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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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