On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations
The authors consider the Cauchy problem for the Navier-Stokes equations by reducing to the integral equation \[ u(x,t)=S(t) u_{0}(x)-\int_{0}^{t}\mathbf{P} S(t-s) \nabla \cdot (u \otimes u) (x,s) ds,\tag{1} \] where \(S(t)=\exp (t \Delta)\) is the heat kernel, \(\mathbf{P}\) is a pseudo-differential operator. Let \(u(x,t)\) be a solution of (1) in \(L^3 (\mathbb{R})\) on \(x\) and in \(C[0,T)\) on \(t\) with initial data \(u_0 \in L^3\), \(w:=u-S(t) u_0\), then \(w \in H \times C[0,T)\). Here \(H\) is a Sobolev space.
scientific article; zbMATH DE number 1485603