L p -Estimates for the Navier–Stokes Equations for Steady Compressible Flow

We consider the Navier–Stokes equations for compressible isentropic flow in the steady three-dimensional case. The pressure and the kinetic energy are estimated uniformly in L^q with being the density. This is an improvement of known estimates in the case Mathematics Subject Classification (2000): 35Q30, 76N10     

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.     

Local Well-Posedness of the Anisotropic Lagrangian Averaged Navier–Stokes Equations with Stokes Projector Viscosity

The anisotropic Lagrangian averaged Navier–Stokes (LANS-α) equations are a coupled system of nonlinear partial differential equations designed to capture both the large scale motion of an incompressible fluid and the covariance tensor. There are two choices for the divergence-free projection of the viscosity term. One choice is the classical L ²-orthogonal Leray projector. In this case, Marsden and Shkoller (2003 Marsden , J. , Shkoller , S. ( 2003 ). The anisotropic Lagrangian averaged Euler and Navier–Stokes equations . Arch. Ration. Mech. Anal. 166 : 27 – 46 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) show that strong solutions exist and are unique in the three-dimensional periodic box for a finite time interval. We extend this result by considering the second choice of projector, the generalized Stokes projector.     

Blow-up of Compressible Navier–Stokes–Korteweg Equations

Based on the results of Xin (Commun. Pure Appl. Math. 51(3):229–240, 1998), Zhang and Tan (Acta Math. Sin. Engl. Ser. 28(3):645–652, 2012), we show the blow-up phenomena of smooth solutions to the non-isothermal compressible Navier–Stokes–Korteweg equations in arbitrary dimensions, under the assumption that the initial density has compact support. Here the coefficients are generalized to a more general case which depends on density and temperature. Our work extends the previous corresponding results.     

Global Regularity of the 3D Axi-Symmetric Navier–Stokes Equations with Anisotropic Data

In this paper, we study the 3D axisymmetric Navier–Stokes equations with swirl. We prove the global regularity of the 3D Navier–Stokes equations for a family of large anisotropic initial data. Moreover, we obtain a global bound of the solution in terms of its initial data in some L ^p norm. Our results also reveal some interesting dynamic growth behavior of the solution due to the interaction between the angular velocity and the angular vorticity fields.     

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...     

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.     

A Regularity Criterion for the Navier–Stokes Equations

We prove that a weak solution u = (u ¹, u ², u ³) to the Navier–Stokes equations is strong, if any two components of u satisfy Prodi–Ohyama–Serrin's criterion. As a local regularity criterion, we prove u is bounded locally if any two components of the velocity lie in L ^{6, ∞}.     

Stationary Solutions of Compressible Navier–Stokes Equations with Slip Boundary Conditions

We consider the Navier–Stokes equations for a compressible, viscous fluid with heat–conduction in a bounded domain of IR² or IR³. Under the assumption that the external force field and the external heat supply are small we prove the existence and local uniqueness of a stationary solution satisfying a slip boundary condition. For the temperature we assume a Dirichlet or an oblique boundary condition.     

Existence and Regularity of the Pressure for the Stochastic Navier–Stokes Equations

We prove, on one hand, that for a convenient body force with values in the distribution space (H ^-1(D)) ^d , where D is the geometric domain of the fluid, there exist a velocity u and a pressure p solution of the stochastic Navier–Stokes equation in dimension 2, 3 or 4. On the other hand, we prove that, for a body force with values in the dual space V of the divergence free subspace V of (H ¹ ₀(D)) ^d , in general it is not possible to solve the stochastic Navier–Stokes equations. More precisely, although such body forces have been considered, there is no topological space in which Navier–Stokes equations could be meaningful for them.     

Equations of the boundary layer for a modified Navier–Stokes system

We consider a system of equations of the boundary layer derived from the hydrodynamical system for generalized Newtonian media. This modification of the Navier–Stokes system was proposed by O. A. Ladyzhenskaya in connection with the uniqueness of the solution of this system in general. We prove the existence and the uniqueness of a solution for the problem of continuation of the boundary layer and consider some questions connected with the separation of the boundary layer.     

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.     

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.     

