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Solutions for semilinear parabolic equations in \(L^ p\) and regularity of weak solutions of the Navier-Stokes system. (English) Zbl 0549.35063
J. Differ. Equations (to appear)
We construct a unique local regular solution in \(L^ q(0,T;L^ p)\) for a class of semilinear parabolic equations which includes the semilinear heat equation \(u_ t-\Delta u=| u|^{\alpha}u (\alpha >0)\) and the Navier-Stokes system. Here p and q are so chosen that the norm of \(L^ q(0,T;L^ p)\) is dimensionless or scaling invariant. The main relation between p and q for the semilinear heat equation is \(1/q=(1/r- 1/p)n/2, p>r\) provided that initial data is in \(L^ r\) with \(r=n\alpha /2>1\), where n is the space dimension. Applying our regular solutions to the Navier-Stokes system, we show that the k/2-dimensional Hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in \(L^ q(0,T;L^ p)\) where \(k=2-q+nq/p\), \(p\geq n\), \(1\leq q<\infty\). We show, moreover, that a turbulent solution is regular if it is in \(C((0,T);L^ n)\).

MSC:
35K55 Nonlinear parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35Q30 Navier-Stokes equations