Stability to the global large solutions of 3-D Navier-Stokes equations

In this paper, we consider the stability to the global large solutions of 3-D incompressible Navier-Stokes equations in the anisotropic Sobolev spaces. In particular, we proved that for any , given a global large solution v∈C([0,∞);H0,s₀(R³)∩L³(R³)) of (1.1) with and a divergence free vector satisfying for some sufficiently small constant depending on , v, and , (1.1) supplemented with initial data v(0)+w₀ has a unique global solution in u∈C([0,∞);H0,s₀(R³)) with ∇u∈L²(R⁺,H0,s₀(R³)). Furthermore, uh is close enough to vh in C([0,∞);H0,s(R³)).     

Helically symmetric solutions to the 3-D Navier-Stokes equations for compressible isentropic fluids

We prove the existence of global weak solutions to the Navier-Stokes equations for compressible isentropic fluids for any γ>1 when the Cauchy data are helically symmetric, where the constant γ is the specific heat ratio. Moreover, a new integrability estimate of the density in any neighborhood of the symmetry axis (the singularity axis) is obtained.     

Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations

For stochastic Navier-Stokes equations in a 3-dimensional bounded domain we first show that if the initial value is sufficiently regular, then martingale solutions are strong on a random time interval and we estimate its length. Then we prove the uniqueness of the strong solution in the class of all martingale solutions. Received November 15, 1995     

Incompressible limit of solutions of multidimensional steady compressible Euler equations

A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity. Another observation is that the incompressibility of the limit for the homentropic Euler flow is directly from the continuity equation, while the incompressibility of the limit for the full Euler flow is from a combination of all the Euler equations. As direct applications of the compactness framework, we establish two incompressible limit theorems for multidimensional steady Euler flows through infinitely long nozzles, which lead to two new existence theorems for the corresponding problems for multidimensional steady incompressible Euler equations.     

Large global solutions to 3-D inhomogeneous Navier-Stokes equations slowly varying in one variable

Motivated by Chemin and Gallagher (2010) [8], we consider the global wellposedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with large initial velocity slowly varying in one space variable. In particular, we proved that when the initial density is close enough to a positive constant, then given divergence free initial velocity field of the type , as that in Chemin and Gallagher (2010) [8] for the classical Navier-Stokes system, we shall prove the global wellposedness of (INS) for ? sufficiently small. The main difficulty here lies in the fact that we will have to obtain the L¹(R⁺;Lip(R³)) estimate for convection velocity in the transport equation of (INS). Toward this and due to the strong anisotropic properties of the approximate solutions, we will have to work in the framework of anisotropic type Besov spaces here.     

Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations

In this paper, we study the zero dissipation limit problem for the one-dimensional compressible Navier-Stokes equations. We prove that if the solution of the inviscid Euler equations is piecewise constants with a contact discontinuity, then there exist smooth solutions to the Navier-Stokes equations which converge to the inviscid solution away from the contact discontinuity at a rate of as the heat-conductivity coefficient κ tends to zero, provided that the viscosity μ is of higher order than the heat-conductivity κ. Without loss of generality, we set μ≡0. Here we have no need to restrict the strength of the contact discontinuity to be small.     

On the stationary solutions of the full compressible Navier-Stokes equations and its stability with respect to initial disturbance

This paper is concerned with the existence, uniqueness and nonlinear stability of stationary solutions to the Cauchy problem of the full compressible Navier-Stokes equations effected by external force of general form in R³.     

Existence and decay of solutions in full space to Navier-Stokes equations with delays

We consider the Navier-Stokes equations with delays in Rⁿ,2≤n≤4. We prove existence of weak solutions when the external forces contain some hereditary characteristics and uniqueness when n=2. Moreover, if the external forces satisfy a time decay condition we show that the solution decays at an algebraic rate.     

Incompressible limits of the Navier-Stokes equations for all time

We study the asymptotic behaviors of the regular solutions to the compressible Navier-Stokes equations for “well-prepared” initial data for all time as the Mach number tends to zero, by deriving a differential inequality with certain decay property. The estimates obtained in this paper are uniform both in time and Mach number.     

