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Approximate solutions to conservation laws via convective parabolic equations. (English) Zbl 0653.35057

The author presents some applications of the theory of compensated compactness to the convergence of approximate solutions of scalar hyperbolic conservation law \[ (*)\quad u_ t+f(u)_ x=0,\quad t\geq 0,\quad x\in R,\quad u(x,0)=u_ 0(x). \] The first approximation is based on the convective parabolic equation \[ (**)\quad u_ t+f(u)_ x=\epsilon \psi (u)_{xx},\quad t\geq 0,\quad x\in R;\quad \epsilon >0,\quad u(x,0)=u_ 0(x). \] The solutions \(u^{\epsilon}\) of (**) converge weakly star in \(L^{\infty}\) to the weak solution of (*) and if f is not affine, the convergence is strong in L p, \(p<\infty\), with the entropy inequality for the limit solution. The latter approximation is based on the modified Lax-Friedrichs difference scheme - formal approximation to the equation \[ u_ t+f(u)_ x=(\Delta t/\lambda \quad 2)\psi (u)_{xx},\quad \lambda =\Delta x/\Delta t,\quad t\to 0. \] These approximations again converge weakly star in \(L^{\infty}\) to the weak solution of (*), verifying the entropy inequality.
Reviewer: A.Doktor

MSC:

35L65 Hyperbolic conservation laws
35K55 Nonlinear parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35A35 Theoretical approximation in context of PDEs
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