Global well-posedness of 2D stochastic Burgers equations with multiplicative noise
This paper deals with the existence and uniqueness of a solution to the 2D stochastic Burger equation given by \[ \begin{aligned} du(t)&=\nu\Delta u dt-(u\cdot\nabla)udt+u(t)\circ dW(t), \\ u(t,x)&=0, \quad t>0, \quad x=(x_1,x_2)\in\partial D.\\ u(0,x)&=u_0(x),\quad x=(x_1,x_2). \end{aligned} \tag{1} \] Here, \(D\subset\mathbb{R}^2\) is a smooth, bounded, open domain, \(W=\sum_{k=1}^\infty\sigma_kB_k(t)\) with \(\sum_{k=1}^\infty\sigma_k^2<\infty\) where \((B_k)_{k\in\mathbb{N}}\) is a sequence of independent, Wiener processes. The parameter \(\nu>0\) stands for the viscosity and \(\circ\) denotes the Stratonovich integral. The authors prove that for any initial condition \(u_0\in\mathbb{H}^1\), such equation \((1)\) has a unique global strong solution \(v\) and \(v\in C([0,T];\mathbb{H}^1)\cap L^2([0,T];\mathbb{H}^1)\). \(\mathbb{H}^1\) denotes the domain of the operator \((-\Delta)^{1/2}\).
scientific article; zbMATH DE number 7462370