Leray Weak Solutions Research Articles

Scaling-invariant Serrin criterion via one velocity component for the Navier–Stokes equations

The classical Ladyzhenskaya–Prodi–Serrin regularity criterion states that if the Leray weak solution u of the Navier–Stokes equations satisfies u\in L^{q}(0,T; L^{p}(\mathbb{R}^{3})) with \frac{2}{q}+\frac{3}{p}\leq 1 , p>3 , then it is regular in \mathbb{R}^{3}\times (0,T) . In this paper, we prove that the Leray weak solution is also regular in \mathbb{R}^{3}\times (0,T) under the scaling-invariant Serrin condition imposed on one component of the velocity, i.e., u_{3}\in L^{q,1}(0,T; L^{p}(\mathbb{R}^{3})) with \frac{2}{q}+\frac{3}{p}\leq 1 , 3<p<+\infty . This result means that if the solution blows up at a time, then all three components of the velocity have to blow up simultaneously.

On the Inertial Range Bounds of K-41-like Magnetohydrodynamics Turbulence.

The spectral slope of magnetohydrodynamic (MHD) turbulence varies depending on the spectral theory considered; is the spectral slope in Kraichnan–Iroshnikov–Dobrowolny (KID) theory, in Marsch–Matthaeus–Zhou and Goldreich–Sridhar theories, also called Kolmogorov-like (K-41-like) MHD theory, the combination of the and scales in Biskamp, and so on. A rigorous mathematical proof to any of these spectral theories is of great scientific interest. Motivated by the 2012 work of A. Biryuk and W. Craig (Physica D 241(2012) 426–438), we establish inertial range bounds for K-41-like phenomenon in MHD turbulent flow through a mathematical rigor; a range of wave numbers in which the spectral slope of MHD turbulence is proportional to is established and the upper and lower bounds of this range are explicitly formulated. We also have shown that the Leray weak solution of the standard MHD model is bonded in the Fourier space, the spectral energy of the system is bounded and its average over time decreases in time.

Smooth Controllability of the Navier–Stokes Equation with Navier Conditions: Application to Lagrangian Controllability

We deal with the 3D Navier–Stokes equation in a smooth simply connected bounded domain, with controls on a non-empty open part of the boundary and a Navier slip-with-friction boundary condition on the remaining, uncontrolled, part of the boundary. We extend the small-time global exact null controllability result in Coron et al. (J Eur Math Soc 22:1625–1673, 2020) from Leray weak solutions to the case of smooth solutions. Our strategy relies on a refinement of the method of well-prepared dissipation of the viscous boundary layers which appear near the uncontrolled part of the boundary, which allows to handle the multi-scale features in a finer topology. As a byproduct of our analysis we also obtain a small-time global approximate Lagrangian controllability result, extending to the case of the Navier–Stokes equations the recent results (Glass and Horsin in J Math Pures Appl (9) 93:61–90, 2010; Glass and Horsin in SIAM J Control Optim 50: 2726–2742, 2012; Horsin and Kavian in ESAIM Control Optim Calc Var 23:1179–1200, 2017) in the case of the Euler equations and the result (Glass and Horsin in ESAIM Control Optim Calc Var 22:1040–1053, 2016) in the case of the steady Stokes equations.

Locally Space-Time Anisotropic Regularity Criteria for the Navier–Stokes Equations in Terms of Two Vorticity Components

In this paper we prove the regularity of Leray weak solutions of the Navier–Stokes equations as long as the vorticity projection to a plane is bounded in the scale critical space $$L^p(0,T;L^q)$$ , $$2/p+3/q=2$$ , $$q \in (3/2,\infty )$$ . The plane may vary in space and time while the unit vector $$v=v(x,t)$$ orthogonal to the plane is locally a Holder function in space with the coefficient 1/2. This extends previous works by Chae and Choe and by Miller. We further show that a generalized form of this criterion improves several other regularity criteria in terms of the vorticity direction known from the literature.

Oscillations and integrability of the vorticity in the 3D NS flows

In the studies of the Navier-Stokes (NS) regularity problem, it has become increasingly clear that a more realistic path to improved a priori bounds is to try to break away from the scaling of the energy-level estimates in the realm of the blow-up-type arguments (the solution in view is regular/smooth up to the possible blow-up time) rather than to try to improve regularity of arbitrary Leray's weak solutions. The present article is a contribution in this direction; more precisely, it is shown--in the context of an algebraic/polynomial-type blow-up profile of arbitrary degree--that a very weak condition on the vorticity direction field (membership in a local $bmo$ space weighted with arbitrary many logarithms)--suffices to break the energy-level scaling in the bounds on the vorticity. At the same time, the obtained bounds transform a 3D NS criticality scenario depicted by the macro-scale long vortex filaments into a no-singularity scenario.

