Mild criticality breaking for the Navier-Stokes equations
Tobias Barker, Christophe Prange
备注：
13 pages. The current version contains an additional theorem regarding the behavior of slightly supercritical Orlicz norms near a potential blow-up time
In this short paper we prove the global regularity of solutions to the Navier-Stokes equations under the assumption that slightly supercritical quantities are bounded. As a consequence, we prove that if a solution $u$ to the Navier-Stokes equations blows-up, then certain slightly supercritical Orlicz norms must become unbounded. This partially answers a conjecture recently made by Terence Tao. The proof relies on quantitative regularity estimates at the critical level and transfer of subcritical information on the initial data to arbitrarily large times. This method is inspired by a recent paper of Aynur Bulut, where similar results are proved for energy supercritical nonlinear Schrödinger equations.
The present paper deals with existence and uniqueness of global mild solutions for the 2D Navier-Stokes equations with impulses. Using the framework of nonautonomous dynamical systems, we extend previous results considering the 2D Navier-Stokes equations with impulse effects and allowing that the nonlinear terms are explicitly time-dependent. Additionally, we present sufficient conditions to obtain dissipativity (boundedness) for solutions starting in bounded sets.
In this paper, we present the result of maximum regularity of the mild solution of the fractional Cauchy problem. As our main result, we investigate the uniqueness of mild solutions for time-fractional Navier-Stokes equations in class $C\left([0,\infty);L^{N}\left( \mathbb{R}^{N}\right)^{N}\right)$ by means of the estimates $L^{p}-L^{q}$ of Giga-Shor inequality and the Gronwall inequality.
This paper investigates the Cauchy problem of the time-space fractional Keller-Segel-Navier- Stokes model, which can describe both memory effect and Lévy process of the system. The local existence and global existence in Lebesgue space are obtained by means of Banach fixed point theorem and Banach implicit function theorem, respectively. In addition, the regularities of local and global mild solutions are improved in fractional homogeneous Sobolev spaces. Furthermore, some properties of mild solutions including mass conservation, decay estimates, stability and self-similarity are established.
In this work, we consider time-fractional Navier-Stokes equations (NSE) with the external forces involving finite delay. Equations are considered on a bounded domain in 3-D space having sufficiently smooth boundary. We transform the system of equations (NSE) to an abstract Cauchy problem and then investigate local existence and uniqueness of the mild solutions. In particular, with some suitable condition on initial datum we establish the global continuation and regularity of the mild solutions. We use semigroup theory, tools of fractional calculus and Banach contraction mapping principle to es…
We first prove decay estimates and spacetime integral bounds for Stokes flows in amalgam spaces $E^r_q$ which connect the classical Lebesgue spaces to the spaces of uniformly locally $r$-integrable functions. Using these estimates, we construct mild solutions of the Navier-Stokes equations in the amalgam spaces satisfying the corresponding spacetime integral bounds. Time-global solutions are constructed for small data in $E^3_q$, $1\le q \le 3$. Our results provide new bounds for the strong solutions classically constructed by Kato and the more recent solutions in uniformly local spaces constr…
This paper is based on a formulation of the Navier-Stokes equations developed by Iyer and Constantin \cite{Cont} , where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. Our contribution is to establish this probabilistic representation formula for mild solutions of the Navier-Stokes equations on $\mathbb{R}^{d} $.
In this paper, we show the optimal decay rate estimates of the space-time derivatives and the joint space-time analyticity of solutions to the Navier-Stokes equations. As it is known from the Hartogs's theorem, for a complex function with two complex variables, the joint analyticity with respect to two variables can be derived from combining of analyticity with respect to each variable. However, as a function of two real variables for space and time, the joint space-time analyticity of solutions to the Navier-Stokes equations cannot be directly obtained from the combination of space analyticit…
Spacetime decay of mild solutions and conditional quantitative transfer of regularity of the incompressible Navier--Stokes Equations from $\mathbb{R}^n$ to bounded domains
We prove unique existence of mild solutions on $L^{\infty}_σ$ for the Navier-Stokes equations in an exterior domain in $ \mathbb{R}^{n}$, $n\geq 2$, subject to the non-slip boundary condition.
