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Stability for systems of conservation laws in several space dimensions. (English) Zbl 0844.35065

The author studies a hyperbolic system of conservation laws in \(m\) spatial variables \[ \partial_t u+ \sum^m_{\alpha= 1} \partial_\alpha F^\alpha(u)= 0,\;x\in \mathbb{R}^m,\;t> 0,\;u(x, 0)= u_0(x),\;x\in \mathbb{R}^m.\tag{1} \] To assure the uniqueness of the solution of (1) usually an entropy condition \[ \partial_t \eta(u)+ \sum^m_{\alpha= 1} \partial_\alpha q^\alpha(u)\leq 0\tag{2} \] is assumed. When the entropy pair \((\eta, q)\) exists, any solution of (1) satisfies automatically (2) and the estimate \(|q(u)|\leq s\eta(u)\). The main question studied in the work is to find sufficient condition for the estimate \[ \int_{|x|< r} |u(x, t)|^p dx\leq c_p \int_{|x|< r+ st} |u_0(x)|^p dx\tag{3} \] with \(1\leq p< \infty\), provided the entropy pair exists. This sufficient condition has the form \[ DF^\alpha\cdot DF^\beta= DF^\beta\cdot DF^\alpha,\tag{4} \] where \(DF^\alpha\) denotes the Jacobian of \(F^\alpha\). More precisely, the fact that (4) implies (3) is established in the case \(n= 2\). The proof is based on the application of Riemann invariants for the system as well as on a construction of suitable entropy pair \((\eta, q)\) satisfying the \(L^p\)-stability.
Reviewer: V.Georgiev (Sofia)

MSC:

35L65 Hyperbolic conservation laws
35B35 Stability in context of PDEs
35L40 First-order hyperbolic systems
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