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The Navier-Stokes equations on $$\mathbb R^n$$ with linearly growing initial data. (English) Zbl 1072.35144
Summary: Consider the equations of Navier-Stokes on $$\mathbb R^n$$ with initial data $$U _0$$ of the form $$U_0(x) = u_0(x) - Mx$$, where $$M$$ is an $$n \times n$$ matrix with constant real entries and $$u_0 \in L^p_\sigma(\mathbb R^n)$$. It is shown that under these assumptions the Navier-Stokes equations admit a unique local solution in $$L^p_\sigma(\mathbb R^n)$$. Moreover, if $$\| e^{tM}\| \leq 1$$ for all $$t \geq 0$$, then this mild solution is even analytic in $$x$$. This is surprising since the underlying semigroup of Ornstein-Uhlenbeck type is not analytic, in contrast to the Stokes semigroup.

##### MSC:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 47H20 Semigroups of nonlinear operators
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