, where it yields Strichartz-like estimates in inviscid models (Euler equations for a compressible flow) and kinetic models (Boltzmann equations), see [14]. Recently, it was used to estimate the number and the strength of collisions in particle dynamics, see [17]. It is therefore a versatile tool, which applies to micro-, meso- and macroscopic descriptions of gases.

2 Dispersive Estimate; The Case $(p,p^*)$

To begin with, we recall that the Burgers equation enjoys an exceptional one-parameter transformation group, a fact already noted in [18] : Let u be an entropy solution of the Cauchy problem for (6) and $\lambda$ be a positive constant. Then the function

$$\begin{aligned} v(t,y)=\frac{1}{\lambda }\,u( t,\lambda y_1,\ldots ,\lambda ^ny_n) \end{aligned}$$

is an entropy solution associated with the initial datum

$$\begin{aligned} v_0(y)=\frac{1}{\lambda }\,u_0(\lambda y_1,\ldots ,\lambda ^ny_n). \end{aligned}$$

The following identities will be used below:

$$\begin{aligned} \int _0^\tau \,\mathrm{d}t\int _{{\mathbb R}^n}v(t,y)^q\mathrm{d}y= & {} \lambda ^{-q-\frac{d(d-1)}{2}}\int _0^\tau \,\mathrm{d}t\int _{{\mathbb R}^n}u(t,y)^q\mathrm{d}y, \end{aligned}$$

(16)

$$\begin{aligned} \int _{{\mathbb R}^n}v_0(y)^q\mathrm{d}y= & {} \lambda ^{-q-\frac{d(d-1)}{2}}\int _{{\mathbb R}^n}u_0(y)^q\mathrm{d}y. \end{aligned}$$

(17)

Let $u_0^\pm$ be the positive and negative parts of the initial datum: $u_0^-\leqq u_0\leqq u_0^+$ with $u_0(x)\in \{u_0^-(x),u_0^+(x)\}$ everywhere. Denote $u_\pm$ the entropy solutions associated with the data $u_0^{\pm }$. By the maximum principle, we have $u_-\leqq u\leqq u_+$ everywhere. Because of $\Vert u(t)\Vert _q\leqq \Vert u_-(t)\Vert _q+\Vert u_+(t)\Vert _q$ and $\Vert u_0\Vert _p=(\Vert u_0^-\Vert _p^p+\Vert u_0^+\Vert _p^p)^{1/p}$, it suffices to proves the estimate for $u_\pm$, that is for initial data that are signed. Moreover since $v(t,y)=-u(t,-y_1,y_2,\ldots ,(-1)^ny_n)$ is the entropy solution associated with $v_0(y)=-u_0(-y_1,y_2,\ldots ,(-1)^ny_n)$, it suffices to treat the case of a non-negative initial data.

We therefore suppose from now on that $u_0\in L^1\cap L^\infty ({\mathbb R}^n)$ and $u_0\geqq 0$, so that $u\geqq 0$ over ${\mathbb R}_+\times {\mathbb R}^n$. We wish to estimate $\Vert u(t)\Vert _q$ in terms of $\Vert u_0\Vert _p$ when $q=p^*=d(1+\frac{p}{n})$. We point out that $p^*>p$.

2.1 A Strichartz-Like Inequality

If $a\in {\mathbb R}$, we define a symmetric matrix

$$\begin{aligned} M(a)=\left( \frac{a^{i+j+p}}{i+j+p}\right) _{0\leqq i,j,\leqq n}. \end{aligned}$$

Remarking that

$$\begin{aligned} M(a)=\int _0^aV(s)\otimes V(s)\,s^{p-1}\,\mathrm{d}s,\qquad V(s)=\begin{pmatrix} 1 \\ \vdots \\ s^n \end{pmatrix}, \end{aligned}$$

we obtain that M(a) is positive definite whenever $a>0$. Obviously,

$$\begin{aligned} \det M(a)=H_{d,p}\,a^{d(p+d-1)}=H_{d,p}\,a^{np^*}, \end{aligned}$$

where the constant

$$\begin{aligned} H_{d,p}=\left\| \frac{1}{i+j+p}\right\| _{0\leqq i,j,\leqq n}>0 \end{aligned}$$

is a Hilbert-like determinant.

Let us form the symmetric tensor

$$\begin{aligned} T(t,y)=M(u(t,y)), \end{aligned}$$

with positive semi-definite values. Its row of index i is given by $(\eta _{i+p}(u), q_{i+p}(u))$, an entropy–flux pair where $\eta _r(s)=\frac{|s|^r}{r}\,$ is convex. In the special case where $p=1$ and $i=0$, it is divergence-free because of (6) itself. Otherwise, it is not divergence-free in general, although it is so whenever u is a classical solution. However the entropy inequality tells us that the opposite of its divergence is a non-negative, hence bounded, measure

$$\begin{aligned} \mu _r=-\mathrm{div}_{t,y}(\eta _r(u), q_r(u))\geqq 0. \end{aligned}$$

The total mass of $\mu _r$ over a slab $(0,\tau )\times {\mathbb R}^n$ is given by

$$\begin{aligned} \Vert \mu _r\Vert =\int _{{\mathbb R}^n}\eta _r(u_0(y))\,\mathrm{d}y-\int _{{\mathbb R}^n}\eta _r(u(\tau ,y))\,\mathrm{d}y\leqq \int _{{\mathbb R}^n}\frac{u_0(y)^r}{r}\,\mathrm{d}y. \end{aligned}$$

Since the latter bound does not depend upon $\tau$, $\mu _r$ is actually a bounded measure over ${\mathbb R}_+\times {\mathbb R}^n$.

