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Well-posedness for the Navier-Stokes equations. (English) Zbl 0972.35084
The existence of a global solution to the Cauchy problem for the Navier-Stokes equations \[ \begin{aligned} &\frac{\partial v}{\partial t}+(v\cdot\nabla)v-\Delta v+\nabla p=0,\qquad \text{div }v=0 \quad \text{in } \mathbb{R}^n\times \mathbb{R}^+\\ &v(x,0)=v_0(x), \qquad x\in \mathbb{R}^n \end{aligned} \tag{1} \] is discussed. It is proved that if \(\text{div }v=0\) and if the norm of \(v_0\) \[ \|v_0\|_1=\sup_{x,R}\Biggl[\text{mes}^{-1}(B(x,R))\int_{Q(x,R)} |w(y,t)|^2 dy dt\Biggr]^{\frac 12} \] is sufficiently small, then the problem (1) has a unique small global solution in the space \(X\) with a norm \[ \|v_0\|_X=\sup_{t>0}\sqrt{t}\|v(\cdot,t)\|_{L^{\infty}(\mathbb{R}^n)}+ \Biggl(\sup_{x,R} \text{mes}^{-1}(B(x,R))\int_{Q(x,R)} |u(y,t)|^2 dy dt\Biggr)^{\frac 12}. \] Here \(Q(x,R)=B(x,R)\times(0,R^2)\) and \(w(x,t)\) is the solution to the Cauchy problem to the heat equation \[ \frac{\partial w}{\partial t}-\Delta v,\quad w(x,0)=v_0. \] A similar result of existence of a local solution is obtained, too. These results are more general than earlier results.

MSC:
35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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