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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