We conclude that the row-wise divergence of T is a (vector-valued) bounded measure, whose total mass is estimated by

$$\begin{aligned} \Vert \mathrm{Div}\,T\Vert _{\mathcal{M}(Q_\tau )}\leqq \sum _{j=0}^n\int _{{\mathbb R}^n}\frac{u_0(y)^{j+p}}{j+p}\,\mathrm{d}y. \end{aligned}$$

(18)

We may therefore apply Compensated Integrability to the tensor T. Theorem 1.3 tells us that $(\det T)^{\frac{1}{d-1}}$ is integrable over $Q_\tau$. Because of

$$\begin{aligned} \Vert T_{0\bullet }(t,\cdot )\Vert _1=\sum _{j=0}^n\int _{{\mathbb R}^n}\frac{u(t,y)^{j+p}}{j+p}\,\mathrm{d}y\leqq \sum _{j=0}^n\int _{{\mathbb R}^n}\frac{u_0(y)^{j+p}}{j+p}\,\mathrm{d}y, \end{aligned}$$

together with (18), Inequality (15) gives

$$\begin{aligned} \int _0^\tau \,\mathrm{d}t\int _{{\mathbb R}^n}u^{p^*}\mathrm{d}y\leqq c_{d,p}\left( \sum _{j=0}^n\int _{{\mathbb R}^n}u_0(y)^{j+p}\,\mathrm{d}y\right) ^{\frac{d}{d-1}}. \end{aligned}$$

(19)

Again, the right-hand side does not depend upon $\tau$, thus the inequality above is true also for $\tau =+\infty$.

The only flaw in the estimate (19) is its lack of homogeneity. To recover a well-balanced inequality, we use the scaling, in particular the formulæ  (16) and (17). Applying (19) to the pair $(v,v_0)$ instead, we get a parametrized inequality

$$\begin{aligned} \left( \int _0^\infty \,\mathrm{d}t\int _{{\mathbb R}^n}u^{p^*}\mathrm{d}y\right) ^{\frac{d-1}{d}}\leqq c_d\lambda ^{\frac{d-1}{2}}\sum _{j=0}^n\lambda ^{-j}\int _{{\mathbb R}^n}u_0(y)^{j+p}\,\mathrm{d}y, \end{aligned}$$

where $\lambda >0$ is up to our choice. In order to minimize the right-hand side, we select the value

$$\begin{aligned} \lambda =\left( \int _{{\mathbb R}^n}u_0(y)^{n+p}\mathrm{d}y/\int _{{\mathbb R}^n}u_0(y)^p\mathrm{d}y\right) ^{\frac{1}{n}}. \end{aligned}$$

The extreme terms, for $j=0$ or n, contribute on an equal footing with

$$\begin{aligned} \left( \int _{{\mathbb R}^n}u_0(y)^{n+p}\mathrm{d}y\right) ^{\frac{1}{2}}\left( \int _{{\mathbb R}^n}u_0(y)^p\mathrm{d}y\right) ^{\frac{1}{2}}. \end{aligned}$$

The other ones, which are

$$\begin{aligned} \left( \int _{{\mathbb R}^n}u_0(y)^{n+p}\mathrm{d}y/\int _{{\mathbb R}^n}u_0(y)^p\mathrm{d}y\right) ^{\frac{1}{2}-\frac{j}{d-1}}\int _{{\mathbb R}^n}u_0^{j+p}\mathrm{d}y, \end{aligned}$$

are bounded by the same quantity, because of Hölder inequality. We end therefore with the fundamental estimate of Strichartz style:

$$\begin{aligned} \left( \int _0^\infty \!\int _{{\mathbb R}^n}u^{p^*}\mathrm{d}y\,\mathrm{d}t\right) ^{\frac{d-1}{d}}\leqq c_d\left( \int _{{\mathbb R}^n}u_0(y)^{p+n}\mathrm{d}y\right) ^{\frac{1}{2}}\left( \int _{{\mathbb R}^n}u_0(y)^p\mathrm{d}y\right) ^{\frac{1}{2}}. \end{aligned}$$

(20)

2.2 Proof of estimate (10)

We shall contemplate (20) as a differential inequality. To this end, we define

$$\begin{aligned} X(t):=\int _{{\mathbb R}^n}u^{p^*}\mathrm{d}y=\Vert u(t)\Vert _{p^*}^{p^*}. \end{aligned}$$

Noticing that $p+n$ is less than $p^*$, and using Hölder inequality, we get

$$\begin{aligned} \int _{{\mathbb R}^n}|w|^{p+n}\mathrm{d}y\leqq \left( \int _{{\mathbb R}^n}|w|^p\mathrm{d}y\right) ^a\left( \int _{{\mathbb R}^n}|w|^{p^*}\mathrm{d}y\right) ^b \end{aligned}$$

for

$$\begin{aligned} a=\frac{p+n}{p+dn}\,,\qquad b=\frac{n^2}{p+dn}\,. \end{aligned}$$

