ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: 35 Naumann, Joachim, Humboldt University of Berlin, Germany Interior Hölder continuity of weak solutions to nonlinear parabolic systems in two spatial dimensions Let Ω ⊂ IR^{2} be open, let 0 < T < +∞ and set Q = Ω × (0, T). We consider the following system of nonlinear PDE’s in Q:

\frac{∂u^{i}}{∂t} − \frac{∂}{∂x_{α}} a_{i}^{α}(x, t, u, ∇u) = f_{i}(x, t)  (i = 1, …, N)  (1)

where u = \{u^{1}, …, u^{N}\}, ∇u = \{\frac{∂u^{i}}{∂x_{α}} \} (= matrix of spatial derivatives; α =

1, 2). The coefficients a_{i}^{α} = a_{i}^{α}(x, t, u, ξ) are assumed to satisfy appropriate Hölder continuity conditions with respect to {t, u, ξ}. Let f ∈ L^{2}(Q; IR^{N}) satisfy the condition

∫_{t_{0}}^{t_{1}} ∫_{Ω′} |f(x, t + h) − f(x, t)|^{2}dxdt ≤ Ch^{2μ} ∀ 0 < h < T − t_{1},

∀ 0 < t_{0} < t_{1} < T, ∀ Ω′ ⊂ \overline{Ω′} ⊂ Ω, where \frac{1}{2} < μ < 1 is “near to 1“.

Then we prove: any weak solution u ∈ W_{2}^{1,0}(Q; IR^{N}) with

esssup_{t∈(0,T)} ∫_{Ω} |u(x, t)|^{2}dx < +∞ is Hölder continuous in Q.