A variational approach to symmetry, monotonicity, and comparison for doubly-nonlinear equations

Abstract

We advance a variational method to prove qualitative properties such as symmetries, monotonicity, upper and lower bounds, sign properties, and comparison principles for a large class of doubly-nonlinear evolutionary problems including gradient flows, some nonlocal problems, and systems of nonlinear parabolic equations.Our method is based on the so-called Weighted-Energy-Dissipation (WED) variational approach. This consists in defining a global parameter-dependent functional over entire trajectories and proving that its minimizers converge to solutions to the target problem as the parameter goes to zero. Qualitative properties and comparison principles can be easily proved for minimizers of the WED functional and, by passing to the limit, for the limiting problem.Several applications of the abstract results to systems of nonlinear PDEs and to fractional/nonlocal problems are presented. Eventually, we present some extensions of this approach in order to deal with rate-independent systems and hyperbolic problems.

A variational approach to Navier–Stokes

“…The rate-dependent case has been analyzed in [1][2][3][4]. See also Liero and Melchionna [33] for a stability result via Γ-convergence [14] and Melchionna [43] for an application to the study of symmetries of solutions.…”

Elliptic-regularization of nonpotential perturbations of doubly-nonlinear gradient flows of nonconvex energies: A variational approach

“…Our interest in the WED formalism lies in the fact that it paves the way to the application of general techniques of the calculus of variation (e.g., Direct Method, relaxation, the Γ-convergence) in the evolutionary setting. Moreover, the WED procedure also brings a new tool to reveal qualitative properties of solutions and comparison principles for evolutionary problems [30]. Furthermore, also in the present paper, this variational formulation brings us a useful technique to check the uniqueness of solutions and structural stability of unperturbed equations from the strict convexity of the WED functionals and Γ-convergence theory, respectively.…”

Variational resolution of outflow boundary conditions for incompressible Navier–Stokes

“…The rate-dependent case has been analysed in [1][2][3][4][5]. See also [35] for a stability result via Γ-convergence [18,39] for an application to the study of symmetries of solutions.…”