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Higher integrability for doubly nonlinear parabolic systems.

Verena Bögelein1, Frank Duzaar2, Christoph Scheven3

  • 1Fachbereich Mathematik, Universität Salzburg, Hellbrunner Str. 34, 5020 Salzburg, Austria.

SN Partial Differential Equations and Applications
|October 24, 2022
PubMed
Summary
This summary is machine-generated.

This study proves higher integrability for spatial gradients in doubly nonlinear parabolic systems. A novel intrinsic scaling method is used, covering a broad parameter range.

Keywords:
Doubly nonlinear parabolic equationHigher integrabilityReverse Hölder inequality

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Area of Science:

  • Partial Differential Equations
  • Nonlinear Analysis
  • Mathematical Physics

Background:

  • Doubly nonlinear parabolic systems are crucial in modeling various physical phenomena.
  • Understanding the regularity of weak solutions, particularly their spatial gradients, is essential for theoretical analysis and applications.
  • Existing methods often face challenges with the differing scaling behaviors of the solution and its gradient.

Purpose of the Study:

  • To establish a local higher integrability result for the spatial gradient of weak solutions to doubly nonlinear parabolic systems.
  • To develop a novel mathematical technique for analyzing these complex systems.
  • To extend the applicability of regularity theory to a wider range of parameters.

Main Methods:

  • Development of a new intrinsic scaling technique.
  • The scaling method intrinsically incorporates both the solution and its spatial gradient.
  • This approach effectively compensates for the disparate scaling properties within the system's equations.

Main Results:

  • A local higher integrability result for the spatial gradient of weak solutions is successfully established.
  • The novel intrinsic scaling technique proves effective in handling the system's complexities.
  • The result is valid for a significant range of system parameters, denoted by and .

Conclusions:

  • The established higher integrability result enhances the understanding of regularity for doubly nonlinear parabolic systems.
  • The intrinsic scaling method offers a powerful new tool for analyzing such equations.
  • This work contributes to the broader field of nonlinear partial differential equations and their applications.