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Analyticity for the Navier-Stokes equations

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Analyticity for the Navier-Stokes equations

Baoxiang Wang (Peking University)

Abstract

We study the Cauchy problem for the incompressible Navier-Stokes equations

ut−∆u+u· ∇u+∇p= 0, divu= 0, u(0, x) =u0.

We show the analyticity of the local solutions of the Navier-Stokes equation with any initial data in critical Besov spaces ˙Bp,qn/p−1(Rn) with 1 < p < ∞, 1 ≤ q ≤ ∞ and the solution is global if u0 is sufficiently small in ˙Bp,qn/p−1(Rn). In the case p = ∞, the analyticity for the local solutions of the Navier-Stokes equation with any initial data in modulation space M∞−1,1(Rn) is obtained. Similar results also hold for the generalized Navier-Stokes equation.

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