Well posedness of nonlinear parabolic systems beyond duality

Bulíček, Miroslav; Burczak, Jan; Schwarzacher, Sebastian

doi:10.1016/j.anihpc.2019.01.004

Bulíček, Miroslav ¹ ; Burczak, Jan ^2,³ ; Schwarzacher, Sebastian ⁴

¹ Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague, Czech Republic
² Mathematical Institute, University of Oxford, UK
³ Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland
⁴ Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague, Czech Republic

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 5, pp. 1467-1500.

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Résumé

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

\partial_{t} u - div (ν (| \nabla u |) \nabla u) = - div f

with a given strictly positive bounded function ν, such that

\lim_{k \to \infty} ν (k) = ν_{\infty}

and

f \in L^{q}

with

q \in (1, \infty)

. The existence, uniqueness and regularity results for

q \geq 2

are by now standard. However, even if a priori estimates are available, the existence in case

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 \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 $L^{q}$ spaces.

MR Zbl

DOI : 10.1016/j.anihpc.2019.01.004

Classification : 35D99, 35K51, 35K61, 35A01, 35A02
Mots-clés : Nonlinear parabolic systems, Weighted estimates, Existence, Uniqueness, Very weak solution, Parabolic Lipschitz approximation, Muckenhoupt weights

Affiliations des auteurs :

Bulíček, Miroslav ¹ ; Burczak, Jan ^{2, 3} ; Schwarzacher, Sebastian ⁴

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     author = {Bul{\'\i}\v{c}ek, Miroslav and Burczak, Jan and Schwarzacher, Sebastian},
     title = {Well posedness of nonlinear parabolic systems beyond duality},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1467--1500},
     publisher = {Elsevier},
     volume = {36},
     number = {5},
     year = {2019},
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     zbl = {1435.35195},
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     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2019.01.004/}
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Bulíček, Miroslav; Burczak, Jan; Schwarzacher, Sebastian. Well posedness of nonlinear parabolic systems beyond duality. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 5, pp. 1467-1500. doi : 10.1016/j.anihpc.2019.01.004. http://archive.numdam.org/articles/10.1016/j.anihpc.2019.01.004/

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