Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGlobal Behavior of Spherically Symmetric Navier-Stokes Equations with Density-Dependent Viscosity
130
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we study a free boundary problem for compressible spherically symmetric Navier-Stokes equations without a solid core. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the weak solution, give some uniform bounds (with respect to time) of the solution and show that it converges to a stationary one as time tends to infinity. Moreover, we obtain the stabilization rate estimates of exponential type in $L^infty$-norm and weighted $H^1$-norm of the solution by constructing some Lyapunov functionals. The results show that such system is stable under the small perturbations, and could be applied to the astrophysics.
rate research
Read More
In this note, by constructing suitable approximate solutions, we prove the existence of global weak solutions to the compressible Navier-Stokes equations with density-dependent viscosity coefficients in the whole space $mathbb{R}^N$, $Ngeq2$ (or exterior domain), when the initial data are spherically symmetric. In particular, we prove the existence of spherically symmetric solutions to the Saint-Venant model for shallow water in the whole space (or exterior domain).
In this paper, we provide rigorous justification of the hydrostatic approximation and the derivation of primitive equations as the small aspect ratio limit of the incompressible three-dimensional Navier-Stokes equations in the anisotropic horizontal viscosity regime. Setting $varepsilon >0$ to be the small aspect ratio of the vertical to the horizontal scales of the domain, we investigate the case when the horizontal and vertical viscosities in the incompressible three-dimensional Navier-Stokes equations are of orders $O(1)$ and $O(varepsilon^alpha)$, respectively, with $alpha>2$, for which the limiting system is the primitive equations with only horizontal viscosity as $varepsilon$ tends to zero. In particular we show that for well prepared initial data the solutions of the scaled incompressible three-dimensional Navier-Stokes equations converge strongly, in any finite interval of time, to the corresponding solutions of the anisotropic primitive equations with only horizontal viscosities, as $varepsilon$ tends to zero, and that the convergence rate is of order $Oleft(varepsilon^fracbeta2right)$, where $beta=min{alpha-2,2}$. Note that this result is different from the case $alpha=2$ studied in [Li, J.; Titi, E.S.: emph{The primitive equations as the small aspect ratio limit of the Navier-Stokes equations: Rigorous justification of the hydrostatic approximation}, J. Math. Pures Appl., textbf{124} rm(2019), 30--58], where the limiting system is the primitive equations with full viscosities and the convergence is globally in time and its rate of order $Oleft(varepsilonright)$.
We study the stationary nonhomogeneous Navier--Stokes problem in a two dimensional symmetric domain with a semi-infinite outlet (for instance, either parabo-loidal or channel-like). Under the symmetry assumptions on the domain, boundary value and external force we prove the existence of at least one weak symmetric solution without any restriction on the size of the fluxes, i.e. the fluxes of the boundary value ${bf a}$ over the inner and the outer boundaries may be arbitrarily large. Only the necessary compatibility condition (the total flux is equal to zero) has to be satisfied. Moreover, the Dirichlet integral of the solution can be finite or infinite depending on the geometry of the domain.
We prove that the energy equality holds for weak solutions of the 3D Navier-Stokes equations in the functional class $L^3([0,T);V^{5/6})$, where $V^{5/6}$ is the domain of the fractional power of the Stokes operator $A^{5/12}$.
It is known that the Maxwell-Klein-Gordon equations in $mathbb{R}^{3+1}$ admit global solutions with finite energy data. In this paper, we present a new approach to study the asymptotic behavior of these global solutions. We show the quantitative energy flux decay of the solutions with data merely bounded in some weighted energy space. We also establish an integrated local energy decay and a hierarchy of $r$-weighted energy decay. The results in particular hold in the presence of large total charge. This is the first result to give a complete and precise description of the global behavior of large nonlinear charged scalar fields.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