Parabolic equations with divergence-free drift in space L_{t}^{l}L_{x}^{q}

In this paper, we study the fundamental solution $\varGamma(t,x;\tau,\xi)$ of the parabolic operator $L_t=\partial_t-\Delta+b(t,x)\cdot\nabla$, where for every $t$, $b(t,\cdot)$ is a divergence-free vector field; and we consider the case that $b$ belongs to the Lebesgue space $L^{\ell}(0,T;L^q(\mathbb{R}^n))$. The regularity of weak solutions to the parabolic equation $L_tu=0$ depends critically on the value of the parabolic exponent $\gamma=2/\ell+n/q$. Without the divergence-free condition on $b$, the regularity of weak solutions has been established when $\gamma\leq1$, and the heat kernel estimate has been obtained as well, except for the case that $\ell=\infty$, $q=n$. The regularity of weak solutions was deemed not true for the critical case $L^{\infty}(0,T;L^n(\mathbb{R}^n))$ for a general $b$, while it is true for the divergence-free case, and a written proof can be deduced from the results in [Yu. A. Semenov, \emph{Regularity theorems for parabolic equations}, J. Funct. Anal. \textbf{231} (2006), no. 2, 375--417]. One of the results obtained in the present paper establishes the Aronson-type estimate for critical and supercritical cases and for vector fields $b$ which are divergence free. We will prove the best possible lower and upper bounds for the fundamental solution one can derive under the current approach. The significance of the divergence-free condition has entered the study of parabolic equations rather recently, mainly because of the discovery of the compensated compactness. The interest for the study of such parabolic equations comes from its connections with Leray's weak solutions of the Navier-Stokes equations and the Taylor diffusion associated with a vector field where the heat operator $L_t$ appears naturally.

Will mathematicians unleash the power of the Navier-Stokes equations?

Since 1934 when Leray proved the global existence of finite energy weak solutions to 3D incompressible Navier-Stokes equations, the regularity and uniqueness of Leray weak solutions has been core in the mathematical theory of incompressible fluid mechanics. This problem is also closely related to the Millionaires problem proposed by Clay Institute, namely, the global regularity or singularity to finite energy solutions of 3D incompressible Navier-Stokes system. The goal of this article is first to introduce the problem and then to survey some progresses on this problem.In Section 1 , following Fefferman, we first present the Millionaires problem concerning 3D incompressible Navier-Stokes equations.We also present the result of Leray on the global existence of finite energy weak solutions to the system. We conclude the introduction byremarking that the reason why we cannot solve the global regularity of 3D Navier-Stokes is that the system is energy super-critical. Whereas the energy lawis so far the only conservation law which we can find for 3D Navier-Stokes system.The goal of Section 2 is to present the main approaches toward the regularity of 3D Navier-Stokes system, namely, partial regularityof suitable weak solutions to 3D Navier-Stokes system, conditional regularity of Leray solutions, and well-posedness of 3D Navier-Stokes system with initial datain the critical spaces.Finally, in the last section, we comment about the possible singularity of 3D Navier-Stokes system.

The optimal upper and lower bounds of convergence rates for the 3D Navier–Stokes equations under large initial perturbation

This paper is concerned with the optimal algebraic convergence rates for Leray weak solutions of the 3D Navier–Stokes equations in Morrey space. It is shown that if the global Leray weak solution u(x,t) of the 3D Navier–Stokes equations satisfies∇u∈Lr(0,∞;M˙p,q(R3)),2r+3p=2,32<p<∞,p≥q>2, then even for the large initial perturbation, every weak solution v(x,t) of the perturbed Navier–Stokes equations converges algebraically to u(x,t) with the optimal upper and lower boundsC1(1+t)−γ2≤‖v(t)−u(t)‖L2≤C2(1+t)−γ2,for large t>1,2<γ<52. The findings are mainly based on the developed Fourier splitting methods and iterative process.

