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Decay of entropy solutions of nonlinear conservation laws. (English) Zbl 0942.35031
The authors study the large time behaviour of periodic entropy solutions to the Cauchy problem for hyperbolic systems of conservation laws \[ u_t+\nabla_x\cdot f(u)=0, \quad u\in{\mathbb R}^m, \quad x\in{\mathbb R}^n,\qquad u(x,0)=u_0(x). \] The general approach is introduced mainly based on scale-invariance of the equations and compactness of the solution operator (under appropriate nondegeneracy conditions). The authors apply this approach to prove decay properties of \(L^\infty\) periodic entropy solutions for multidimensional scalar conservation laws and for some important hyperbolic systems of conservation laws. Conservation laws with relaxation are considered as well.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35L65 Hyperbolic conservation laws
35B10 Periodic solutions to PDEs
35L45 Initial value problems for first-order hyperbolic systems
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