Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSelf-similar solutions for compressible Navier-Stokes equations
88
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We construct forward self-similar solutions (expanders) for the compressible Navier-Stokes equations. Some of these self-similar solutions are smooth, while others exhibit a singularity do to cavitation at the origin.
rate research
Read More
The energy equalities of compressible Navier-Stokes equations with general pressure law and degenerate viscosities are studied. By using a unified approach, we give sufficient conditions on the regularity of weak solutions for these equalities to hold. The method of proof is suitable for the case of periodic as well as homogeneous Dirichlet boundary conditions. In particular, by a careful analysis using the homogeneous Dirichlet boundary condition, no boundary layer assumptions are required when dealing with bounded domains with boundary.
In this paper, it is shown that there does not exist a non-trivial Lerays backward self-similar solution to the 3D Navier-Stokes equations with profiles in Morrey spaces $dot{mathcal{M}}^{q,1}(mathbb{R}^{3})$ provided $3/2<q<6$, or in $dot{mathcal{M}}^{q,l}(mathbb{R}^{3})$ provided $6leq q<infty$ and $2<lleq q$. This generalizes the corresponding results obtained by Nev{c}as-Rr{a}uv{z}iv{c}ka-v{S}ver{a}k [19, Acta.Math. 176 (1996)] in $L^{3}(mathbb{R}^{3})$, Tsai [25, Arch. Ration. Mech. Anal. 143 (1998)] in $L^{p}(mathbb{R}^{3})$ with $pgeq3$,, Chae-Wolf [3, Arch. Ration. Mech. Anal. 225 (2017)] in Lorentz spaces $L^{p,infty}(mathbb{R}^{3})$ with $p>3/2$, and Guevara-Phuc [11, SIAM J. Math. Anal. 12 (2018)] in $dot{mathcal{M}}^{q,frac{12-2q}{3}}(mathbb{R}^{3})$ with $12/5leq q<3$ and in $L^{q, infty}(mathbb{R}^3)$ with $12/5leq q<6$.
In this paper we consider the barotropic compressible quantum Navier-Stokes equations with a linear density dependent viscosity and its limit when the scaled Planck constant vanish. Following recent works on degenerate compressible Navier-Stokes equations, we prove the global existence of weak solutions by the use of a singular pressure close to vacuum. With such singular pressure, we can use the standard definition of global weak solutions which also allows to justify the limit when the scaled Planck constant denoted by $epsilon$ tends to 0.
This paper addresses the construction and the stability of self-similar solutions to the isentropic compressible Euler equations. These solutions model a gas that implodes isotropically, ending in a singularity formation in finite time. The existence of smooth solutions that vanish at infinity and do not have vacuum regions was recently proved and, in this paper, we provide the first construction of such smooth profiles, the first characterization of their spectrum of radial perturbations as well as some endpoints of unstable directions. Numerical simulations of the Euler equations provide evidence that one of these endpoints is a shock formation that happens before the singularity at the origin, showing that the implosion process is unstable.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