The inequality (20) therefore implies that

$$\begin{aligned} \left( \int _0^\infty X(t)\,\,\mathrm{d}t\right) ^{\frac{2n}{d}}\leqq c_d \Vert u_0\Vert _p^{p(1+a)} X(0)^b. \end{aligned}$$

Considering the solution $w(t,y)=u(t+\tau ,y)$, whose initial datum is $u(\tau ,\cdot )$, we also have

$$\begin{aligned} \left( \int _\tau ^\infty X(t)\,\mathrm{d}t\right) ^{\frac{2n}{db}}\leqq c_d \Vert u(\tau )\Vert _p^{p\frac{1+a}{b}} X(\tau )\leqq c_d\Vert u_0\Vert _p^{p\frac{1+a}{b}} X(\tau ). \end{aligned}$$

(21)

Let us denote

$$\begin{aligned} Y(\tau ):=\int _\tau ^\infty X(t)\,\mathrm{d}t. \end{aligned}$$

We recast (21) as

$$\begin{aligned} Y^\rho +c_d\Vert u_0\Vert _p^\mu Y'\leqq 0,\qquad \rho :=\frac{2n}{db}\,\quad \mu :=p\,\frac{1+a}{b}\,. \end{aligned}$$

Remark that $\rho =2\frac{p+dn}{dn}>2$. Multiplying by $Y^{-\rho }$ and integrating, we infer that

$$\begin{aligned} t+c_d\Vert u_0\Vert _p^\mu Y(0)^{1-\rho }\leqq c_d\Vert u_0\Vert _p^\mu Y(t)^{1-\rho }. \end{aligned}$$

This provides a first decay estimate

$$\begin{aligned} Y(t)\leqq c_d\Vert u_0\Vert _p^{\frac{\mu }{\rho -1}}t^{-\frac{1}{\rho -1}}. \end{aligned}$$

Remarking that $t\mapsto X(t)$ is a non-increasing function, so that

$$\begin{aligned} \frac{\tau }{2}\,X(\tau )\leqq Y\left( \frac{\tau }{2}\right) , \end{aligned}$$

we deduce the ultimate decay result

$$\begin{aligned} X(t)\leqq c_d\Vert u_0\Vert _p^{\frac{\mu }{\rho -1}}t^{-\frac{\rho }{\rho -1}}. \end{aligned}$$

Restated in terms of a Lebesgue norm of u(t), this says that

$$\begin{aligned} \Vert u(t)\Vert _{p^*}\leqq c_d\Vert u_0\Vert _p^{\alpha (p,{p^*})}\, t^{-\beta (p,{p^*})}, \end{aligned}$$

(22)

where $\alpha (p,q)$ and $\beta (p,q)$ are given in (8) and (9). This is precisely the special case of (10) under consideration here.

3 General Pairs (p, q) Where $p<q\leqq \infty$

Because of (11) and of the Hölder inequality, it will be enough to prove (10) when $q=+\infty$. Once again, it is sufficient to treat the case of non-negative data / solutions.

3.1 An Estimate for $(u-\ell )_+$

Let $\ell >0$ be a given number. We denote $w_\ell$ the entropy solution of (6) associated with the initial datum $(u_0-\ell )_++\ell =\max \{u_0,\ell \}$. The function $z_\ell :=w_\ell -\ell$ is an entropy solution of a modified conservation law

$$\begin{aligned} \partial _tz_\ell +\sum _{k=1}^n\partial _k\frac{(z_\ell +\ell )^{k+1}}{k+1}=0. \end{aligned}$$

This is not exactly the Burgers equation for $z_\ell$. However the $(n+2)$-uplet

$$\begin{aligned} \left( 1,X+\ell ,\ldots ,\ldots ,\frac{(X+\ell )^{n+1}}{n+1}\right) \end{aligned}$$

is a basis of ${\mathbb R}_{n+1}[X]$. We pass from this basis to $(1,X,\ldots ,\frac{X^{n+1}}{n+1})$ by a triangular matrix with unit diagonal. There exists therefore a change of coordinates

$$\begin{aligned} \left( {\begin{array}{c}t\\ y'\end{array}}\right) =P\left( {\begin{array}{c}t\\ y\end{array}}\right) =\begin{pmatrix} 1 &{} 0 \\ \vdots &{} Q \end{pmatrix}\left( {\begin{array}{c}t\\ y\end{array}}\right) , \end{aligned}$$

where Q is a unitriangular matrix, such that $z_\ell$ obeys the Burgers equation in the new coordinates:

$$\begin{aligned} \frac{\partial z_\ell }{\partial t}+\sum _{k=1}^n\frac{\partial }{\partial y_k'}\,\frac{z_\ell ^{k+1}}{k+1}=0. \end{aligned}$$

We may therefore apply (22) to $z_\ell$ :

$$\begin{aligned} \left( \int _{{\mathbb R}^n}z_\ell (t,y')^{p^*}\mathrm{d}y'\right) ^{\frac{1}{p^*}}\leqq c_d\left( \int _{{\mathbb R}^n}z_\ell (0,y')^p\,\mathrm{d}y'\right) ^{\frac{\alpha (p,{p^*})}{p}}\, t^{-\beta (p,{p^*})}. \end{aligned}$$

