This paper is devoted to establish the existence of weak solutions to the scalar conservation laws (1) ut+f(u)x=0, by compensated compactness theory. He shows that the unique weak entropic solution to (1) can be obtained by replacing the usual viscous approximation by means of the porous media operator, (2) ut+f(u)x=ε(|u|m−1u)xx, m>1, ε>0. The advantage of using the method is that it provides an approximating solution which, in some situations, coincides with the exact solution of (1) outside a compact set (in the space variable, for fixed time), while the perturbation effects of the usual viscosity always lead to undesired modifications of the far fields.
Convergence of approximate solutions to scalar conservation laws by degenerate diffusion
MARCATI, PIERANGELO
1989-01-01
Abstract
This paper is devoted to establish the existence of weak solutions to the scalar conservation laws (1) ut+f(u)x=0, by compensated compactness theory. He shows that the unique weak entropic solution to (1) can be obtained by replacing the usual viscous approximation by means of the porous media operator, (2) ut+f(u)x=ε(|u|m−1u)xx, m>1, ε>0. The advantage of using the method is that it provides an approximating solution which, in some situations, coincides with the exact solution of (1) outside a compact set (in the space variable, for fixed time), while the perturbation effects of the usual viscosity always lead to undesired modifications of the far fields.Pubblicazioni consigliate
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