Long Time Behavior of Weak Solutions to Navier–Stokes–Poisson System

Ducomet et?al. (Discrete Contin Dyn Syst 11(1): 113?C130, 2004) showed the existence of global weak solutions to the Navier?CStokes?CPoisson system. We study the global behavior of such a solution. This is done by (1) proving uniqueness of a solution to the stationary system; (2) by showing convergence of a weak solution to the stationary solution. In (1) we consider only the case with repulsion. We prove our result in the case of a bounded domain with smooth boundary in ${\mathbb{R}^3}$ and also in the case of the whole space ${\mathbb{R}^3}$ .     

On the Local Smoothness of Solutions of the Navier–Stokes Equations

We consider the Cauchy problem for incompressible Navier–Stokes equations with initial data in , and study in some detail the smoothing effect of the equation. We prove that for T < ∞ and for any positive integers n and m we have , as long as stays finite.     

On the Characterization of the Navier–Stokes Flows with the Power-Like Energy Decay

We study the energy decay of the turbulent solutions to the Navier–Stokes equations in the whole three-dimensional space. We show as the main result that the solutions with the energy decreasing at the rate \({O(t^{-\alpha}), t \rightarrow \infty, \alpha \in [0, 5/2]}\) , are exactly characterized by their initial conditions belonging into the homogeneous Besov space \({\dot{B}^{-\alpha}_{2, \infty}}\) . Similarly, for a solution u and \({p \in [1, \infty]}\) the integral \({\int_{0}^{\infty} \|t^{\alpha/2} u(t)\|^p \frac{1}{t} dt}\) is finite if and only if the initial condition of u belongs to the homogeneous Besov space \({\dot{B}_{2, p}^{-\alpha}}\) . For the case \({\alpha \in (5/2, 9/2]}\) we present analogical results for some subclasses of turbulent solutions.     

Time-Periodic Solutions to the Full Navier–Stokes–Fourier System

We consider the full Navier–Stokes–Fourier system describing the motion of a compressible viscous and heat conducting fluid driven by a time-periodic external force. We show the existence of at least one weak time periodic solution to the problem under the basic hypothesis that the system is allowed to dissipate the thermal energy through the boundary. Such a condition is in fact necessary, as energetically closed fluid systems do not possess non-trivial (changing in time) periodic solutions as a direct consequence of the Second law of thermodynamics.     

Asymptotic Stability of Landau Solutions to Navier–Stokes System

It is known that the three-dimensional Navier–Stokes system for an incompressible fluid in the whole space has a one parameter family of explicit stationary solutions that are axisymmetric and homogeneous of degree −1. We show that these solutions are asymptotically stable under any L ²-perturbation.     

Maximal Regularity of the Time - Periodic Linearized Navier–Stokes System

Time-periodic solutions to the linearized Navier–Stokes system in the n-dimensional whole-space are investigated. For time-periodic data in L ^q -spaces, maximal regularity and corresponding a priori estimates for the associated time-periodic solutions are established. More specifically, a Banach space of time-periodic vector fields is identified with the property that the linearized Navier–Stokes operator maps this space homeomorphically onto the L ^q -space of time-periodic data.     

On the Regularity Conditions of Suitable Weak Solutions of the 3D Navier–Stokes Equations

Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier–Stokes equations in a space-time domain containing z₀=(x₀, t₀)z_{0}=(x_{0}, t_{0}), and let Q_z0,r = B_x0,r ×(t₀ -r², t₀)Q_{z_{0},r}= B_{x_{0},r} \times (t_{0} -r^{2}, t_{0}) be a parabolic cylinder in the domain. We show that if either $\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r})$\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r}) with $\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r})$\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r}) with \frac3g + \frac2a ￡ 2\frac{3}{\gamma} + \frac{2}{\alpha} \leq 2, where L^{γ, α}_x,t denotes the Serrin type of class, then z₀ is a regular point for ν. This refines previous local regularity criteria for the suitable weak solutions.     

On Spatial Decay Estimates for Derivatives of Vorticities with Application to Large Time Behavior of the Two-Dimensional Navier–Stokes Flow

In this paper we establish spatial decay estimates for derivatives of vorticities solving the two-dimensional vorticity equations equivalent to the Navier–Stokes equations. As an application we derive asymptotic behaviors of derivatives of vorticities at time infinity. It is well known by now that the vorticity behaves asymptotically as the Oseen vortex provided that the initial vorticity is integrable. We show that each derivative of the vorticity also behaves asymptotically as that of the Oseen vortex.      

Fine Properties of Self-Similar Solutions of the Navier–Stokes Equations

We study the solutions of the nonstationary incompressible Navier–Stokes equations in , of self-similar form , obtained from small and homogeneous initial data a(x). We construct an explicit asymptotic formula relating the self-similar profile U(x) of the velocity field to its corresponding initial datum a(x).     

A Regularity Criterion for the Weak Solutions to the Navier–Stokes–Fourier System

We show that any weak solution to the full Navier–Stokes–Fourier system emanating from the data belonging to the Sobolev space W ^3,2 remains regular as long as the velocity gradient is bounded. The proof is based on the weak-strong uniqueness property and parabolic a priori estimates for the local strong solutions.     