Remarking that the time variable is unchanged, and the Jacobian of the change of variable $y\mapsto y'$ at fixed time equals one, we actually have

$$\begin{aligned} \Vert z_\ell (t)\Vert _{p^*}\leqq c_d\Vert z_\ell (0)\Vert _p^{\alpha (p,{p^*})}\, t^{-\beta (p,{p^*})}. \end{aligned}$$

Finally, the maximum principle tells us that $u\leqq w_\ell$. The inequality above is therefore an estimate of the positive part of $u-\ell$ :

$$\begin{aligned} \Vert (u-\ell )_+(t)\Vert _{p^*}\leqq c_d\Vert (u_0-\ell )_+\Vert _p^{\alpha (p,{p^*})}\, t^{-\beta (p,{p^*})}. \end{aligned}$$

(23)

3.2 An iteration à la De Giorgi

We now prove the $L^p$–$L^\infty$ estimate, in the special case where $\Vert u_0\Vert _p=1$. We recall that $u_0$ is non-negative.

For the moment, we fix an arbitrary constant $B>0$, which we will choose large enough in the end of the proof. Then we define the following sequences for $k\in {\mathbb N}$ :

$$\begin{aligned} t_k=1-2^{-k},\quad \ell _k=Bt_k,\quad w_k=(u-\ell _k)_+,\quad a_k=\Vert w_k(t_k)\Vert _p. \end{aligned}$$

Remark that the sequences $\ell _k$ and $w_k$ are increasing and decreasing, respectively. Since $t_0=0$, we have $a_0=\Vert u_0\Vert _p=1$.

For each value of k, we apply (23) in order to estimate $\Vert w_{k+1}(t_{k+1})\Vert _{p^*}$ in terms of $\Vert w_{k+1}(t_k)\Vert _p$. For the sake of simplicity, we write $\alpha ,\beta$ for $\alpha (p,p^*)$ and $\beta (p,p^*)$. We get

$$\begin{aligned} \Vert w_{k+1}(t_{k+1})\Vert _{p^*}\leqq & {} c_{d,p}\Vert w_{k+1}(t_k)\Vert _p^{\alpha }\, (t_{k+1}-t_k)^{-\beta }\\= & {} c_{d,p}2^{\beta (k+1)}\Vert w_{k+1}(t_k)\Vert _p^{\alpha }\leqq c_{d,p}2^{\beta (k+1)}a_k^{\alpha }. \end{aligned}$$

With Hölder inequality, we also have that

$$\begin{aligned} a_{k+1}=\Vert w_{k+1}(t_{k+1})\Vert _{p}\leqq \Vert w_{k+1}(t_{k+1})\left\| _{p^*}\Vert \mathbf{1}_{\{y\,:\,w_{k+1}(t_{k+1},y)>0\}}\right\| _r \end{aligned}$$

where

$$\begin{aligned} \frac{1}{p}=\frac{1}{p^*}+\frac{1}{r}\,. \end{aligned}$$

Remark that $r>1$. Combining both inequalities, we obtain

$$\begin{aligned} a_{k+1}\leqq c_{d,p}2^{\beta (k+1)}a_k^\alpha \left| \{y\,:\,w_{k+1}(t_{k+1},y)>0\}\right| ^{\frac{1}{r}}. \end{aligned}$$

Observing that $w_{k+1}>0$ implies $w_k>B2^{-k-1}$, we infer

$$\begin{aligned} a_{k+1}\leqq c_{d,p}2^{\beta (k+1)}a_k^\alpha \left| \{y\,:\,w_k(t_{k+1},y)>B2^{-k-1}\}\right| ^{\frac{1}{r}}. \end{aligned}$$

We now use the Chebychev Inequality

$$\begin{aligned} \left| \{y\,:\,w_k(t_{k+1},y)>B2^{-k-1}\}\right| ^{\frac{1}{p}}\leqq B^{-1}2^{k+1}\Vert w_k(t_{k+1})\Vert _p\leqq B^{-1}2^{k+1}\Vert w_k(t_{k})\Vert _p \end{aligned}$$

to deduce that

$$\begin{aligned} a_{k+1}\leqq c_{d,p}B^{-\frac{p}{r}}2^{(\beta +\frac{p}{r})(k+1)}a_k^{\alpha +\frac{p}{r}}=C2^{Ck}a_k^{1+\delta }B^{-\gamma }. \end{aligned}$$

We have set $\delta =\alpha -\frac{p}{p^*}$ and $\gamma =\frac{p}{r}$ .

By a direct computation, we verify that $\delta$ is positive:

$$\begin{aligned} \alpha -\frac{p}{p^*}=\frac{p^*h(p^*)-ph(p)}{p^*h(p)}=2\,\frac{p^*-p}{p^*h(p)}>0. \end{aligned}$$

The sequence $b_k:=B^{-\frac{\gamma }{\delta }}a_k$, which starts with $b_0=B^{-\frac{\gamma }{\delta }}$, therefore satisfies a recurrence relation

$$\begin{aligned} b_{k+1}\leqq C 2^{Ck}b_k^{1+\delta }. \end{aligned}$$

It is known that if $b_0$ is small enough, that is if B is large enough, then $b_k\rightarrow 0+$ as $k\rightarrow +\infty$. Equivalently, $a_k\rightarrow 0+$.

