Abstract.
Consider the equations of Navier-Stokes on ℝn with initial data U0 of the form U0(x)=u0(x)−Mx, where M is an n×n matrix with constant real entries and u0 ∈ Lp σ (ℝn). It is shown that under these assumptions the equations of Navier-Stokes admit a unique local solution in Lp σ (ℝn). Moreover, if ||etM||≤1 for all t≥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.
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Communicated by V. Šverák
Acknowledgement It is our pleasure to thank G. METAFUNE, E. PRIOLA and A. RHANDI for fruitful discussions on the Ornstein-Uhlenbeck
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Hieber, M., Sawada, O. The Navier-Stokes Equations in ℝn with Linearly Growing Initial Data. Arch. Rational Mech. Anal. 175, 269–285 (2005). https://doi.org/10.1007/s00205-004-0347-0
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DOI: https://doi.org/10.1007/s00205-004-0347-0