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Register a new userWell-posedness of thermal layer equations for inviscid compressible flows
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A semi-explicit formula of solution to the boundary layer system for thermal layer derived from the compressible Navier-Stokes equations with the non-slip boundary condition when the viscosity coefficients vanish is given, in particular in three space dimension. In contrast to the inviscid Prandtl system studied by [7] in two space dimension, the main difficulty comes from the coupling of the velocity field and the temperature field through a degenerate parabolic equation. The convergence of these boundary layer equations to the inviscid Prandtl system is justified when the initial temperature goes to a constant. Moreover, the time asymptotic stability of the linearized system around a shear flow is given, and in particular, it shows that in three space dimension, the asymptotic stability depends on whether the direction of tangential velocity field of the shear flow is invariant in the normal direction respective to the boundary.
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In this paper, we are concerned with the motion of electrically conducting fluid governed by the two-dimensional non-isentropic viscous compressible MHD system on the half plane, with no-slip condition for velocity field, perfect conducting condition for magnetic field and Dirichlet boundary condition for temperature on the boundary. When the viscosity, heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate, there is a boundary layer that is described by a Prandtl-type system. By applying a coordinate transformation in terms of stream function as motivated by the recent work cite{liu2016mhdboundarylayer} on the incompressible MHD system, under the non-degeneracy condition on the tangential magnetic field, we obtain the local-in-time well-posedness of the boundary layer system in weighted Sobolev spaces.
In this paper, the initial-boundary value problem of the 1D full compressible Navier-Stokes equations with positive constant viscosity but with zero heat conductivity is considered. Global well-posedness is established for any $H^1$ initial data. The initial density is required to be nonnegative, which is not necessary to be uniformly away from vacuum. This not only generalizes the well-known result of Kazhikhov--Shelukhin (Kazhikhov, A.~V.; Shelukhin, V.~V.: emph{Unique global solution with respect to time of initial boundary value problems for one-dimensional equations of a viscous gas}, J.,Appl.,Math.,Mech., bf41 rm(1977), 273--282.) from the heat conductive case to the non-heat conductive case, and the initial vacuum is allowed.
We consider the flow of an upper convected Maxwell fluid in the limit of high Weissenberg and Reynolds number. In this limit, the no-slip condition cannot be imposed on the solutions. We derive equations for the resulting boundary layer and prove the well-posedness of these equations. A transformation to Lagrangian coordinates is crucial in the argument.
We prove the existence of relative finite-energy vanishing viscosity solutions of the one-dimensional, isentropic Euler equations under the assumption of an asymptotically isothermal pressure law, that is, $p(rho)/rho = O(1)$ in the limit $rho to infty$. This solution is obtained as the vanishing viscosity limit of classical solutions of the one-dimensional, isentropic, compressible Navier--Stokes equations. Our approach relies on the method of compensated compactness to pass to the limit rigorously in the nonlinear terms. Key to our strategy is the derivation of hyperbolic representation formulas for the entropy kernel and related quantities; among others, a special entropy pair used to obtain higher uniform integrability estimates on the approximate solutions. Intricate bounding procedures relying on these representation formulas then yield the required compactness of the entropy dissipation measures. In turn, we prove that the Young measure generated by the classical solutions of the Navier--Stokes equations reduces to a Dirac mass, from which we deduce the required convergence to a solution of the Euler equations.
Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation begin{eqnarray*} u_t+u_{xxx}+epsilon |partial_x|^{2alpha}u+(u^2)_x=0, u(0)=phi, end{eqnarray*} where $0<epsilon,alphaleq 1$ and $u$ is a real-valued function, we show that it is globally well-posed in $H^s (s>s_alpha)$, and uniformly globally well-posed in $H^s (s>-3/4)$ for all $epsilon in (0,1)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the KdV equation if $epsilon$ tends to 0.
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