We demonstrate that the problem of existence of Leray self-similar blow up solutions in a generalized mild Navier-Stokes system with the fractional Laplacian $(-Δ)^{γ/2}$ can be stated as a fixed point problem for a "renormalization" operator. We proceed to construct {\it a-priori} bounds, that is a renormalization invariant precompact set in an appropriate weighted $L^p$-space. As a consequence of a-priori bounds, we prove existence of renormalization fixed points for $d \ge 2$ and $d<γ<2 d+2$, and existence of non-trivial Leray self-similar mild solutions in $C^\infty([0,T),(H^k)^d \cap (L^p…
We study mild solutions to the Navier-Stokes equation on the $n$-dimensional hyperbolic space $\mathbb{H}^n$, $n \geq 2$. We use dispersive and smoothing estimates proved by Pierfelice on a class of complete Riemannian manifolds to extend the Fujita-Kato theory of mild solutions from $\mathbb{R}^n$ to $\mathbb{H}^n$. This includes well-posedness results for $L^n$ initial data and $L^n \cap L^p$ initial data for $1 < p < n$, global in time results for small initial data, and time decay results for the $L^n$ and $L^p$ norms of both $u$ and $\nabla u$. Due to the additional exponential time decay…
In this note, boundary Type I blowups of suitable weak solutions to the Navier-Stokes equations are discussed. In particular, it has been shown that, under certain assumptions, the existence of non-trivial mild bounded ancient solutions in half space leads to the existence of suitable weak solutions with Type I blowup on the boundary.
In this paper, we consider the uniqueness of solutions to the 3d Navier-Stokes equations with initial vorticity given by $ω_0 = αe_z δ_{x = y = 0}$, where $δ_{x=y= 0}$ is the one dimensional Hausdorff measure of an infinite, vertical line and $α\in \mathbb R$ is an arbitrary circulation. This initial data corresponds to an idealized, infinite vortex filament. One smooth, mild solution is given by the self-similar Oseen vortex column, which coincides with the heat evolution. Previous work by Germain, Harrop-Griffiths, and the first author implies that this solution is unique within a class of m…
We show that the pressure associated with a distributional solution of the Navier-Stokes equations on the whole space satisfies a local expansion defined as a distribution if and only if the solution is mild. This gives a new perspective on Lemarié-Rieusset's "equivalence theorem." Here, the Leray projection operator composed with a gradient is defined without using the Littlewood-Paley decomposition. Prior sufficient conditions for the local expansion assumed spatial decay or estimates on the gradient and imply the considered solution is mild. An important tool is an explicit description of t…
This paper is concerned with the blowup criterion for mild solution to the incompressible Navier-Stokes equation in higher spatial dimensions $d \geq 4$. By establishing an $ε$ regularity criterion, we show that if the mild solution $u$ with initial data in $\dot B^{-1+d/p}_{p,q}(\mathbb{R}^d) $, $d<p,\,q<\infty$ becomes singular at a finite time $T_*$, then $$ \limsup_{t\to T_*} \|u(t)\|_{\dot B^{-1+d/p}_{p,q}(\mathbb{R}^d)} = \infty. $$ The corresponding result in 3D case has been obtained by I.Gallagher, G.S.KochandF.Planchon. As a by-product, we also prove a regularity criterion for the Le…
In this short survey paper, we focus on some new developments in the study of the regularity or potential singularity formation for solutions of the 3D Navier-Stokes equations. Some of the motivating questions are: Are certain norms accumulating/concentrating on small scales near potential blow-up times? At what speed do certain scale-invariant norms blow-up? Can one prove explicit quantitative regularity estimates? Can one break the criticality barrier, even slightly? We emphasize that these questions are closely linked together. Many recent advances for the Navier-Stokes equations are direct…
An old problem asks whether bounded mild ancient solutions of the 3 dimensional Navier-Stokes equations are constants. While the full 3 dimensional problem seems out of reach, in the works \cite{KNSS, SS09}, the authors expressed their belief that the following conjecture should be true. For incompressible axially-symmetric Navier-Stokes equations (ASNS) in three dimensions: \textit{bounded mild ancient solutions are constant}. Understanding of such solutions could play useful roles in the study of global regularity of solutions to the ASNS. In this article, we essentially prove this conjectur…
We establish a Liouville theorem for bounded mild ancient solutions to the axi-symmetric incompressible Navier-Stokes equations on $(-\infty, 0] \times (\mathbb{R}^2 \times \mathbb{T}^1)$. This is a step forward to completely solve the conjecture on $(-\infty, 0] \times \mathbb{R}^3$ which was made in \cite{KNSS} to describe the potential singularity structures of the Cauchy problem.
The initial value problem of the incompressible Navier-Stokes equations with non-zero forces in $L^{n,\infty}(\mathbb{R}^n)$ is investigated. Even though the Stokes semigroup is not strongly continuous on $L^{n,\infty}(\mathbb{R}^n)$, with the qualitative condition for the external forces, it is clarified that the mild solution of the Naiver-Stokes equations satisfies the differential equations in the topology of $L^{n,\infty}(\mathbb{R}^n)$. Inspired by the conditions for the forces, we characterize the maximal complete subspace in $L^{n,\infty}(\mathbb{R}^n)$ where the Stokes semigroup is st…
We prove a uniqueness result for weak solutions to the Vlasov-Navier-Stokes system in two dimensions, both in the whole space and in the periodic case, under a mild decay condition on the initial distribution function. The main result is achieved by combining methods from optimal transportation (introduced in this context by G. Loeper) with the use of Hardy's maximal function, in order to obtain some fine Wassestein-like estimates for the difference of two solutions of the Vlasov equation.
