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\(L^2\)-contraction of large planar shock waves for multi-dimensional scalar viscous conservation laws. (English) Zbl 1417.35129

Summary: We consider a \(L^2\)-contraction (a \(L^2\)-type stability) of large viscous shock waves for the multi-dimensional scalar viscous conservation laws, up to a suitable shift by using the relative entropy methods. Quite different from the previous results, we find a new way to determine the shift function, which depends both on the time and space variables and solves a viscous Hamilton-Jacobi type equation with source terms. Moreover, we do not impose any conditions on the anti-derivative variables of the perturbation around the shock profile. More precisely, it is proved that if the initial perturbation around the viscous shock wave is suitably small in \(L^2\)-norm, then the \(L^2\)-contraction holds true for the viscous shock wave up to a suitable shift function. Note that BV-norm or the \(L^\infty\)-norm of the initial perturbation and the shock wave strength can be arbitrarily large. Furthermore, as the time \(t\) tends to infinity, the \(L^2\)-contraction holds true up to a (spatially homogeneous) time-dependent shift function. In particular, if we choose some special initial perturbations, then \(L^2\)-convergence of the solutions towards the associated shock profile can be proved up to a time-dependent shift.

MSC:

35Q35 PDEs in connection with fluid mechanics
76L05 Shock waves and blast waves in fluid mechanics
35L65 Hyperbolic conservation laws
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