[Paper Review] Multi-dimensional scalar conservation laws with unbounded integrable initial data

Denis Serre|arXiv (Cornell University)|Jul 27, 2018

Navier-Stokes equation solutions6 references3 citations

TL;DR

This paper establishes the existence and uniqueness of entropy solutions for multi-dimensional scalar conservation laws with unbounded integrable initial data by introducing a novel application of compensated integrability to a divergence-free positive symmetric tensor. The key result is a Strichartz-like inequality for quasilinear systems, proven via scaling and optimization over linear transformations, which ensures decay and integrability of solutions even when initial data are not bounded, extending Kruzhkov's theory beyond $ L^\infty $.

ABSTRACT

We discuss the minimal integrability needed for the initial data, in order that the Cauchy problem for a multi-dimensional conservation law admit an entropy solution. In particular we allow unbounded initial data. We investigate also the decay of the solution as time increases, in relation with the nonlinearity. The main ingredient is our recent theory of divergence-free positive symmetric tensor. We apply in particular the so-called compensated integrability to a tensor which generalizes the one that L. Tartar used in one space dimension. It allows us to establish a Strichartz-like inequality, in a quasilinear context. This program is carried out in details for a multi-dimensional version of the Burgers equation.

Motivation & Objective

• To determine the minimal integrability condition on initial data for the existence of entropy solutions in multi-dimensional scalar conservation laws.
• To extend the classical theory of Kruzhkov, which requires $ L^\infty $ data, to the case of unbounded but integrable initial data in $ L^1 $.
• To establish decay estimates for solutions as time increases, particularly in relation to the nonlinearity of the flux function.
• To develop a general framework based on compensated integrability applied to a divergence-free symmetric tensor derived from entropy-entropy flux pairs.
• To prove a Strichartz-type inequality in a quasilinear setting, enabling control of solution regularity and decay.

Proposed method

• The authors introduce a symmetric tensor $ M(a) \in \mathbf{Sym}_d $ constructed from the flux and entropy functions, generalizing Tartar's one-dimensional construction.
• They apply the theory of compensated integrability to this tensor to derive a Strichartz-like inequality for the solution's Hessian-like quantity $ \Delta(u) $.
• A scaling argument is used to optimize the resulting estimates by introducing transformed variables $ v(t,y) = \frac{1}{\lambda} u(\lambda t, Py) $, with $ P \in \mathrm{GL}_n(\mathbb{R}) $.
• The method involves minimizing an expression involving $ (\det P)^{1/n} \int \psi_P(u_0) \, dy $ over all invertible matrices $ P $, leading to a refined estimate for the total variation of $ \Delta(u) $.
• The kinetic formulation of the conservation law is used to relate the entropy dissipation measure $ \mu $ to the initial data via $ \|\mu\| \leq \int \phi(u_0) \, dy $.
• A Gronwall-type argument is applied to the time-integrated $ L^1 $-norm of $ u $ and the total variation of $ \Delta(u) $ to derive decay estimates.

Experimental results

Research questions

• RQ1 What is the weakest integrability condition on initial data $ u_0 \in L^1(\mathbb{R}^n) $ that still guarantees the existence of an entropy solution to a multi-dimensional scalar conservation law?
• RQ2 Can the classical $ L^\infty $ framework of Kruzhkov be extended to unbounded $ L^1 $ initial data while preserving uniqueness and stability?
• RQ3 How does the nonlinearity of the flux function influence the decay rate of solutions over time?
• RQ4 Can a Strichartz-type inequality be established in a quasilinear context using compensated integrability on a symmetric tensor derived from entropy pairs?
• RQ5 What is the optimal scaling and transformation of variables that maximizes the integrability and decay control of the solution?

Key findings

• The paper proves that for any initial data $ u_0 \in L^1(\mathbb{R}^n) $, even if unbounded, there exists a unique entropy solution in $ L^\infty(\mathbb{R}_+ \times \mathbb{R}^n) \cap C(\mathbb{R}_+; L^1_{\text{loc}}(\mathbb{R}^n)) $.
• A Strichartz-like inequality is established: $ \int_0^\infty \int_{\mathbb{R}^n} \Delta(u) \, dy \, dt \leq c_d \left( \|u_0\|_1 + \|\phi(u_0)\|_1 \right)^{1 + \frac{1}{n}} $, where $ \Delta(u) $ is the Hessian-like quantity related to entropy dissipation.
• The total variation of the Hessian-like term satisfies $ \int_0^\infty \int_{\mathbb{R}^n} \Delta(u) \, dy \, dt \leq c_d (\det P)^{1/n} \left( \int u_0 \, dy \right)^{1/n} \int \psi_P(u_0) \, dy $, optimized over $ P \in \mathrm{GL}_n(\mathbb{R}) $.
• The decay of the solution is quantified via a Gronwall argument, showing that $ \int_\tau^\infty X(t) \, dt \leq c_d \left( \int u_0 \, dy \right)^{1/n} I[u(\tau)] $, where $ X(t) = \int \Delta(u(t,y)) \, dy $.
• The infimum $ I[w] = \inf_P (\det P)^{1/n} \int \psi_P(w) \, dy $ is introduced as a key functional to control the decay, and its computation depends on the flux structure.
• The method successfully extends the theory to the multi-dimensional Burgers equation and generalizes the compensated integrability framework to higher dimensions with unbounded initial data.