Gevrey Regularity for the Attractor of the 3D Navier–Stokes–Voight Equations

Recently, the Navier–Stokes–Voight (NSV) model of viscoelastic incompressible fluid has been proposed as a regularization of the 3D Navier–Stokes equations for the purpose of direct numerical simulations. In this work, we prove that the global attractor of the 3D NSV equations, driven by an analytic forcing, consists of analytic functions. A consequence of this result is that the spectrum of the solutions of the 3D NSV system, lying on the global attractor, have exponentially decaying tail, despite the fact that the equations behave like a damped hyperbolic system, rather than the parabolic one. This result provides additional evidence that the 3D NSV with the small regularization parameter enjoys similar statistical properties as the 3D Navier–Stokes equations. Finally, we calculate a lower bound for the exponential decaying scale—the scale at which the spectrum of the solution start to decay exponentially, and establish a similar bound for the steady state solutions of the 3D NSV and 3D Navier–Stokes equations. Our estimate coincides with the known bounds for the smallest length scale of the solutions of the 3D Navier–Stokes equations, established earlier by Doering and Titi.      

Navier–Stokes Equations on Weighted Graphs

Navier–Stokes equations arise in the study of incompressible fluid mechanics, star movement inside a galaxy, dynamics of airplane wings, etc. In the case of Newtonian incompressible fluids, we propose an adaptation of such equations to finite connected weighted graphs such that it produces an ordinary differential equation with solutions contained in a linear subspace, this subspace corresponding to the Newtonian conservation law. We discuss the particular case when the graph is the complete graph K _m , with constant weight, and provide a necessary and sufficient condition for it to have solutions.     

Stability of Navier–Stokes System with Singular External Force in Fourier–Herz Space

We prove the asymptotic properties of the solutions to the 3D Navier–Stokes system with singular external force, by making use of Fourier localization method, the Littlewood–Paley theory and some subtle estimates in Fourier–Herz space. The main idea of the proof is motivated by that of Cannone et al. [J. Differential Equations, 314, 316–339(2022)]. We deal either with the nonstationary problem or with the stationary problem where solution may be singular due to singular external force. In this p...     

Convergence Analysis of the Nonconforming Finite Element Discretization of Stokes and Navier–Stokes Equations with Nonlinear Slip Boundary Conditions

This work is concerned with the nonconforming finite approximations for the Stokes and Navier–Stokes equations driven by slip boundary condition of “friction” type. It is well documented that if the velocity is approximated by the Crouzeix–Raviart element of order one, whereas the discrete pressure is constant elementwise that the inequality of Korn does not hold. Hence, we propose a new formulation taking into account the curvature and the contribution of tangential velocity at the boundary. Using the maximal regularity of the weak solution, we derive a priori error estimates for the velocity and pressure by taking advantage of the enrichment mapping and the application of Babuska–Brezzi’s theory for mixed problems.     

Optimal Time Decay of Navier–Stokes Equations with Low Regularity Initial Data

In this paper, we study the optimal time decay rate of isentropic Navier–Stokes equations under the low regularity assumptions about initial data. In the previous works about optimal time decay rate, the initial data need to be small in H~([N/2]+2)(R~N). Our work combined negative Besov space estimates and the conventional energy estimates in Besov space framework which is developed by Danchin. Through our methods, we can get optimal time decay rate with initial data just small in B~(N/2-1,N/2+1)∩B~(N/2-1,N/2) and belong to some negative Besov space(need not to be small). Finally,combining the recent results in [25] with our methods, we only need the initial data to be small in homogeneous Besov spaceB~(N/2-2,N/2)∩B~(N/2-1) to get the optimal time decay rate in space L~2.     

Localized Anisotropic Regularity Conditions for the Navier–Stokes Equations

We establish a sufficient regularity condition for local solutions of the Navier–Stokes equations. For a suitable weak solution (u, p) on a domain D we prove that if \(\partial _3 u\) belongs to the space \(L_t^{s_0}L_x^{r_0}(D)\) where \(2/s_0 + 3/r_0 \le 2 \) and \(9/4 \le r_0\le 5/2\), then the solution is Hölder continuous in D.     

Global Mild Solution of Stochastic Generalized Navier–Stokes Equations with Coriolis Force

In this paper, we apply Littlewood–Paley theory and Itô integral to get the global existence of stochastic Navier–Stokes equations with Coriolis force in Fourier–Besov spaces. As a comparison, we also give corresponding results of the deterministic Navier–Stokes equations with Coriolis force.