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Regularity and large time behaviour of solutions of a conservation law without convexity. (English) Zbl 0616.35054

This paper deals with the question of smoothness and asymptotic behaviour of weak solutions of the non-linear conservation law: \((\partial u/\partial t)^{(x,t)}+(\partial /\partial x)^{f(u(x,t))}=0\) that satisfy the initial condition \(u(x,0)=u_ 0(x)\) where \(u_ 0: {\mathbb{R}}\to {\mathbb{R}}\) and \(f: {\mathbb{R}}\to {\mathbb{R}}\) are given functions and the unknown \(u: {\mathbb{R}}\times [0,\infty]\to {\mathbb{R}}\). By a ”weak solution” of the Cauchy problem is meant a bounded measurable function u with distributional derivatives \(\partial u/\partial t\) and \(\partial u/\partial x\) that are locally finite Borel Measures that satisfy the entropy condition: \((\partial /\partial t)(\eta (u(x,t)))+(\partial /\partial x)(q(u(x,t)))\geq 0\) for all concave functions \(\eta\) : \({\mathbb{R}}\to {\mathbb{R}}\) where \(q: {\mathbb{R}}\to {\mathbb{R}}\) is defined in terms of \(\eta\) by the formula \(q(u)=\int^{u}_{0}f'(s)\eta '(s)ds.\)
One of the usual hypotheses in the literature on this problem is that of convexity of f which helps make use of the Fenchel duality theory. There has also been many investigations that do not use the convexity hypothesis. In this paper, the author not only does not assume that f is convex but on the other hand imposes the hypotheses: \(f(0)=f'(0)=f''(0)=0\) and \(uf''(u)<0\) for all \(u\neq 0\) so that f has a single point of inflexion at the origin.
The following result is proved regarding smoothness of solutions. ”If the Cauchy datum \(u_ 0\) belongs to a certain non-empty open dense subset of \(C^{\infty}({\mathbb{R}})\), then every solution is also \(C^{\infty}\) except on a meagre set S consisting of a finite number of points, a finite number of straight line segments and a finite number of \(C^{\infty}\) arcs”. The nature of the singularities of the solutions across these line segments and \(C^{\infty}\) arcs is also examined in detail. Also established are decay estimates for the quantity f(u(x,t))- \(u(x,t)f'(u(x,t))\) associated with the solution u for various hypotheses on the initial data \(u_ 0\) and in some cases, the corresponding estimates for u are derived from these.
Reviewer: P.Ramankutty

MSC:

35L65 Hyperbolic conservation laws
35B65 Smoothness and regularity of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
35L45 Initial value problems for first-order hyperbolic systems
35L60 First-order nonlinear hyperbolic equations
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References:

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