Doubly nonlinear parabolic equations

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School of Science | Doctoral thesis (article-based) | Defence date: 2020-05-15
Degree programme
50 + app. 134
Aalto University publication series DOCTORAL DISSERTATIONS, 68/2020
This thesis is devoted to the study of the existence and regularity of weak solutions to some doubly nonlinear parabolic equations. Such equations are nonlinear both with respect to the solution and its gradient, and are thus more difficult to handle than the parabolic p-Laplace equation and the porous medium equation, which are special cases of the equations considered here. Much of the work in this thesis concerns the so-called diffusive shallow medium equation, which presents an additional technical difficulty due to the given profile function appearing in the equation. For the diffusive shallow medium equation we use a De Giorgi type iteration to prove that weak solutions are locally bounded. We show that bounded weak solutions are locally Hölder continuous using the method of intrinsic scaling introduced originally by DiBenedetto. We prove the existence of a bounded solution to a Cauchy-Dirichlet problem associated with the diffusive shallow medium equation by introducing regularized problems for which the existence of solutions is easy to prove by means of Galerkin's method. In order to show that these solutions converge to a solution of the original problem, we have proved some new compactness results which might be useful also for other nonlinear equations. There are various definitions for weak solutions appearing in the literature, and in this thesis we have devoted special attention to showing that the adopted definitions allow rigorous proofs of the existence and regularity results without any additional assumptions. For the diffusive shallow medium equation, a definition which is natural in this sense has not previously been used. For doubly singular parabolic equations we have proved a range of regularity results such as local boundedness, an integral Harnack inequality, local Hölder continuity, expansion of positivity and a Harnack inequality in properly scaled space-time cylinders.
The public defense on 15th May 2020 at 12:00 will be organized via remote technology. Link: Zoom Quick Guide:
Supervising professor
Kinnunen, Juha, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland
Thesis advisor
Kinnunen, Juha, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland
doubly nonlinear equation, regularity, Hölder continuity, local boundedness, existence, weak solution, Harnack estimate
Other note
  • [Publication 1]: Thomas Singer, Matias Vestberg. Local boundedness of weak solutions to the Diffusive Wave Approximation of the Shallow Water equations. Journal of Differential Equations, Volume 266, Issue 6, pages 3014-3033, March 2019.
    Full text in Acris/Aaltodoc:
    DOI: 10.1016/j.jde.2018.08.051 View at publisher
  • [Publication 2]: Thomas Singer, Matias Vestberg. Local Hölder continuity of weak solutions to a diffusive shallow medium equation. Nonlinear Analysis, Volume 185, pages 306-335, August 2019.
    DOI: 10.1016/ View at publisher
  • [Publication 3]: Verena Bögelein, Nicholas Dietrich, Matias Vestberg. Existence of solutions to a diffusive shallow medium equation. Submitted to a journal, 35 pages, Available at: arXiv:2001.07942, January 2020.
  • [Publication 4]: Vincenzo Vespri, Matias Vestberg. An Extensive Study of the Regularity of Solutions to Doubly Singular equations. Advances in Calculus of Variations, 2020, 39 pages.
    DOI: 10.1515/acv-2019-0102 View at publisher