We have therefore found a constant $B>0$ such that

$$\begin{aligned} \Vert (u-\ell _k)_+(1)\Vert _p\leqq \Vert (u-\ell _k)_+(t_k)\Vert _p=a_k\rightarrow 0+. \end{aligned}$$

Since $\ell _k\rightarrow B$, this means exactly that $\Vert u(1)\Vert _\infty \leqq B$.

3.3 End of the proof of dispersive estimates

Let $u_0\in L^1\cap L^\infty ({\mathbb R}^n)$ be non-negative. For two positive parameters $\lambda ,\mu$, the entropy solution associated with the datum

$$\begin{aligned} v_0(y)=\frac{1}{\lambda }u_0(\mu \lambda y_1,\ldots ,\mu \lambda ^ny_n) \end{aligned}$$

is the function

$$\begin{aligned} v(t,y)=\frac{1}{\lambda }u(\mu t,\mu \lambda y_1,\ldots ,\mu \lambda ^ny_n). \end{aligned}$$

If

$$\begin{aligned} \lambda ^{p+\frac{n(n+1)}{2}}\mu ^n=\int _{{\mathbb R}^n}u_0(y)^p\,\mathrm{d}y, \end{aligned}$$

(24)

then $\Vert v_0\Vert _p=1$ and we may apply the previous section to conclude: $\Vert v(1)\Vert _\infty \leqq B$. In terms of u, this writes

$$\begin{aligned} \Vert u(\mu )\Vert _\infty \leqq B\lambda . \end{aligned}$$

Calculating $\lambda$ from (24) gives

$$\begin{aligned} \Vert u(\mu )\Vert _\infty \leqq B\left( \mu ^{-n}\Vert u_0\Vert _p^p\right) ^{\frac{2}{n^2+n+2p}}, \end{aligned}$$

which is nothing but the dispersive estimate (10) for $q=+\infty$.

There remains to pass from $q=+\infty$ to every $q\in [p,+\infty ]$. We do that by applying the Hölder inequality. Writing

$$\begin{aligned} \frac{1}{q}=\frac{1-\theta }{p}+\frac{\theta }{\infty }\,, \end{aligned}$$

we have

$$\begin{aligned} \Vert u(t)\Vert _q\leqq \Vert u(t)\Vert _p^{1-\theta }\Vert u(t)\Vert _\infty ^\theta \leqq \Vert u_0\Vert _p^{1-\theta }\left( Bt^{-\beta (p,\infty )}\Vert u_0\Vert _p^{\alpha (p,\infty )}\right) ^\theta . \end{aligned}$$

We conclude by using the relations (11).

4 The $L^p$-Semi-group for Finite Exponents

We now prove Theorem 1.2. We start with a remark about $L^p$-spaces.

Lemma 4.1

Let $a\in L^p({\mathbb R}^n)$ be given. There exists a sequence $(b_m)_{m\geqq 0}$ in $(L^p\cap L^\infty )({\mathbb R}^n)$, converging towards a in $L^p({\mathbb R}^n)$, such that $b_m-a\in L^1({\mathbb R}^n)$ and

$$\begin{aligned} \lim _{m\rightarrow +\infty }\Vert b_m-a\Vert _1=0. \end{aligned}$$

Proof

Recall that

$$\begin{aligned} L^p({\mathbb R}^n)=(L^1\cap L^p)({\mathbb R}^n)+(L^p\cap L^\infty )({\mathbb R}^n). \end{aligned}$$

Decomposing our function as $a=a_1+a_\infty$ where

$$\begin{aligned} a_1\in (L^1\cap L^p)({\mathbb R}^n),\qquad a_\infty \in (L^p\cap L^\infty )({\mathbb R}^n), \end{aligned}$$

we may form the sequence of bounded functions $b_m:=a_\infty +\pi _m\circ a_1$, where $\pi _m$ is the projection from ${\mathbb R}$ onto the interval $[-m,m]$. Because of

$$\begin{aligned} \Vert b_m\Vert _p\leqq \Vert a_\infty \Vert _p+\Vert \pi _m\circ a_1\Vert _p\leqq \Vert a_\infty \Vert _p+\Vert a_1\Vert _p, \end{aligned}$$

this sequence is bounded in $L^p({\mathbb R}^n)$. In addition, $b_m-a=\pi _m\circ a_1-a_1\in L^1\cap L^p({\mathbb R}^n)$, and

$$\begin{aligned} \Vert b_m-a\Vert _1= & {} \Vert \pi _m\circ a_1-a_1\Vert _1{\mathop {\longrightarrow }\limits ^{m\rightarrow +\infty }}0,\\ \Vert b_m-a\Vert _p= & {} \Vert \pi _m\circ a_1-a_1\Vert _p{\mathop {\longrightarrow }\limits ^{m\rightarrow +\infty }}0. \end{aligned}$$

$\square$

Let $u_0\in L^p({\mathbb R}^n)$ be given. In order to define $S_tu_0$, we consider a sequence $b_m$ that approximates $u_0$ in the sense of Lemma 4.1. Remark that we do not care about the construction of $b_m$, as we only use the properties stated in the lemma.