The Incompressible Navier–Stokes Equations on Non-compact Manifolds

We shall prove dispersive and smoothing estimates for Bochner type laplacians on some non-compact Riemannian manifolds with negative Ricci curvature, in particular on hyperbolic spaces. These estimates will be used to prove Fujita-Kato type theorems for the incompressible Navier-Stokes equations. We shall also discuss the uniqueness of Leray weak solutions in the two dimensional case.

Stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations

This paper is devoted to the investigation of stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations. For a Leray weak solution of the Navier–Stokes equations in a critical Besov space, it is shown that the Leray weak solution is uniformly stable with respect to a small perturbation of initial velocity and external forcing. If the perturbation is not small, the perturbed weak solution converges asymptotically to the original weak solution as the time tends to the infinity. Additionally, an energy equality and weak–strong uniqueness for the three-dimensional Navier–Stokes equations are derived. The findings are mainly based on the estimations of the nonlinear term of the Navier–Stokes equations in a Besov space framework, the use of special test functions and the energy estimate method.

Regimes of nonlinear depletion and regularity in the 3D Navier–Stokes equations

The periodic 3D Navier–Stokes equations are analyzed in terms of dimensionless, scaled, L2m-norms of vorticity Dm (1 ⩽ m < ∞). The first in this hierarchy, D1, is the global enstrophy. Three regimes naturally occur in the D1 − Dm plane. Solutions in the first regime, which lie between two concave curves, are shown to be regular, owing to strong nonlinear depletion. Moreover, numerical experiments have suggested, so far, that all dynamics lie in this heavily depleted regime []; new numerical evidence for this is presented. Estimates for the dimension of a global attractor and a corresponding inertial range are given for this regime. However, two more regimes can theoretically exist. In the second, which lies between the upper concave curve and a line, the depletion is insufficient to regularize solutions, so no more than Leray's weak solutions exist. In the third, which lies above this line, solutions are regular, but correspond to extreme initial conditions. The paper ends with a discussion on the possibility of transition between these regimes.

A Note on the Regularity Criterion of Weak Solutions of Navier‐Stokes Equations in Lorentz Space

This paper is concerned with the regularity of Leray weak solutions to the 3D Navier‐Stokes equations in Lorentz space. It is proved that the weak solution is regular if the horizontal velocity denoted by satisfies The result is obvious and improved that of Dong and Chen (2008) on the Lebesgue space.

Regularity criterion for weak solutions to the Navier–Stokes equations in terms of the pressure in the class [formula omitted

The goal of this paper is to establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier–Stokes equations in R 3 . It is proved that if the pressure belongs to L 2 ( 0 , T ; B . ∞ , ∞ − 1 ( R 3 ) ) , then the weak solution is actually regular. This improves a result established in a recent work by the second author Gala (2011) [19].

Remarks on the Regularity Criteria of Three-dimensional Navier-Stokes Equations in Margin Case

In the study of the regularity criteria for Leray weak solutions to threedimensional Navier-Stokes equations, two sufficient conditions such that the horizontal velocity ũ satisfies ũ∈ L2(0,T;BMO(R)) or ũ∈ L2/1+r(0,T; Ḃr ∞,∞(R )) for 0< r< 1 are considered. AMS Subject Classifications: 35Q35 Chinese Library Classifications: O175.24

Remark on a regularity criterion in terms of pressure for the Navier-Stokes equations

In this note we establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier-Stokes equation in R d \mathbb {R}^{d} . It is known that if a Leray weak solution u u belongs to L 2 1 − r ( ( 0 , T ) ; L d r ) \ \ for some \ \ 0 ≤ r ≤ 1 , \begin{equation} L^{\frac {2}{1-r}}\left ( \left ( 0,T\right ) ;L^{\frac {d}{r}}\right ) \text { \ \ for some \ \ }0\leq r\leq 1, \end{equation} then u u is regular. It is proved that if the pressure p p associated to a Leray weak solution u u belongs to L 2 2 − r ( ( 0 , T ) ; M . 2 , d r ( R d ) d ) , \begin{equation} L^{\frac {2}{2-r}}\left ( \left ( 0,T\right ) ;\overset {.}{\mathcal {M}}_{2,\frac { d}{r}}\left ( \mathbb {R}^{d}\right ) ^{d}\right ) , \end{equation} where M . 2 , d r ( R d ) \overset {.}{\mathcal {M}}_{2,\frac {d}{r}}\left ( \mathbb {R}^{d}\right ) is the critical Morrey-Campanato space (a definition is given in the text) for 0 &gt; r &gt; 1 0&gt;r&gt;1 , then the weak solution is actually regular. Since this space M . 2 , d r \overset {.}{\mathcal {M}}_{2,\frac {d}{r}} is wider than L d r L^{\frac {d}{r}} and X . r \overset {.}{X}_{r} , the above regularity criterion (0.2) is an improvement of Zhou’s result.

