JournalsaihpcVol. 36, No. 5pp. 1467–1500

Well posedness of nonlinear parabolic systems beyond duality

  • Miroslav Bulíček

    Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague, Czech Republic
  • Jan Burczak

    Mathematical Institute, University of Oxford, UK, Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland
  • Sebastian Schwarzacher

    Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague, Czech Republic
Well posedness of nonlinear parabolic systems beyond duality cover
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Abstract

We develop a methodology for proving well-posedness in optimal regularity spaces for a wide class of nonlinear parabolic initial–boundary value systems, where the standard monotone operator theory fails. A motivational example of a problem accessible to our technique is the following system

tudiv(ν(u)u)=divf\partial _{t}u−\mathrm{div}\left(\nu (|\mathrm{∇}u|)\mathrm{∇}u\right) = −\mathrm{div}f

with a given strictly positive bounded function ν, such that limkν(k)=ν\mathrm{\lim }_{k\rightarrow \infty }⁡\nu (k) = \nu _{\infty } and fLqf \in L^{q} with q(1,)q \in (1,\infty ). The existence, uniqueness and regularity results for q2q \geq 2 are by now standard. However, even if a priori estimates are available, the existence in case q(1,2)q \in (1,2) was essentially missing. We overcome the related crucial difficulty, namely the lack of a standard duality pairing, by resorting to proper weighted spaces and consequently provide existence, uniqueness and optimal regularity in the entire range q(1,)q \in (1,\infty ).

Furthermore, our paper includes several new results that may be of independent interest and serve as the starting point for further analysis of more complicated problems. They include a parabolic Lipschitz approximation method in weighted spaces with fine control of the time derivative and a theory for linear parabolic systems with right hand sides belonging to Muckenhoupt weighted LqL^{q} spaces.

Cite this article

Miroslav Bulíček, Jan Burczak, Sebastian Schwarzacher, Well posedness of nonlinear parabolic systems beyond duality. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 5, pp. 1467–1500

DOI 10.1016/J.ANIHPC.2019.01.004