To begin with, $u_m(t):=S_tb_m$ is well-defined and belongs to $L^\infty ({\mathbb R}^n)$. Because of (10), we have

$$\begin{aligned} \Vert u_m(t)\Vert _q\leqq c_{d,p,q}\Vert b_m\Vert _p^{\alpha (p,q)}\,t^{-\beta (p,q)}\leqq C_{p,q}(u_0)\,t^{-\beta (p,q)}. \end{aligned}$$

(25)

The sequence $(u_m)_{m>0}$ is thus bounded in $C_0(\tau ,\infty ;L^q({\mathbb R}^n))$ for every $q\in [p,\infty )$ and every $\tau >0$.

The contraction property gives us

$$\begin{aligned} \Vert u_m(t)-u_\ell (t)\Vert _1\leqq \Vert b_m-b_\ell \Vert _1{\mathop {\longrightarrow }\limits ^{m,\ell \rightarrow +\infty }}0. \end{aligned}$$

Let r, q be exponents satisfying $p\leqq r<q<\infty$. By Hölder inequality, we have

$$\begin{aligned} \Vert u_m(t)-u_\ell (t)\Vert _r\leqq \Vert u_m(t)-u_\ell (t)\Vert _1^\theta (\Vert u_m(t)\Vert _q+\Vert u_\ell (t)\Vert _q)^{1-\theta }, \end{aligned}$$

where $\theta \in (0,1]$. With (25), we infer that

$$\begin{aligned} \Vert u_m(t)-u_\ell (t)\Vert _r{\mathop {\longrightarrow }\limits ^{m,\ell \rightarrow +\infty }}0, \end{aligned}$$

uniformly over $(\tau ,\infty )$.

We have thus proved that $(u_m)_{m>0}$ is a Cauchy sequence in $C_0(\tau ,\infty ;L^r({\mathbb R}^n))$, hence is convergent in this space. If $b_m'$ is another approximating sequence for $u_0$, and $u_m'$ the corresponding solution of the Cauchy problem, we may form an approximating sequence $c_m$ in the sense of Lemma 4.1, by alterning $b_1,b_1',b_2,b_2',\ldots$. The sequence $u_1,u_1',u_2,u_2',\ldots$ will be convergent in the sense above. This shows that the limit of $u_m$ does not depend upon the precise sequence $(b_m)_{m>0}$ chosen above. Thus we may set

$$\begin{aligned} S_tu_0:=\lim _{m\rightarrow +\infty }u_m(t), \end{aligned}$$

which defines a

$$\begin{aligned} u\in C_b({\mathbb R}_+;L^p({\mathbb R}^n))\bigcap \bigcap _{p<r<\infty }C_0(0,+\infty ;L^r({\mathbb R}^n)). \end{aligned}$$

There remains to prove that u is an entropy solution of (6). For this, we use the fact that $u_m$ is itself an entropy solution, and the convergence stated above ensures that every monomial $(u_m)^j$ in the flux $f(u_m)$, converges towards $u^j$ in $L^1_\mathrm{loc}$.

The fact that $u(0)=u_0$ follows from $u_m(0)=b_m$, the $L^p$-convergence $b_m\rightarrow u_0$, and the uniform convergence $u_m(t)\rightarrow u(t)$ in $L^p({\mathbb R}^n)$.

5 Other “Monomial” Scalar Conservation Laws

We consider in this section conservation laws whose fluxes are monomial. Denoting $m_k(s)=\frac{s^{k+1}}{k+1}$ , they bear the form

$$\begin{aligned} \partial _tu+\partial _1m_{k_1}(u)+\cdots +\partial _nm_{k_n}(u)=0, \end{aligned}$$

(26)

where $0<k_1<\cdots <k_n$ are integers. The time derivative may be written as well $\partial _tm_{k_0}(u)$ with $k_0=0$.

As before, we may restrict to non-negative initial datum $u_0$ that belong to $L^1\cap L^\infty ({\mathbb R}^n)$. Given an exponent $p\geqq 1$, our symmetric tensor is now $T(t,y)=M(u(t,y))$, where

$$\begin{aligned} M(a):=\left( m_{p+k_i+k_j-1}(a)\right) _{0\leqq i,j\leqq n}. \end{aligned}$$

Notice that M(a) is symmetric, and its upper-left entry is $\frac{a^p}{p}$ . Because of

$$\begin{aligned} M(a)=\int _0^as^{p-1}V(s)\otimes V(s)\,\mathrm{d}s,\qquad V(s):=\begin{pmatrix} s^{k_0} \\ \vdots \\ s^{k_n} \end{pmatrix}, \end{aligned}$$

is positive definite whenever $a>0$. We have

$$\begin{aligned} \det M(a)=\Delta (p,\mathbf {k})a^N,\qquad N=dp+2K,\quad K:=\sum _0^nk_i. \end{aligned}$$

As above, the lines of T are made of entropy–entropy flux pairs of the equation (26). Its row-wise divergence is therefore a vector-valued bounded measure. Compensated integrability yields again an inequality

$$\begin{aligned} \left( \int _0^\infty \!\,\mathrm{d}t\int _{{\mathbb R}^n}u(t,y)^Q\mathrm{d}y\right) ^{\frac{n}{d}}\leqq c_{d,p,\mathbf {k}}\sum _{j=0}^n\int _{{\mathbb R}^n}u_0(y)^{p+k_j}\mathrm{d}y,\qquad Q:=\frac{N}{n}\,. \end{aligned}$$

