Self-similar Solutions of the Navier–Stokes Equations on Weak Weighted Lorentz Spaces

In the present paper, we prove the existence of global solutions for the Navier–Stokes equations in Rnwhen the initial velocity belongs to the weighted weak Lorentz space Λn,∞(u) with a sufficiently small norm under certain restriction on the weight u. At the same time, self-similar solutions are induced if the initial velocity is, besides, a homogeneous function of degree-1. Also the uniqueness is discussed.     

Remarks On The Regularity Of Weak Solutions Of The Navier–Stokes Equations

In this paper we extend the results of Foias–Guillopé–Temam on the regularity and a priori estimates for the weak solutions of the Navier–Stokes equations. More specifically, we obtain upperbounds for the temporal averages of the Gevrey class norm for the weak solutions of the equations. The estimates are obtained first by getting integrated version of Foias–Temam's local in time estimate for Gevrey class norms of strong solutions and next by an induction procedure. We also strengthen slightly the local in time Gevrey class regularization of strong solutions.     

On Boundary Regularity of the Navier–Stokes Equations

Abstract

We study boundary regularity of weak solutions of the Navier–Stokes equations in the half-space in dimension n ≥ 3. We prove that a weak solution u which is locally in the class L ^{p, q} with 2/p + n/q = 1, q > n near boundary is Hölder continuous up to the boundary. Our main tool is a pointwise estimate for the fundamental solution of the Stokes system, which is of independent interest.     

On the Stationary Navier–Stokes Equations in the Half-Plane

We consider the stationary incompressible Navier–Stokes equation in the half-plane with inhomogeneous boundary condition. We prove the existence of strong solutions for boundary data close to any Jeffery–Hamel solution with small flux evaluated on the boundary. The perturbation of the Jeffery–Hamel solution on the boundary has to satisfy a nonlinear compatibility condition which corresponds to the integral of the velocity field on the boundary. The first component of this integral is the flux which is an invariant quantity, but the second, called the asymmetry, is not invariant, which leads to one compatibility condition. Finally, we prove the existence of weak solutions, as well as weak–strong uniqueness for small data and provide numerical simulations.     

On Type I Singularities of the Local Axi-Symmetric Solutions of the Navier–Stokes Equations

Local regularity of axially symmetric solutions to the Navier–Stokes equations is studied. It is shown that under certain natural assumptions there are no singularities of Type I.     

Global Solutions of Multidimensional Approximate Navier–Stokes Equations of a Viscous Gas

Considering the simplified Navier–Stokes equations for the motion of a viscous gas under the adherence condition, we define a weak solution and prove an existence theorem by means of a priori estimates.     

The Convergence of the Navier–Stokes–Poisson System to the Incompressible Euler Equations

ABSTRACT

The combining quasineutral and inviscid limit of the Navier–Stokes–Poisson system in the torus 𝕋 ^d , d ≥ 1 is studied. The convergence of the Navier–Stokes–Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data.     

On the Weak Solutions to Steady-State Mixed Navier—Stokes/Darcy Model

Acta Mathematica Sinica, English Series - In this paper, for the mixed Navier—Stokes/Darcy model with Beavers—Joseph—Saffman’s interface condition, we first establish an a...     

Decay of Solutions to 2-Dimensional Navier Stokes Equations

In this paper we want to establish sharp rates of both L~2 and L~∞ decay of glo-bal solutions to the initial value problems for 2-dimensional incompressible Navier-Stokes equations, with, initial data U_0(x)∈L_1∩L~2. U_t + U·U -△U + p = 0,·U = 0,U(x,0) = U_0(x), (1)where U = U(x,t) = (U_1(x,t),U_2(x,t)) is a real vector valued function, △is 2-di-mensional Laplace operator, is gradient operator. We will present a simple method for establishing the decay results.     

Concentrations Problem for Solutions to Compressible Navier–Stokes Equations

Doklady Mathematics - A three-dimensional initial-boundary value problem for the isentropic equations of the dynamics of a viscous gas is considered. The concentration phenomenon is that, for...     

A Hopf-Bifurcation Theorem for the Vorticity Formulation of the Navier–Stokes Equations in ℝ3

We prove a Hopf-bifurcation theorem for the vorticity formulation of the Navier–Stokes equations in ?³ in case of spatially localized external forcing. The difficulties are due to essential spectrum up to the imaginary axis for all values of the bifurcation parameter which a priori no longer allows to reduce the problem to a finite dimensional one.     

Global L~2 Stability of the Nonhomogeneous Incompressible Navier–Stokes Equations

In this paper, the problem of the global L^2 stability for large solutions to the nonhomogeneous incompressible Navier-Stokes equations in 3D bounded or unbounded domains is studied. By delicate energy estimates and under the suitable condition of the large solutions, it shows that if the initial data are small perturbation on those of the known strong solutions, the large solutions are stable.     

On the Suitable Weak Solutions to the Boussinesq Equations in a Bounded Domain

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On Energy Cascades in the Forced 3D Navier–Stokes Equations

We show—in the framework of physical scales and \((K_1,K_2)\)-averages—that Kolmogorov’s dissipation law combined with the smallness condition on a Taylor length scale is sufficient to guarantee energy cascades in the forced Navier–Stokes equations. Moreover, in the periodic case we establish restrictive scaling laws—in terms of Grashof number—for kinetic energy, energy flux, and energy dissipation rate. These are used to improve our sufficient condition for forced cascades in physical scales.     

Nonexistence of singular pseudo-self-similar solutions of the Navier–Stokes system

We show that there are no singular pseudo-self-similar solutions of the Navier-Stokes system with finite energy. Received March 8, 2000 / Published online February 5, 2001     

An Approach to Time Integration of the Navier–Stokes Equations

Computational Mathematics and Mathematical Physics - An approach to the time integration of the Navier–Stokes equations for a compressible heat-conducting gas is developed. According to this...     

Weak Solutions to the Cahn-Hilliard Equation with Degenerate Diffusion Mobility in ℝN

This paper is concerned with a popular form of Cahn-Hilliard equation which plays an important role in understanding the evolution of phase separation. We get the existence and regularity of a weak solution to nonlinear parabolic, fourth order Cahn-Hilliard equation with degenerate mobility M(u)=u^m(1-u)^m which is allowed to vanish at 0 and 1. The existence and regularity of weak solutions to the degenerate Cahn-Hilliard equation are obtained by getting the limits of Cahn-Hilliard equation with non-degenerate mobility. We explore the initial value problem with compact support and obtain the local non-negative result. Further, the above derivation process is also suitable for the viscous Cahn-Hilliard equation with degenerate mobility.     

Weak and Strong Solutions of the 3D Navier–Stokes Equations and Their Relation to a Chessboard of Convergent Inverse Length Scales

Journal of Nonlinear Science - Using the scale invariance of the Navier–Stokes equations to define appropriate space-and-time-averaged inverse length scales associated with weak solutions of...