Regularity Criteria in Terms of the Pressure for the Navier–Stokes Equations in the Critical Morrey–Campanato Space

In this paper, we establish a Serrin-type regularity criterion in terms of the pressure for Leray weak solutions to the Navier–Stokes equation in ℝ^3 . It is proved that the solution is regular if the associate pressure satifies p \in L^{\frac 2{2-r}} ((0, T); \dot{\mathcal M}_{2,\frac 3 r} (ℝ^3)) \quad \text{or}\quad ∇p \in L^{\frac 2{3-r}} ((0, T); \dot{\mathcal M}_{2,\frac 3 r} (ℝ^3)) for 0 &lt; r &lt; 1 , where \dot{\mathcal M}_{2,\frac 3 r} is the critical Morrey–Campanto space. Regularity criteria for the 3D MHD equations are also given.

Leray weak solutions of the incompressible Navier Stokes system on exterior domains via the artificial compressibility method

In this paper we study the Leray weak solutions of the incompressible Navier Stokes equation in an exterior domain.We describe, in particular, an hyperbolic version of the so called artificial compressibility method investigated by J.L.Lions and Temam. The convergence of these type of approximation show in general a lack of strong convergence due to the presence of acoustic waves. In this paper we face this difficulty by taking care of the dispersive nature of these waves by means of the Strichartz estimates or waves equations satisfied by the pressure. We actually decompose the pressure in different acoustic components, each one of them satisfies a specific initial boundary value problem. The strong convergence analysis of the velocity field will be achieved by using the associated Leray-Hodge decomposition.

A note on the regularity criterion in terms of pressure for the Navier–Stokes equations

In this note we establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier–Stokes equation in R d . Here we call u a Leray weak solution if u is a weak solution of finite energy, i.e. u ∈ L ∞ ( ( 0 , T ) ; L 2 ) ∩ L 2 ( ( 0 , T ) ; H . 1 ) . It is known that if a Leray weak solution u belongs to (0.1) L 2 1 − r ( ( 0 , T ) ; L d r ) for some 0 ≤ r ≤ 1 , then u is regular (see [J. Serrin, On the interior regularity of weak solutions of the Navier–Stokes equations, Arch. Ration. Mech. Anal. 9 (1962) 187–195]). We succeed in proving the regularity of the Leray weak solution u in terms of pressure under the condition (0.2) p ∈ L 2 2 − r ( ( 0 , T ) ; X . r ( R d ) d ) , where X . r ( R d ) is the multiplier space (a definition is given in the text) for 0 ≤ r ≤ 1 . Since this space X . r is wider than L d r , the above regularity criterion (0.2) is an improvement on Zhou’s result [Y. Zhou, On regularity criteria in terms of pressure for the Navier–Stokes equations in R 3 , Proc. Amer. Math. Soc. 134 (2006) 149–156].

A quasineutral type limit for the Navier–Stokes–Poisson system with large data

In this paper we investigate a quasineutral type limit for the Navier–Stokes–Poisson system. We prove that the projection of the approximating velocity fields on the divergence-free vector field is relatively compact and converges to a Leray weak solution of the incompressible Navier–Stokes equation. By exploiting the wave equation structure of the density fluctuation we achieve the convergence of the approximating sequences by means of a dispersive estimate of the Strichartz type.

A DISPERSIVE APPROACH TO THE ARTIFICIAL COMPRESSIBILITY APPROXIMATIONS OF THE NAVIER–STOKES EQUATIONS IN 3D

In this paper we study how to approximate the Leray weak solutions of the incompressible Navier–Stokes equations. In particular we describe an hyperbolic version of the so-called artificial compressibility method investigated by J. L. Lions and Temam. By exploiting the wave equation structure of the pressure of the approximating system we achieve the convergence of the approximating sequences by means of dispersive estimates of Strichartz type. We prove that the projection of the approximating velocity fields on the divergence free vectors is relatively compact and converges to a Leray weak solution of the incompressible Navier–Stokes equation.