Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in \(\mathbb{R}^ 3\). (English) Zbl 0865.35101

The author investigates the global solvability of the incompressible Navier-Stokes equation in \(\mathbb{R}^3\). For that purpose he introduces the homogeneous Besov spaces \(\dot B^\alpha_{p,q}\). The author proves that a global solution with initial data \(u_0\) exists and is unique in the space \(H^s(\mathbb{R}^3)\), \(s>1/2\), if \(|u_0|_{\dot B^{-1/4}_{4,\infty}}\) is smaller than some universal constant, and the solution is in \(L^p\), \(p>3/2\), if \(u_0\in L^p\cap\dot B^{-(1-3/2p)}_{2p,\infty}\) and \(|u_0|_{\dot B^{-(1-3/2p)}_{2p,\infty}}\) is smaller than some constant depending on \(p\). In the latter result \(u_0\in L^p\cap\dot B^{-(1-3/2p)}_{2p,\infty}\) may be replaced by \(u_0\in L^p\cap L^3\), and if \(p>3\) by \(L^2\cap L^p\).
Some regularity results of solutions are also established. In the proofs, the author transforms the original problem into an integral equation, and uses the exact expression of the heat kernel instead of the semigroup theory to obtain the necessary estimates.


35Q30 Navier-Stokes equations
58D25 Equations in function spaces; evolution equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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