Strong Traces for Solutions of Multidimensional Scalar Conservation Laws

By Alexis Vasseur. 2001

226 citations recorded by the source corpus.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/a5d68010bc2b98cfc509979cb85313751e7d3866

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In Applied Mathematics

In Astronomy and Astrophysics

Citations in the corpus

Relative Entropy and the Stability of Shocks and Contact Discontinuities for Systems of Conservation Laws with non-BV Perturbations (2010, background): “Let us emphasize that this notion of trace is more restrictive than the strong trace introduced in [43], which is known to be verified for bounded solutions of scalar conservation laws.”
ANALYSIS AND APPROXIMATION OF A SCALAR CONSERVATION LAW WITH A FLUX FUNCTION WITH DISCONTINUOUS COEFFICIENTS (2003, methodology): “This underlines its relation with the study of the behavior, as ε→ 0, of the solution uε of the problem ∂uε ∂t + ∂ ∂x A(uε) + zε(x)B(uε) = 0 , where zε → δ0, performed by Vasseur.(33) Besides, degenerate parabolic equations of the kind ut +(k(x)g(u))x−A(u)xx = 0, where the coefficient k is discontin”
Strong Traces for Solutions to Scalar Conservation Laws with General Flux (2007, methodology): “The method has been generalized, in [18], in the case of trace for arbitrary domain Ω of the multidimensional scalar conservation laws involving a ”non degenerate” flux function verifying: L({ξ | τ + ζ.”
EXISTENCE OF STRONG TRACES FOR QUASI-SOLUTIONS OF MULTIDIMENSIONAL CONSERVATION LAWS (2007, background): “Observe that the Vasseur’s result in [ 26 ] directly follows from the indicated precompactness property, with no regularity restrictions on the flux vector and the weaker non-degeneracy assumption.”
The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation (2014, unknown): “Vasseur [33] and G.”
CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME AND STABILITY FOR CONSERVATION LAWS WITH A DISCONTINUOUS SPACE-TIME DEPENDENT FLUX (2004, methodology): “2 can probably be extended to the case where k(x, t) varies smoothly on each Rm by replacing the kinetic approach used in [36] by the compensated compactness method used herein, but this is outside the scope of the present paper.”
Gauss‐Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws (2007, methodology): “As examples, we refer the reader to [15, 16, 17, 19] for the stability of Riemann solutions, which may contain rarefaction waves, contact discontinuities, and/or vacuum states, in the class of entropy solutions of the Euler equations for gas dynamics; to [15, 18] for the decay of periodic entropy so”
On the well-posedness of entropy solutions to conservation laws with a zero-flux boundary condition (2007, unknown): “In Section 2 we state some technical assumptions, introduce the concepts of domains with Lipschitz deformable boundaries and traces, and recall Vasseur’s result from [22].”
Criteria on Contractions for Entropic Discontinuities of Systems of Conservation Laws (2015, background): “Strong trace property in the case of scalar conservation laws have been widely studied [15, 20, 28, 29, 41, 40, 52].”
Chapter 4 – Notes on Hyperbolic Systems of Conservation Laws and Transport Equations (2007, background): “Indeed, when g is sufficiently regular and satisfies suitable assumptions of genuine nonlinearity, Vasseur proved in [39] that any entropy solution has a strong trace at 0.”
On the theory of divergence-measure fields and its applications (2001, methodology): “Theorem 6.1 was generalized with slightly strong nonlinearity condition by Vasseur [ 45 ], with the aid of the generalized Gauss-Green theorem and normal traces (Theorem 2.1) combined with the kinetic formulation of Lions-Perthame-Tadmor [31].”