arXiv 2402.114822024-02math.PRmath-phmath.APmath.MP

具有涡量界的三维随机Navier-Stokes方程的鞅适定弱解

Martingale Suitable Weak Solutions of $3$-D Stochastic Navier-Stokes Equations with Vorticity Bounds

Weiquan Chen, Zhao Dong

AI总结：

本文为带一般非线性噪声的三维随机Navier-Stokes方程构造鞅适定弱解，建立两种随机局部能量不等式，并给出涡量及其梯度的范数界。

摘要：

本文为具有一般非线性噪声的三维不可压缩随机Navier-Stokes方程构造了鞅适定弱解。在确定性情形中，众所周知，“适定弱解”是满足两种不同类型局部能量不等式的Leray-Hopf弱解。在随机情形中，我们应用“鞅解”的思想，避免变换到随机系统，并证明了两种局部能量不等式的新的随机版本。特别地，在加性噪声和线性乘性噪声情形下，OU过程和指数公式在我们对局部能量不等式的表述中不起作用。这与处理加性噪声情形的文献[FR02, Rom10]不同。此外，我们成功地应用了“几乎处处上鞅”的概念来描述这种局部能量行为。为了与文献[DR00]中提出的著名“耗散弱解”建立联系，我们导出了一个局部能量等式，并将该概念自然地推广到随机情形。对于解的进一步正则性，在初始涡量为有限正则符号测度的情形下，我们能够给出涡量的$L^\infty\big([0,T];L^1(Ω\times\mathbb T^3)\big)$范数界以及涡量梯度的$L^{\frac{4}{3+δ}}\big(Ω\times[0,T]\times\mathbb T^3\big)$范数界。

英文摘要：

In this paper, we construct martingale suitable weak solutions for $3$-dimensional incompressible stochastic Navier-Stokes equations with generally non-linear noise. In deterministic setting, as widely known, ``suitable weak solutions'' are Leray-Hopf weak solutions enjoying two different types of local energy inequalities (LEIs). In stochastic setting, we apply the idea of ``martingale solution", avoid transforming to random system, and show new stochastic versions of the two local energy inequalities. In particular, in additive and linear multiplicative noise case, OU-processes and the exponential formulas DO NOT play a role in our formulation of LEIs. This is different to \cite{FR02,Rom10} where the additive noise case is dealt. Also, we successfully apply the concept of ``a.e. super-martingale'' to describe this local energy behavior. To relate the well-known ``dissipative weak solutions" come up with in \cite{DR00}, we derive a local energy equality and extend the concept onto stochastic setting naturally. For further regularity of solutions, we are able to bound the $L^\infty\big([0,T];L^1(Ω\times\mathbb T^3)\big)$ norm of the vorticity and $L^{\frac{4}{3+δ}}\big(Ω\times[0,T]\times\mathbb T^3\big)$ norm of the gradient of the vorticity, in case that the initial vorticity is a finite regular signed measure.

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