TY  - GEN 
T1  - On evolutionary problems with a-priori bounded gradients
A1  - Bulíček, Miroslav
A1  - Hruška, David
A1  - Málek, Josef
Y1  - 2021
KW  - $\infty$-Laplacian
KW  - a~priori bounded gradient
KW  - existence
KW  - nonlinear parabolic equation
KW  - renormalized solution
KW  - uniqueness
KW  - weak solution
N2  - 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 \mbox{$L^1$-coercivity}. Applying higher differentiability techniques in space and time, choosing a special weighted norm (equivalent to the Euclidean norm in $\mathbb{R}^d$), incorporating finer properties of integrable functions and using the concept of renormalized solution, 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  $L^\infty$-bound on the gradient of the unknown solution.
M1  - Preprint project = NCMM
M1  - Preprint year = 2021
M1  - Preprint number = 02
M1  - Preprint ID = NCMM/2021/02
ER  -