We prove that suitable weak solutions of the Navier-Stokes equations exhibit Type I singularities if and only if there exists a non-trivial mild bounded ancient solution satisfying a Type I decay condition. The main novelty is in the reverse direction, which is based on the idea of zooming out on a regular solution to generate a singularity. By similar methods, we prove a Liouville theorem for ancient solutions of the Navier-Stokes equations bounded in $L^3$ along a backward sequence of times.
In this paper, we consider the Liouville property for ancient solutions of the incompressible Navier-Stokes equations. In 2D and the 3D axially symmetric case without swirl, we prove sharp Liouville theorems for smooth ancient mild solutions: velocity fields $v$ are constants if vorticity fields satisfy certain condition and $v$ are sublinear with respect to spatial variables, and we also give counterexamples when $v$ are linear with respect to spatial variables. The condition which vorticity fields need to satisfy is $\lim\limits_{|x|\rightarrow +\infty}|w(x,t)|=0$ and $\lim\limits_{r\rightar…
Lei and Lin have recently given a proof of a global mild solution of the three-dimensional Navier-Stokes equations in function spaces based on the Wiener algebra. An alternative proof of existence of these solutions was then developed by Bae, and this new proof allowed for an estimate of the radius of analyticity of the solutions at positive times. We adapt the Bae proof to prove existence of the Lei-Lin solution in the spatially periodic setting, finding an improved bound for the radius of analyticity in this case.
We study the Navier-Stokes equations on a smooth bounded domain $D\subset \mathbb R^d$ ($d=2$ or 3), under the effect of an additive fractional Brownian noise. We show local existence and uniqueness of a mild $L^p$-solution for $p>d$.
In the paper \cite{KNSS:1}, the authors make the following conjecture: {\it any bounded ancient mild solution of the 3D axially symmetric Navier-Stokes equations is constant.} And it is proved in the case that the solution is swirl free. Our purpose of this paper is to improve their result by allowing that the solution can grow with a power smaller than 1 with respect to the distance to the origin. Also, we will show that such a power is optimal to prove the Liouville type theorem since we can find counterexamples for the Navier-Stokes equations such that the Liouville theorem fails if the sol…
This short note serves as an introduction to the papers arXiv:1711.01651 and arXiv:1711.04486 by Maekawa, Miura and Prange. These two works deal with the existence of mild solutions on the one hand and local energy weak solutions on the other hand to the Navier-Stokes equations in the half-space $\R^3_+$. We emphasize a concentration result for (sub)critical norms near a potential singularity. The contents of these notes were presented during the X-EDP seminar at IHÉS in October 2017.
We consider the conditional regularity of mild solution $v$ to the incompressible Navier-Stokes equations in three dimensions. Let $e \in \mathbb{S}^2$ and $0 < T^\ast < \infty$. J. Chemin and P. Zhang \cite{CP} proved the regularity of $v$ on $(0,T^\ast]$ if there exists $p \in (4, 6)$ such that $$\int_0^{T^\ast}\|v\cdot e\|^p_{\dot{H}^{\frac{1}{2}+\frac{2}{p}}}dt < \infty.$$ J. Chemin, P. Zhang and Z. F. Zhang \cite{CPZ} extended the range of $p$ to $(4, \infty)$. In this article we settle the case $p \in [2, 4]$. Our proof also works for the case $p \in (4,\infty)$.
For any divergence free initial datum $u_0$ with $\|u_0\|_\infty+\|\nabla u_0\|_{L^p}+\|\nabla^2 u_0\|_{L^p}<\infty$ for some $p>d\ (d\ge 2)$, the well-posedness and smoothness are proved for incompressible Navier-Stokes equations on $\mathbb{R}^d$ or $\mathbb{T}^d:=\mathbb{R}^d/\mathbb{Z}^d,$ up to a time explicitly given by the initial datum and three constants coming from the upper bounds of the heat kernel and the Riesz transform. A mild well-posedness is also proved for $L^p$-bounded initial data. The blow-up is proved for both type solutions with finite maximal time.
We prove the time analyticity for weak solutions of inhomogeneous parabolic equations with measurable coefficients in the half space with either the Dirichlet boundary condition or the conormal boundary condition under the assumption that the solution and the source term have the exponential growth of order $2$ with respect to the space variables. We also obtain the time analyticity for bounded mild solutions of the incompressible Navier-Stokes equations in the half space with the Dirichlet boundary condition. Our work is an extension of the recent work by Zhang [Proc. Amer. Math. Soc. 148 (20…