Blow-up of critical norms for the 3-D Navier-Stokes equations

Let u =(uh, u3) be a smooth solution of the 3-D Navier-Stokes equations in R3× [0, T). It was proved that if u3 ∈ L∞(0, T;˙B-1+3/p p,q(R3)) for 3 p, q ∞ and uh∈ L∞(0, T; BMO-1(R3)) with uh(T) ∈ VMO-1(R3), then u can be extended beyond T. This result generalizes the recent result proved by Gallagher et al.(2016), which requires u ∈ L∞(0, T;˙B-1+3/pp,q(R3)). Our proof is based on a new interior regularity criterion in terms of one velocity component, which is independent of interest.     

The zero-velocity limit solutions of the Navier-Stokes equations of compressible fluid revisited

For an isentropic fluid flow driven by a potential force ∇F, the set of the socalled rest states (zero-velocity stationary solutions) is investigated. A sufficient condition is found, in terms of the potentialF, for the problem to possess at most two distinct solutions with given mass and total energy.

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Sunto Com'è noto, per un fluido isentropico viscoso, soggetto ad una forza esterna di tipo potenzialeF, solo la quiete è possibile. La struttura topologica dell’insieme di tutte le soluzioni di quiete è studiata come l’ω-limite delle traiettorie. è provato che se gli insiemi di livello superiore diF sono connessi, edF è localmente Lipschitziana, allora la soluzione (densità) è univocamente determinata dalla sua massa. Inoltre, se l’energia potenzialee è assegnata, la condizione geometrica sulle superficie di livello può essere rilassata.

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The work was supported by Grant 201/98/1450 GA ČR and Grant A1019703 GA AV ČR.     

Uniqueness for some Leray-Hopf solutions to the Navier-Stokes equations

We exhibit simple sufficient conditions which give weak-strong uniqueness for the 3D Navier-Stokes equations. The main tools are trilinear estimates and energy inequalities. We then apply our result to the framework of Lorentz, Morrey and Besov over Morrey spaces so as to get new weak-strong uniqueness classes and so uniqueness classes for solutions in the Leray-Hopf class. In the last section, we give a uniqueness and regularity result. We obtain new uniqueness classes for solutions in the Leray-Hopf class without energy inequalities but sufficiently regular.     

Analytical solutions to the Navier-Stokes equations for non-Newtonian fluid

The pressureless Navier-Stokes equations for non-Newtonian fluid are studied. The analytical solutions with arbitrary time blowup, in radial symmetry, are constructed in this paper. With the previous results for the analytical blowup solutions of the N-dimensional （N ≥ 2） Navier-Stokes equations, we extend the similar structure to construct an analytical family of solutions for the pressureless Navier-Stokes equations with a normal viscosity term （μ（ρ）｜ u｜^α u）.     

A regularity criterion for the solutions of 3D Navier-Stokes equations

In terms of two partial derivatives of any two components of velocity fields, we give a new criterion for the regularity of solutions of the Navier-Stokes equation in R³. More precisely, let u=(u¹,u²,u³) be a weak solution in (0,T)×R³. Then u becomes a classical solution if any two functions of _∂1u¹, _∂2u² and _∂3u³ belong to Lθ(0,T;Lr(R³)) provided with , .     

Martingale and stationary solutions for stochastic Navier-Stokes equations

Summary We prove the existence of martingale solutions and of stationary solutions of stochastic Navier-Stokes equations under very general hypotheses on the diffusion term. The stationary martingale solutions yield the existence of invariant measures, when the transition semigroup is well defined. The results are obtained by a new method of compactness.     

On the uniqueness and stability of solutions of extremal problems for the stationary Navier-Stokes equations

We consider inverse extremal problems for the stationary Navier-Stokes equations. In these problems, one seeks an unknown vector function occurring in the Dirichlet boundary condition for the velocity and the solution of the considered boundary value problem on the basis of the minimization of some performance functional. We derive new a priori estimates for the solutions of the considered extremal problems and use them to prove theorems of the local uniqueness and stability of solutions for specific performance functionals.