Existence of Solutions with the Prescribed Flux of the Navier–Stokes System in an Infinite Cylinder

The existence and uniqueness of a solution to the nonstationary Navier–Stokes system having a prescribed flux in an infinite cylinder is proved. We assume that the initial data and the external forces do not depend on x₃ and find the solution (u, p) having the following form

On the Stability in Weak Topology of the Set of Global Solutions to the Navier–Stokes Equations

Let X be a suitable function space and let ${\mathcal{G} \subset X}$ be the set of divergence free vector fields generating a global, smooth solution to the incompressible, homogeneous three-dimensional Navier–Stokes equations. We prove that a sequence of divergence free vector fields converging in the sense of distributions to an element of ${\mathcal{G}}$ belongs to ${\mathcal{G}}$ if n is large enough, provided the convergence holds “anisotropically” in frequency space. Typically, this excludes self-similar type convergence. Anisotropy appears as an important qualitative feature in the analysis of the Navier–Stokes equations; it is also shown that initial data which do not belong to ${\mathcal{G}}$ (hence which produce a solution blowing up in finite time) cannot have a strong anisotropy in their frequency support.     

On the Static-Limit Solutions to the Navier—Stokes Equations of Compressible Flow

We consider the zero-velocity stationary problem of the Navier--Stokes equations of compressible isentropic flow describing the distribution of the density r \varrho of a fluid in a spatial domain W ì R^N \Omega \subset {\rm R}^N driven by a time-independent potential external force [(f)\vec] = \triangledown F \vec f = \triangledown F . We study the structure of the set of all solutions to the stationary problem having a prescribed mass m > 0 and a prescribed energy. Cardinality of the solution set depends on m and it is either continuum or at most two. Conditions on m for distinguishing these cases have been found. Uniqueness for the stationary system is also studied.     

Fast Decays of Strong Global Solutions of the Navier–Stokes Equations

In the paper we study the asymptotic dynamics of strong global solutions of the Navier Stokes equations. We are concerned with the question whether or not a strong global solution w can pass through arbitrarily large fast decays. Avoiding results on higher regularity of w used in other papers we prove as the main result that for the case of homogeneous Navier–Stokes equations the answer is negative: If [0, 1/4) and δ₀ > 0, then the quotient remains bounded for all t ≥ 0 and δ∈[0, δ₀]. This result is not valid for the non-homogeneous case. We present an example of a strong global solution w of the non-homogeneous Navier–Stokes equations, where the exterior force f decreases very quickly to zero for while w passes infinitely often through stages of arbitrarily large fast decays. Nevertheless, we show that for the non-homogeneous case arbitrarily large fast decays (measured in the norm cannot occur at the time t in which the norm is greater than a given positive number.      

A Multi-Fluid Compressible System as the Limit of Weak Solutions of the Isentropic Compressible Navier–Stokes Equations

This paper mainly concerns the mathematical justification of a viscous compressible multi-fluid model linked to the Baer-Nunziato model used by engineers, see for instance Ishii (Thermo-fluid dynamic theory of two-phase flow, Eyrolles, Paris, 1975), under a “stratification” assumption. More precisely, we show that some approximate finite-energy weak solutions of the isentropic compressible Navier–Stokes equations converge, on a short time interval, to the strong solution of this viscous compressible multi-fluid model, provided the initial density sequence is uniformly bounded with corresponding Young measures which are linear convex combinations of m Dirac measures. To the authors’ knowledge, this provides, in the multidimensional in space case, a first positive answer to an open question, see Hillairet (J Math Fluid Mech 9:343–376, 2007), with a stratification assumption. The proof is based on the weak solutions constructed by Desjardins (Commun Partial Differ Equ 22(5–6):977–1008, 1997) and on the existence and uniqueness of a local strong solution for the multi-fluid model established by Hillairet assuming initial density to be far from vacuum. In a first step, adapting the ideas from Hoff and Santos (Arch Ration Mech Anal 188:509–543, 2008), we prove that the sequence of weak solutions built by Desjardins has extra regularity linked to the divergence of the velocity without any relation assumption between λ and μ. Coupled with the uniform bound of the density property, this allows us to use appropriate defect measures and their nice properties introduced and proved by Hillairet (Aspects interactifs de la m’ecanique des fluides, PhD Thesis, ENS Lyon, 2005) in order to prove that the Young measure associated to the weak limit is the convex combination of m Dirac measures. Finally, under a non-degeneracy assumption of this combination (“stratification” assumption), this provides a multi-fluid system. Using a weak–strong uniqueness argument, we prove that this convex combination is the one corresponding to the strong solution of the multi-fluid model built by Hillairet, if initial data are equal. We will briefly discuss this assumption. To complete the paper, we also present a blow-up criterion for this multi-fluid system following (Huang et al. in Serrin type criterion for the three-dimensional viscous compressible flows, arXiv, 2010).     

On the Existence of Periodic Solutions of the Navier–Stokes Equations in a Thin Domain Using the Topological Degree

We use the method of the topological degree, the theory of fractional powers of positive operators, and the Grisvard formula together with results proved by G. Raugel and G. R. Sell to study the periodic solutions of the incompressible Navier–Stokes equations in a thin three-dimensional domain.     

On the Rate of Decay of the Oseen Semigroup in Exterior Domains and its Application to Navier–Stokes Equation

We prove L_p-L_q estimates of the Oseen semigroup in n-dimensional exterior domains which refine and improve those obtained by Kobayashi and Shibata [15]. As an application, we give a globally in time stability theory for the stationary Navier–Stokes flow whose velocity at infinity is a non-zero constant vector. We thus extend the result of Shibata [21]. In particular, we find an optimal rate of convergence of solutions of the non-stationary problem to those of the corresponding stationary problem.