The conservation law is invariant under the scaling

$$\begin{aligned} u\longmapsto v(t,y):=\frac{1}{\lambda }u(t,\lambda ^{k_1}y_1,\ldots ,\lambda ^{k_n}y_n). \end{aligned}$$

Applying the estimate above to v, we obtain a parametrized inequality :

$$\begin{aligned} \left( \int _0^\infty \!\,\mathrm{d}t\int _{{\mathbb R}^n}u(t,y)^Q\mathrm{d}y\right) ^{\frac{n}{d}}\leqq c_{d,p,\mathbf {k}}\lambda ^{\frac{K}{d}}\sum _{j=0}^n\lambda ^{-k_j}\int _{{\mathbb R}^n}u_0(y)^{p+k_j}\mathrm{d}y. \end{aligned}$$

We now choose

$$\begin{aligned} \lambda =\left( \int _{{\mathbb R}^n}u_0(y)^{p+k_n}\mathrm{d}y \, / \, \int _{{\mathbb R}^n}u_0(y)^{p}\mathrm{d}y\right) ^{\frac{1}{k_n}} \end{aligned}$$

and obtain a Strichartz-like estimate:

$$\begin{aligned} \left( \int _0^\infty \!\,\mathrm{d}t\int _{{\mathbb R}^n}u(t,y)^Q\mathrm{d}y\right) ^{\frac{n}{d}}\leqq c_{d,p,\mathbf {k}}\left( \int _{{\mathbb R}^n}u_0(y)^{p+k_n}\mathrm{d}y \right) ^\theta \left( \int _{{\mathbb R}^n}u_0(y)^{p}\mathrm{d}y\right) ^{1-\theta }, \end{aligned}$$

where

$$\begin{aligned} \theta :=\frac{K}{dk_n}\in (0,1). \end{aligned}$$

Applying this calculation to the interval $(\tau ,+\infty )$, and using the decay of the $L^p$-norm, we infer that

$$\begin{aligned} \left( \int _\tau ^\infty \!\,\mathrm{d}t\int _{{\mathbb R}^n}u(t,y)^Q\mathrm{d}y\right) ^{\frac{n}{d}}\leqq c_{d,p,\mathbf {k}}\left( \int _{{\mathbb R}^n}u(\tau ,y)^{p+k_n}\mathrm{d}y \right) ^\theta \left( \int _{{\mathbb R}^n}u_0(y)^{p}\mathrm{d}y\right) ^{1-\theta }. \end{aligned}$$

(27)

We may now continue the analysis with a Gronwall argument, provided $p+k_n\in (p,Q]$. We leave the interested reader to check the details. Our first dispersion estimate is

$$\begin{aligned} \Vert u(t)\Vert _Q\leqq c_{d,p}t^{-\beta (p)}\Vert u_0\Vert _p^{\alpha (p)}, \end{aligned}$$

(28)

whenever $p\geqq nk_n-2K$ (remark that for the Burgers equation, this restriction is harmless).

At this stage, it seems that we miss an argument in order to carry out the DeGiorgi technique, because the conservation law satisfied by $u-\ell$ will be a different one. Whether it can be done here and for general conservation laws is left for a future work. What we can do at least is to combine the estimates (28) in order to cover pairs (p, q) of finite exponents. For instance, starting from a pair (p, Q) as above and chosing $p_1=Q$, we have a corresponding $Q_1$ such that (28) applies with $(p_1,Q_1)$ instead of (p, Q). We infer

$$\begin{aligned} \Vert u(t)\Vert _{Q_1}\leqq c_{d,Q}(t/2)^{-\beta (Q)}\Vert u(t/2)\Vert _Q^{\alpha (Q)}\leqq c_{d,p}t^{-\beta (Q)-\alpha (Q)\beta (p)}\Vert u_0\Vert _p^{\alpha (p)\alpha (Q)}. \end{aligned}$$

Because the iteration $p\rightarrow Q$ defines a sequence which tends to $+\infty$, and using the Hölder inequality to fill the gaps, we deduce the dispersion inequalities for the monomial conservation law as follows:

Theorem 5.1

For the scalar conservation law (26) with monomial fluxes, there exist finite constants $c_{d,p,q}$ such that whenever $p\geqq nk_n-2K$, $q\in [p,\infty )$ and $u_0\in L^p\cap L^\infty ({\mathbb R}^n)$, we have

$$\begin{aligned} \Vert u(t)\Vert _q\leqq c_{d,p,q}t^{-\beta (p,q)}\Vert u_0\Vert _p^{\alpha (p,q)}. \end{aligned}$$

The exponents are given by the formula

$$\begin{aligned} \alpha (p,q)=\frac{h(q)}{h(p)}\,,\qquad h(p):=1+\frac{K}{p}\qquad \hbox {and}\qquad \beta (p,q)=n\left( \frac{\alpha (p,q)}{p}-\frac{1}{q}\right) . \end{aligned}$$

As in the case of the Burgers equation, we can use these estimates in order to define the semi-group over $L^p$-spaces.

