On evolutionary problems with a-priori bounded gradients

We study a nonlinear evolutionary partial differential equation that can be viewed as a generalization of the heat equation where the temperature gradient is a priori bounded but the heat flux provides merely L 1 -coercivity. Applying higher differentiability techniques in space and time, choosing a...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2023-09, Vol.62 (7), Article 188
Main Authors: Bulíček, Miroslav, Hruška, David, Málek, Josef
Format: Article
Language:English
Subjects:
Analysis
Calculus of Variations and Optimal Control
Optimization
Coercivity
Control
Heat flux
Mathematical and Computational Physics
Mathematical functions
Mathematics
Mathematics and Statistics
Nonlinear systems
Nonlinearity
Parameters
Partial differential equations
Systems Theory
Theoretical
Thermodynamics
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Summary:We study a nonlinear evolutionary partial differential equation that can be viewed as a generalization of the heat equation where the temperature gradient is a priori bounded but the heat flux provides merely L 1 -coercivity. Applying higher differentiability techniques in space and time, choosing a special weighted norm (equivalent to the Euclidean norm in R d ), incorporating finer properties of integrable functions and flux truncation techniques, we prove long-time and large-data existence and uniqueness of weak solution, with an L 1 -integrable flux, to an initial spatially-periodic problem for all values of a positive model parameter. If this parameter is smaller than 2 / ( d + 1 ) , where d denotes the spatial dimension, we obtain higher integrability of the flux. As the developed approach is not restricted to a scalar equation, we also present an analogous result for nonlinear parabolic systems in which the nonlinearity, being the gradient of a strictly convex function, gives an a-priori L ∞ -bound on the gradient of the unknown solution.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-023-02524-4