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Structure of entropy solutions for multi-dimensional scalar conservation laws. (English) Zbl 1036.35127

The authors of this paper study the structure of entropy solutions of scalar conservation laws in \(n\) space dimensions having the form \[ \partial_tu+ \text{div}_xf(u)=0. \] A bounded measurable entropy solution \(u\) is characterized by dissipation of entropy \[ -\mu_{\eta ,q}\equiv \partial_t\eta (u)+ \text{div}_xq(u)\leq 0 \] in the space \(\mathcal D^{\prime }_{t,x}\) for any convex entropy-entropy flux pair \((\eta (u), q(u))\in \mathbb R \times \mathbb R^n\) compatible with the given flux function \(f(v)\in \mathbb R^n\), that is, \(q^{\prime }(v)=\eta ^{\prime }(v)f ^{\prime }(v)\) and \(\eta ^{\prime \prime }(v)\geq \) for all \(v\in \mathbb R^n\).
An entropy solution \(u\) of the equation under consideration is not necessarily in the class of BV functions, even if the conservation law is genuinely nonlinear. As it is known, smooth solutions cannot exist in general. The weak formulation of the considered problem allows for singularities at the expense of the rigidity. Here the Cauchy problem is ill-posed. The notion of entropy solution restores the right amount of rigidity for existence and uniqueness. It is shown that this rigidity also survives in the form of a regularizing effect on the structure of \(u\).
The main result is that finite entropy dissipation in combination with genuine nonlinearity is indeed enough for a structure result. The authors obtain BV-like structure for entropy solutions without using BV control. This result highlights the regularizing effect of genuine nonlinearity in a qualitative way. It is based on the locally finite rate of entropy dissipation. The proof of this result relies on the geometric classification of blow-ups in the framework of the kinetic formulation.

MSC:

35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
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