## A note on $$W^{1,p}$$ estimates for quasilinear parabolic equations.(English)Zbl 1025.35007

The authors study interior $$W^{1,p}$$ estimates for weak solutions to quasilinear parabolic equations in divergence form $u_t - \operatorname {div}a(x,t, \nabla u) = 0$ where $$a$$ satisfies some hypotheses, in particular that it is a Carathéodory function, i.e. that $$a(x,t,\zeta)$$ in $$(x,t)$$ is measurable and it is continuous with respect to $$\zeta$$ for each $$x.$$
They extend the elliptic case studied by L. Caffarelli and the first author in [Commun. Pure Appl. Math. 51, 1-21 (1998; Zbl 0906.35030)] to the parabolic case. The basic argument is to estimate the level sets of $$\nabla_x u$$ and obtain the requested growth of their measure to have the integrability property. An important tool to obtain this results is a form of the Calderon-Zygmund covering property.

### MSC:

 35B45 A priori estimates in context of PDEs 35K55 Nonlinear parabolic equations 42B25 Maximal functions, Littlewood-Paley theory

Zbl 0906.35030
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