Existence and Uniqueness of Global Strong Solutions for One-Dimensional Compressible Navier–Stokes Equations (original) (raw)

Regularity of 1D compressible isentropic Navier–Stokes equations with density-dependent viscosity

Journal of Differential Equations, 2008

In this paper, we consider one-dimensional compressible isentropic Navier-Stokes equations with the viscosity depending on density and with the free boundary. The viscosity coefficient μ is proportional to ρ θ with θ > 0, where ρ is the density. The existence, uniqueness, regularity of global weak solutions in H 1 ([0, 1]) have been established by Xin and Yao in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint]. Furthermore, under certain assumptions imposed on the initial data, we improve the regularity result obtained in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint] by driving some new a priori estimates.

Inflow problem for the one-dimensional compressible Navier–Stokes equations under large initial perturbation

Journal of Differential Equations, 2014

This paper is concerned with the inflow problem for the one-dimensional compressible Navier-Stokes equations. For such a problem, Matsumura and Nishihara showed in [A. Matsumura and K. Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a onedimensional system of compressible viscous gas. Comm. Math. Phys. 222 (2001), 449-474] that there exists boundary layer solution to the inflow problem and both the boundary layer solution, the rarefaction wave, and the superposition of boundary layer solution and rarefaction wave are nonlinear stable under small initial perturbation. The main purpose of this paper is to show that similar stability results for the boundary layer solution and the supersonic rarefaction wave still hold for a class of large initial perturbation which can allow the initial density to have large oscillation. The proofs are given by an elementary energy method and the key point is to deduce the desired lower and upper bounds on the density function.

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.

1D and 2D GLOBAL STRONG SOLUTIONS OF NAVIER STOKES EXISTENCE AND UNIQUENESS

2022

Consider the Navier-Stokes equation for a one-dimensional and two-dimensional compressible viscous liquid. It is a well-known fact that there is a strong solution locally in time when the initial data is smooth and the initial density is limited down by a positive constant. In this article, under the same hypothesis, I show that the density remains uniformly limited in time from the bottom by a positive constant, and therefore a strong solution exists globally in time. In addition, most existing results are obtained with a positive viscosity factor, but current results are true even if the viscosity factor disappears with density. Finally, I prove that this solution is unique in a class of weak solutions that satisfy the usual entropy inequalities. The point of this work is the new entropy-like inequalities that Bresch and Desjardins introduced into the shallow water system of equations. This discrepancy gives the density additional regularity (assuming such regularity exists first)...

Global strong solutions to compressible Navier–Stokes system with degenerate heat conductivity and density-depending viscosity

Communications in Mathematical Sciences

We consider the compressible Navier-Stokes system where the viscosity depends on density and the heat conductivity is proportional to a positive power of the temperature under stress-free and thermally insulated boundary conditions. Under the same conditions on the initial data as those of the constant viscosity and heat conductivity case ([Kazhikhov-Shelukhin. J. Appl. Math. Mech. 41 (1977)], we obtain the existence and uniqueness of global strong solutions. Our result can be regarded as a natural generalization of the Kazhikhov's theory for the constant heat conductivity case to the degenerate and nonlinear one under stress-free and thermally insulated boundary conditions.

Blowup of solutions for the compressible Navier–Stokes equations with density-dependent viscosity coefficients

Nonlinear Analysis: Theory, Methods & Applications, 2013

We consider blowup of classical solutions to compressible Navier-Stokes equations with revised Maxwell's law which can be regarded as a relaxation to the classical Newtonian flow. For this new model, we show that for some special large initial data, the life span of any C 1 solution must be finite. This shows much difference with Newtonian flow where the global existence of C 1 solutions is still an open and famous problem in fluid dynamics. We exploit the property of finite propagation speed and the methods from Sideris (1985) to prove our results.

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.

On compressible Navier-Stokes equations with density dependent viscosities in bounded domains

Journal de Mathématiques Pures …, 2007

The present note extends to smooth enough bounded domains recent results about barotropic compressible Navier-Stokes systems with density dependent viscosity coefficients. We show how to get the existence of global weak solutions for both classical Dirichlet and Navier boundary conditions on the velocity, under appropriate constraints on the initial density profile and domain curvature. An additional turbulent drag term in the momentum equation is used to handle the construction of approximate solutions.

Global well-posedness of regular solutions to the three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities and vacuum

Advances in Mathematics, 2021

In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations with degenerate viscosities is considered. By introducing some new variables and making use of the " quasi-symmetric hyperbolic"-"degenerate elliptic" coupled structure to control the behavior of the fluid velocity, we prove the global-in-time well-posedness of regular solutions with vacuum for a class of smooth initial data that are of small density but possibly large velocities. Here the initial mass density is required to decay to zero in the far field, and the spectrum of the Jacobi matrix of the initial velocity are all positive. The result here applies to a class of degenerate density-dependent viscosity coefficients, is independent of the BD-entropy, and seems to be the first on the global existence of smooth solutions which have large velocities and contain vacuum state for such degenerate system in three space dimensions.