estimates for quantities advected by a compressible flow (original) (raw)
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Communications on Pure and Applied Mathematics, 2010
We establish the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. For the Navier-Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup-norm of solutions with respect to the physical viscosity coefficient may not be directly controllable and, furthermore, convex entropy-entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy-type estimates with respect to the viscosity coefficient for the solutions of the Navier-Stokes equations and establish the existence of measure-valued solutions of the isentropic Euler equations generated by the Navier-Stokes equations. Based on the uniform energy-type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier-Stokes equations for weak entropy-entropy flux pairs, generated by compactly supported C 2 test functions, are confined in a compact set in H −1 , which lead to the existence of measure-valued solutions that are confined by the Tartar-Murat commutator relation. A careful characterization of the unbounded support of the measure-valued solution confined by the commutator relation yields the reduction of the measure-valued solution to a Delta mass, which leads to the convergence of solutions of the Navier-Stokes equations to a finite-energy entropy solution of the isentropic Euler equations.
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,
A priori estimates in terms of the maximum norm for the solutions of the Navier–Stokes equations
Journal of Differential Equations, 2004
In this paper, we consider the Cauchy problem for the incompressible Navier-Stokes equations with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial velocity field. For illustrative purposes, we first derive corresponding a priori estimates for certain parabolic systems. Because of the pressure term, the case of the Navier-Stokes equations is more difficult, however. r 2004 Elsevier Inc. All rights reserved.
Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow
Indiana University Mathematics Journal, 1995
We derive a detailed, pointwise description of the asymptotic behavior of solutions of the Cauchy problem for the Navier-Stokes equations of compressible flow in several space dimensions, with initial data in L 1 ∩ H k(n) . We show that, asymptotically, the solution decomposes into the sum of two terms, one of which dominates in L p for p > 2, the other for p < 2. The dominant term for p > 2 has constant density and divergence-free momentum field, decaying at the rate of a heat kernel. Thus, as measured in L p for p > 2, all smooth, small-amplitude solutions of the Navier-Stokes equations are asymptotically incompressible. When p < 2, the dominant term reflects instead the spreading effect of convection, and decays more slowly than a heat kernel; in fact, the solution may grow without bound in L p for p near 1. These features of the solution do not arise in one dimensional flow, nor are they apparent from previously known L 2 decay rates.
A bound from below for the temperature in compressible Navier–Stokes equations
Monatshefte für Mathematik, 2009
We consider the full system of compressible Navier-Stokes equations for heat conducting fluid. We show that the temperature is uniformly positive for t ≥ t 0 (for any t 0 > 0) for any solutions with finite initial entropy. The assumptions on the viscosity and conductivity coefficients are minimal (for instance, the solutions constructed by E. Feireisl in [2] verify all the requirements).
High Regularity of Solutions of Compressible Navier-Stokes Equations
Advances in Differential Equations, 2007
We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R 3. The initial density may vanish in an open subset of Ω or to be positive but vanish at space infinity. We first prove the local existence of solutions (ρ (j) , u (j)) in C([0, T * ]; H 2(k−j)+3 × D 1 0 ∩ D 2(k−j)+3 (Ω)), 0 ≤ j ≤ k, k ≥ 1 under the assumptions that the data satisfy compatibility conditions and that the initial density is sufficiently small. To control the nonnegativity or decay at infinity of density, we need to establish a boundary value problem of (k+1)-coupled elliptic system which may not be in general solvable. The smallness condition of initial density is necessary for the solvability, which is not necessary in case that the initial density has positive lower bound. Secondly, we prove the global existence of smooth radial solutions of isentropic compressible Navier-Stokes equations on a bounded annulus or a domain which is the exterior of a ball under a smallness condition of initial density.
Regularity estimates for scalar conservation laws in one space dimension
Journal of Hyperbolic Differential Equations, 2018
We deal with the regularizing effect that, in scalar conservation laws in one space dimension, the nonlinearity of the flux function [Formula: see text] has on the entropy solution. More precisely, if the set [Formula: see text] is dense, the regularity of the solution can be expressed in terms of [Formula: see text] spaces, where [Formula: see text] depends on the nonlinearity of [Formula: see text]. If moreover the set [Formula: see text] is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that [Formula: see text] for every [Formula: see text] and that this can be improved to [Formula: see text] regularity except an at most countable set of singular times. Finally, we present some examples that show the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces.
Global solutions of the compressible navier-stokes equations with larger discontinuous initial data
Communications in Partial Differential Equations, 2000
We prove the global existence of weak solutions to the Navier-Stokes equations for compressible, heat-conducting flow in one space dimension with large, discontinuous initial data, and we obtain a-priori estimates for these solutions which are independent of time, sufficient to determine their asymptotic behavior. In particular, we show that, as time goes to infinity, the solution tends to a constant state determined by the initial mass and the initial energy, and that the magnitudes of singularities in the solution decay to zero.