Corollary 5.1

The semi-group $(S_t)_{t\geqq 0}$ for equation (26) extends by continuity as a continuous semi-group over $L^p({\mathbb R}^n)$ for every $p\in [1,+\infty )$ such that $p\geqq nk_n-2K$. It maps $L^p({\mathbb R}^n)$ into $L^q({\mathbb R}^n)$ for every $q\in [p,\infty )$. If $u_0\in L^p({\mathbb R}^n)$, then the function $u(t,y):=(S_tu_0)(y)$ is an entropy solution with initial datum $u_0$.

6 Compensated Integrability for General Fluxes f

We consider now a multi-dimensional conservation law of the most general form (1). Following the ideas developped in the Burgers and monomial cases, we begin by considering a signed, bounded initial datum: $u_0\in L^1\cap L^\infty ({\mathbb R}^n)$, $u_0\geqq 0$. If $a\in {\mathbb R}_+$, we define a symmetric matrix

$$\begin{aligned} M_g(a)=\int _0^ag(s)Z'(s)\otimes Z'(s)\,\mathrm{d}s, \end{aligned}$$

where $Z(s)=(f_0(s)=s,f_1(s),\ldots ,f_n(s))$ and g is some positive function. This matrix is positive definite under the non-degeneracy condition that Z([0, a]) is not contained in an affine hyperplane. We denote

$$\begin{aligned} \Delta _g(a):=(\det M_g(a))^{\frac{1}{n}}\geqq 0. \end{aligned}$$

Let us define $T(t,y):=M_g(u(t,y))$. Because of $u\in L^\infty ({\mathbb R}_+;L^1\cap L^\infty ({\mathbb R}^n))$, the tensor T is integrable over $(0,\tau )\times {\mathbb R}^n$. Each row of T is made of entropy–entropy flux pairs $(F_i,Q_i)$. Since $F_i$ might not be convex, we cannot estimate the measure $\mu _i=-\partial _tF_i(u)-\mathrm{div}_yQ_i(u)$ directly by the integral of $F_i(u_0)$. To overcome this difficulty, we define a convex function $\phi _g$ over ${\mathbb R}_+$ by

$$\begin{aligned} \phi _g(0)=\phi _g'(0)=0,\qquad \phi _g''(s)=|F''(s)|, \end{aligned}$$

where $F=(F_0,\ldots ,F_n)$. Remark that $|F'|\leqq \phi _g'$ and $|F|\leqq \phi _g$. Let $\Phi _g$ be the entropy flux associated with the entropy $\phi _g$. Then the measure $\nu _g:=-\partial _t\phi _g(u)-\mathrm{div}_y\Phi _g(u)$ is non-negative and a bound of its total mass is as usual

$$\begin{aligned} \Vert \nu _g\Vert \leqq \int _{{\mathbb R}^n}\phi _g(u_0(y))\,\mathrm{d}y. \end{aligned}$$

We now use the kinetic formulation of (1), a notion for which we refer to [13], Theorem 3.2.1. Recall the definition of the kinetic function $\chi (\xi ;a)$, whose value is $\mathrm{sgn}\,a$ if $\xi$ lies between 0 and a, and is 0 otherwise. There exists a non-negative bounded measure $m(t,y,\xi )$ such that the function $w(t,y,\xi )=\chi (\xi ;u(t,y))$ satisfies

$$\begin{aligned} \partial _tw+f'(\xi )\cdot \nabla _yw=\frac{\partial }{\partial \xi }\,m,\qquad w(0,y;\xi )=\chi (\xi ;u_0(y)). \end{aligned}$$

If $(\eta ,q)$ is an entropy–entropy flux pair, then the measure $\mu =-\partial _t\eta -\mathrm{div}_yq$ is given by

$$\begin{aligned} \mu =\int _{\mathbb R}\eta ''(\xi )dm(\xi ). \end{aligned}$$

We deduce that the vector-valued measure $\mu =(\mu _0,\ldots ,\mu _n)$ satisfies $|\mu |\leqq \nu _g$. This yields the estimate

$$\begin{aligned} \Vert \mu \Vert \leqq \int _{{\mathbb R}^n}\phi _g(u_0(y))\,\mathrm{d}y. \end{aligned}$$

We may therefore apply the compensated integrability, which gives here that

$$\begin{aligned}&\int _0^\tau \,\mathrm{d}t\int _{{\mathbb R}^n}\Delta _g(u(t,y))\,\mathrm{d}y \\&\quad \leqq c_d\left( \Vert F(u_0)\Vert _1+\Vert F(u(\tau ))\Vert _1+\int _{{\mathbb R}^n}\phi _g(u_0(y))\,\mathrm{d}y\right) ^{1+\frac{1}{n}}. \end{aligned}$$

Because of $|F|\leqq \phi _g$ and $\Vert \phi _g(u(\tau ))\Vert _1\leqq \Vert \phi _g(u_0)\Vert _1$, we end up with an analog of (20):

$$\begin{aligned} \int _0^\infty \,\mathrm{d}t\int _{{\mathbb R}^n}\Delta _g(u(t,y))\mathrm{d}y\leqq c_d\Vert \phi _g(u_0)\Vert _1^{1+\frac{1}{n}}. \end{aligned}$$

(29)

Whether (29) can be used to prove dispersive estimates depends on the amount of nonlinearity of the equation (1). We leave discussion of this question for a future work.