Abstract
We study the viscous limit problem for a general system of conservation laws. We prove that if the solution of the underlying inviscid problem is piecewise smooth with finitely many noninteracting shocks satisfying the entropy condition, then there exist solutions to the corresponding viscous system which converge to the inviscid solutions away from shock discontinuities at a rate of ε1 as the viscosity coefficient ε vanishes.
Highlights
We consider the relation between the solutions, uε, of the system of viscous conservation laws uεt f uε x ε B uε uεx x, uε ∈ Rn, x ∈ R, t ≥ 0, ε > 0, 1.1 and the distributional solution, u, of the corresponding system of conservation laws without viscosity ut f u x 0, u ∈ Rn, x ∈ R, t > 0
When the Euler flow contains a single shock, Hoff and Liu 1 studied the isentropic case, they established the limit process from the solutions of the compressible Navier-Stokes equations to the single shock-wave solution of the corresponding compressible Euler system so-called p-system. They show that the solutions to the isentropic Navier-Stokes equations with shock data exist and converge to the inviscid shocks as the viscosity vanishes, uniformly away from the shocks
Yu 6 revealed the rich structure of nonlinear wave interactions due to the presence of shocks and initial layers by a detailed pointwise analysis
Summary
We consider the relation between the solutions, uε, of the system of viscous conservation laws uεt f uε x ε B uε uεx x, uε ∈ Rn, x ∈ R, t ≥ 0, ε > 0, 1.1 and the distributional solution, u, of the corresponding system of conservation laws without viscosity ut f u x 0, u ∈ Rn, x ∈ R, t > 0. Goodman and Xin 2 gave a very detailed description of the asymptotic behavior of solutions for the general viscous systems as the viscosity tends to zero, via a method of matching asymptotics This method can be applied to the Navier-Stokes equations 1.1 , such as 3–5. As far as rarefaction wave is concerned, Xin in 7 has obtained that the solutions for the isentropic Navier-Stokes equations with weak centered rarefaction wave data exist for all time and converge to the weak centered rarefaction wave solution of the corresponding Euler system, as the viscosity tends to zero, uniformly away from the initial discontinuity. In this paper, motivated by Goodman and Xin’s work 2 , we establish that the piecewise smooth solutions, u, of 1.2 , with finitely many noninteracting shocks satisfying the entropy condition, are strong limits as ε → 0 of solutions, uε, of 1.1 when the matrix LBR is positive definite. We use O 1 to denote any positive bounded function which is independent of ε
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