اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe primitive equations approximation of the anisotropic horizontally viscous Navier-Stokes equations
82
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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 $Oleft(varepsilonright)$.
قيم البحث
اقرأ أيضاً
Consider the anisotropic Navier-Stokes equations as well as the primitive equations. It is shown that the horizontal velocity of the solution to the anisotropic Navier-Stokes equations in a cylindrical domain of height $varepsilon $ with initial data
$u_0=(v_0,w_0)in B^{2-2/p}_{q,p}$, $1/q+1/ple 1$ if $qge 2$ and $4/3q+2/3ple 1$ if $qle 2$, converges as $varepsilon to 0$ with convergence rate $mathcal{O} (varepsilon )$ to the horizontal velocity of the solution to the primitive equations with initial data $v_0$ with respect to the maximal-$L^p$-$L^q$-regularity norm. Since the difference of the corresponding vertical velocities remains bounded with respect to that norm, the convergence result yields a rigorous justification of the hydrostatic approximation in the primitive equations in this setting. It generalizes in particular a result by Li and Titi for the $L^2$-$L^2$-setting. The approach presented here does not rely on second order energy estimates but on maximal $L^p$-$L^q$-estimates for the heat equation.
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 $A^{5/12}$.
We show that non-uniqueness of the Leray-Hopf solutions of the Navier--Stokes equation on the hyperbolic plane observed in arXiv:1006.2819 is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on the hyperbolic spac
es of higher dimension. We also describe the corresponding general Hamiltonian setting of hydrodynamics on complete Riemannian manifolds, which includes the hyperbolic setting.
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 solu
tion, 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.
In this paper, we construct global weak solutions{`a} la Hoff (i.e. intermediate regularity) for the compressible Navier-Stokes system governing a barotropic fluid with a pressure law p($rho$) = a$rho$ $gamma$ where a > 0 and $gamma$ $ge$ d/(4 -- d))
and with an anisotropic fourth order symmetric viscous tensor with smooth coefficients under the assumption that the norms of the initial data ($rho$0 -- M, u0) $in$ L 2$gamma$ T d x (H 1 (T d)) d are sufficiently small, where M denotes the total mass of the fluid. We consider periodic boundary conditions for simplicity i.e. a periodic box $Omega$ = T d with d = 2, 3 with |$Omega$| = 1. The main technical contribution of our paper is the extension of the Hoff solutions framework by relaxing the integrability needed for the initial density which is usually assumed to be L $infty$ (T d). In this way, we are able to cover the case of viscous tensors that depend on the time and space variables. Moreover, when comparing to the results known for the global weak solutions{`a} la Leray (i.e. obtained assuming only the basic energy bounds), we obtain a relaxed condition on the range of admissible adiabatic coefficients $gamma$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
