Boundary layers in parabolic perturbations of scalar conservation laws

Abstract

We consider the problem of estimating the boundary layer thickness for vanishing viscosity solutions of boundary value problems for parabolic perturbations of a scalar conservation law in a space strip in R d . For the boundary layer thickness δ(ε) we obtain that one can take δ(ε) = ε r , for any r < 1/2, arbitrarily close to 1/2. Mathematics Subject Classification (2000). Primary: 35L65; Secondary: 35B35,35B40.

Boundary-layer effects for the 2-D Boussinesq equations with vanishing diffusivity limit in the half plane

“…The BL-thickness for scalar conservation laws was also defined by Frid and Shelukhin [13] in a similar manner. The BL-thickness for the one-dimensional cylindrical compressible Navier-Stokes equations with vanishing shear viscosity limit was studied in [12,24], where the weighted Sobolev space W 1,1 g (Ω) in (1.…”

“…[36]), since it holds that lim inf n→∞ δ n (ε) = √ ε. To make the analysis of BL-thickness simpler, as that in [12,13,24], we restrict ourselves to the special case of vanishing initial data:…”

“…The methods used in [3] and [16] are both the so-called "asymptotic expansion" method which is based on the inner scaled variable Y = y/ε q where y > 0 is the flow domain and q > 0 is the scaling exponent. As it was pointed out in [13] that the completely different results obtained in [3] and [16] imply that, to apply the asymptotic expansion method, one should first determine a suitable choice of the inner scaled variable Y = y/ε q or the value of boundary-layer thickness δ(ε) = ε q . Thus, in order to shed some light on the further mathematical analysis of boundary layers for the Boussinesq equations, in this paper we try to estimate the boundary-layer thickness from the mathematical aspect.…”

“…[2,3,16,[40][41][42]44]). It also arises in the theory of hyperbolic systems when the parabolic equations with small viscosities are applied as perturbations, see, for example, [13,15,17,18,37,43] and the references therein. Under the assumption of analytic initial data, Caflisch and Sammartino [2,3] constructed a local in time solution, independent of the viscosity ε, for the Navier-Stokes equations, and proved that the Navier-Stokes solution tends asymptotically (as ε → 0) to an Euler solution outside a boundary layer and to a solution of Prandtl's equations within the boundary layer.…”

A Shock Layer in Parabolic Perturbations of a Scalar Conservation Law

“…The same approach has been applied to define a boundary layer for scalar conservation laws [2] and a shear-viscosity boundary layer in a boundary-value problem for the Navier-Stokes equations of compressible fluids [1].…”