Challenges for the WIDE variational principle

Nonlinear evolution equations are ubiquitous in applications and pose a variety of challenging mathematical questions, from well-posedness, to regularity. Variational methods provide far-reaching modeling and analytical tools. Among these, the Weighted Inertia- Energy-Dissipation (WIDE) approach has already proved successful in a range of classical situations. The WIDE variational approach consists in the minimization of a parameter-dependent, global-in-time functional and a passage to the limit in the parameter. One aims at proving that the minimizers of such parameter-dependent functional converge to solutions of the limiting nonlinear evolution problem. This global variational approach paves the way to applying to nonlinear evolution problems the tools of the calculus of variations such as the Direct Method, Gamma-convergence, and relaxation. The outcome will be novel existence and approximation theories. The aim of the project is to assess the applicability of the WIDE variational principle out of its comfort zone, by targeting a number of hot and demanding research fronts. These include generalized doubly nonlinear flows, stochastic partial differential equations, conservations laws and fluids, and optimal control problems. The program will be developed by a new international research team coordinated by Ulisse Stefanelli, with the cooperation of Goro Akagi (Sendai), Takeshi Fukao (Kyoto), and Hao Wu (Shanghai). Keywords: Evolution equations, global variational methods, generalized doubly nonlinear flows, stochastic PDEs, conservation laws, fluids, optimal control.

Nonlinear evolution equations arise ubiquitously in scientific applications and pose a variety of challenging mathematical questions, from well-posedness, to regularity. Variational analysis offers a powerful set of tools for both modeling and analysis. Among the variational methods, the Weighted Inertia-Energy-Dissipation (WIDE) approach has proven particularly effective in addressing diverse problems. From a technical perspective, the WIDE variational approach involves minimizing a parameter-dependent, global-in-time functional and then passing to the limit in the parameter. The goal is to show that minimizers of this functional converge to solutions of the limiting nonlinear differential problem. This results in a novel variational approximation technique, paving the way for the application of the calculus of variations to nonlinear evolution equations. The project aimed at challenging the applicability of the WIDE out of the comfort zone, by applying it to a number of active research fronts. We have been able to ascertain the amenability of the WIDE approach for doubly nonlinear flows, stochastic partial differential equations, Navier-Stokes equations, and some optimal control problems. In all of these settings, we could derive new approximation and existence results. The program has benefited from the collaboration with an interational group of researchers, including Goro Akagi (Tohoku), Takeshi Fukao (Ryukoku), Luca Scarpa (Politecnico Milano), and Hao Wu (Fudan).