Regularizing the Navier-Stokes Equations via Additive Type Noise plus Small Linear Multiplicative Noise

Introduction

The incompressible Navier-Stokes equations are well-known for its wide applications in Fluid Mechanics and the longstanding regularity and uniqueness issues in mathematics. Over the years, we have series of breakthroughs coming up on the negative side, concerning non-uniqueness of weak solutions with the absent of external force term (see for example [5, 7, 8, 12]) and non-uniqueness of Leray-Hopf weak solutions with specific non-zero external force (see [1, 2]). See also the more recent breakthrough [14] on non-unique smooth solutions arising from specific (large) initial values in the critical space $BMO^{-1}$. These work suggest that the equations may “break down” at least under certain theoretical circumstance. Over the decades, however, people have been striving for the positive direction, trying to figure out circumstances in which the equations have unique solutions. Among these, the most famous part may probably be the theory of small solutions in the following critical spaces:

For global well-posedness results on small initial values in the above spaces (except $\dot{B}^{-1}_{\infty,\infty}$), we refer to [FK64,Che92], [22], [CMP94] and [23] respectively. The $BMO^{-1}$ space is the largest critical space where existence results are available and is commonly considered as the largest one where one can safely apply the Picard iteration or fixed point argument to construct solutions [23]. The end point space $\dot{B}^{-1}_{\infty,\infty}$ on the right-hand-side is the largest possible critical space in the sense that all scaling invariant spaces for the Navier-Stokes equations are embedded in it [9, 4]. However, it turns out to be “too large” for the equations: in [6], J. Bourgain and N. Pavlović show the famous result of norm inflation phenomenon, namely initial values arbitrarily small in $\dot{B}^{-1}_{\infty,\infty}$ can produce solutions that become arbitrarily large in the $\dot{B}^{-1}_{\infty,\infty}$-norm by arbitrarily short time. So this means that the equations are locally (in time) ill-posed in this space in a very strong sense.

A challenging and active direction that combine this well-posedness issue with probability theory is the so-called regularization by noise effect. It was motivated by a classical result [Ver81] in stochastic analysis that established well-posedness of the stochastic differential equation:

when $b$ is just a bounded function and $\sigma$ satisfies certain conditions which include the simplest and probably most interesting case $\sigma\equiv\operatorname{const.} \ne0$. Such noise term is independent of the solution itself and is said to be of “additive type”. In this direction, people try to “restore” the well-posedness of Navier-Stokes equations by asking whether one can introduce a reasonable noise term so that, in appropriate sense, the resulting stochastic equations have unique solutions for general initial values. Here, “in appropriate sense” often means “almost surely” (i.e. with probability one) or “with high probability” (i.e. with probability being arbitrarily close to one if the noise parameters are large enough). Regularization by additive noise for Navier-Stokes remains open and challenging, but we should mention that there are interesting results like “almost sure absent of spacial singularity for each fixed time” [18], “continuity of solution map on the initial values via Markov selection” [DD03, FR08], etc. We also mention here some famous results on regularization effect by other noise forms: see [20 RZZ14] for linear multiplicative noise; [17, 3] for transport noise and [21] for a non-linear and non-local type of noise.

Our work: In this paper, we construct a specific type of additive noise (by which we mean that the noise term is independent of the solutions themselves) so that the system is globally well-posed with high probability in the critical Besov space $\dot{B}^{-1+d/q}_{q,\infty}$ when $1 \le q < \infty$. This answers the open question of regularization by additive noise for the first time, although the general additive noise case still remains open. As a further result, we show that if a small linear multiplicative noise is added additionally into the equations, then the regularizing effect can be strengthen in the sense that: for general data in the largest critical space $\dot{B}^{-1}_{\infty,\infty}$, one can establish well-posedness theorem on any large enough time interval $[0,T]$ with high probability. This can be viewed as a restoration of the local ill-posedness result [6] on the deterministic side.

Main idea: Our strategy is rather simple compared to the mentioned literature above. The key is just that we need to find suitable solution spaces so that, when the solution is viewed as the fixed point of a bilinear map, the map should be continuous on the space and, at the same time, we should be able to control the stochastic Stokes flow in the space via the additive part of the noise that we impose into the Navier-Stokes equations. For initial values in $\dot{B}^{-1+d/q}_{q,\infty}$ when $1 \le q < \infty$, this is relatively simple as we just need to look at solution space like $L^p_t\dot{B}^{-1+2/p+d/q}_{q,\infty}$ (which is the $L^p$ version of the commonly known mild solution space $L^\infty_t\dot{B}^{-1+d/q}_{q,\infty}$). The solutions have $*$-weak continuous paths in space $\dot{B}^{-1+d/q}_{q,\infty}$ as supposed. For initial values in $\dot{B}^{-1}_{\infty,\infty}$, we need to impose a further linear multiplicative noise (with arbitrarily small but non-zero intensity) into the system, so that a corrector is impose into the bilinear map via Duhamel principle and the map become continuous on the modified solution space like $L^p_t\dot{B}^{-1+2/p+\delta}_{\infty,\infty}$ ($\delta>0$ is small). Consequently, the solution $u$ have a “modified” $*$-weak continuous paths in the sense that $u \in C_{w^*}([0,T];\dot{B}^{-1}_{\infty,\infty})+C_{w^*}([0,T];\dot{B}^{-1+\delta}_{\infty,\infty})$.

Main results

We study Cauchy problem of the $d$-dimensional incompressible stochastic Navier-Stokes equations posed on whole space $\mathbb{R}^d$ ($d \ge3$):

where initial value $u_0 \in\dot{B}^{-1+d/q}_{q,\infty}$ ($1 \le q \le\infty$) is divergence-free in weak sense. The system is driven by a stochastic noise term $N$ which will be specified in different situations. In unified form, our noise term consists of an additive part and a “small” linear multiplicative noise:

where $\varepsilon_{\mathrm{lin}}\in(-\nu^{1/2},\nu^{1/2})$ is constant and $\{B,B^j\}_j$ is an independent family of one dimensional standard Brownian motions given on a probability basis $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\geq0},\mathbb{P})$. The coefficients $G_j:\Omega\times[0,+\infty)\longrightarrow\dot{B}^{-1+d/q}_{q,\infty}$ would be constructed to be divergence-free and adaptive with respect to $\{\mathcal{F}_t\}$. Stochastic integrals would always be understood in the sense of Itô integral in this paper. We shall first define the solutions via Duhamel principal. So let’s denote $(\varepsilon\in\mathbb{R})$

By subtracting the equation of $g$ from (1), which is the stochastic Stokes equations

we see that $v:=u-g$ solves

i.e.

The Leray projector $\mathbb{P}=\operatorname{Id}+\nabla(-\Delta)^{-1}\operatorname{div}$ is defined via Fourier multiplier. This motivates the following definition.