Incompressible limit for a viscous compressible fluid (original) (raw)
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On the incompressible limit of the compressible navier-stokes equations
Communications in Partial Differential Equations, 1995
Many interesting problems in classical physics involve the behavior of solutions of nonlinear hyperbolic systems as certain parameter and coefficients becomes infinite. Quite often, the limiting solution (when it exits) satisfies a completely different nonlinear partial differential equation. The incompressible limit of the compressible Navier-Stokes equations is one physical problem involving dissipation when snch a singular limiting process is interesting. In this article we study the time-discretized compressible Navier-Stokes equation and consider the incompressible limit as the Mach number tends to zero. For 1'-law gas, 1 < l' =5 2, D =5 4, we show that the solutions (Po J.£f'./€) of the compressible Navier-St.okes system converge to the solution (I, v) of the incompressible Navier-Stokes system. Furthermore we also prove that the limit also satisfies the Leray energy inequality.
Advances in Applied Mathematics, 1991
We study the slightly compressible Navier-Stokes equations. We first consider the Cauchy problem, periodic in space. Under appropriate assumptions on the initial data, the solution of the compressible equations consists-to first order-of a solution of the incompressible equations plus a function which is highly oscillatory in time. We show that the highly oscillatory part (the sound waves) can be described by wave equations, at least locally in time. We also show that the bounded derivative principle is valid; i.e., the highly oscillatory part can be suppressed by initialization. Besides the Cauchy problem, we also consider an initial-boundary value problem. At the inflow boundary, the viscous term in the Navier-Stokes equations is important. We consider the case where the compressible pressure is prescribed at inflow. In general, one obtains a boundary layer in the pressure; in the velocities a boundary layer is not present to first approximation. 8
Remarks on the inviscid limit for the compressible flows
We establish various criteria, which are known in the incompressible case, for the validity of the inviscid limit for the compressible Navier-Stokes flows considered in a general domain Ω in R n with or without a boundary. In the presence of a boundary, a generalized Navier boundary condition for velocity is assumed, which in particular by convention includes the classical no-slip boundary conditions. In this general setting we extend the Kato criteria and show the convergence to a solution which is dissipative "up to the boundary". In the case of smooth solutions, the convergence is obtained in the relative energy norm.
Communications in Mathematical Physics, 1987
In this paper we study the system (1.1), (1.3), which describes the stationary motion of a given amount of a compressible heat conducting, viscous fluid in a bounded domain Ω of R n , n^2. Here u(x) is the velocity field, p(x) is the density of the fluid, ζ(x) is the absolute temperature,/(x) and h(x) are the assigned external force field and heat sources per unit mass, and p(p t ζ) is the pressure. In the physically significant case one has g = 0. We prove that for small data (/, g, h) there exists a unique solution (u, p, ζ) of problem (1.1), (1.3) 1? in a neighborhood of (0,m, ζ 0); for arbitrarily large data the stationary solution does not exist in general (see Sect. 5). Moreover, we prove that (for barotropic flows) the stationary solution of the Navier-Stokes equations (1.8) is the incompressible limit of the stationary solutions of the compressible Navier-Stokes equations (1.7), as the Mach number becomes small. Finally, in Sect. 5 we will study the equilibrium solutions for system (4.1). For a more detailed explanation see the introduction.
Remarks on the rate of decay of solutions to linearized compressible Navier-Stokes equations
2015
We consider the Lp−Lq estimates of solutions to the Cauchy problem of linearized compressible Navier–Stokes equation. Especially, we investigate the diffusion wave property of the compressible Navier–Stokes flows, which was studied by D. Hoff and K. Zumbrum and Tai-P. Liu and W. Wang. 1. Introduction. In this paper, we consider the Cauchy problem of the following linearized compressible Navier-Stokes equations: ρt + γdiv v = 0 in (0,∞) × Rn,(1.1) vt − α∆v − β∇div v + γ∇ρ = 0 in (0,∞) × Rn,
An Introduction to Compressible Flows with Applications
SpringerBriefs in Mathematics, 2019
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Applications of Mathematics
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