On the Local Smoothness of Solutions of the Navier-stokes Equations
نویسنده
چکیده
We consider the Cauchy problem for incompressible Navier-Stokes equations ut + u∇xu − ∆xu + ∇xp = 0, divu = 0 in Rd×R+ with initial data a ∈ L(R), and study in some detail the smoothing effect of the equation. We prove that for T < ∞ and for any positive integers n and m we have tD t D n xu ∈ L(R × (0, T )), as long as the ‖u‖Ld+2 x,t (Rd×(0,T )) stays finite.
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