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Weighted Inertia-Dissipation-Energy approach to doubly nonlinear wave equations

Goro Akagi, Verena Bögelein, Alice Marveggio, Ulisse Stefanelli

TL;DR

The paper extends the variational WIDE framework to genuinely doubly nonlinear hyperbolic equations of the form $\rho u_{tt}+g(u_t)-\Delta u+f(u)=0$ by employing parameter-dependent convex functionals with an exponential time weight. It develops a full approximation scheme using Moreau–Yosida regularizations, proves existence and strong solvability of the Euler–Lagrange system, and establishes convergence as $\varepsilon\to0$ (causal limit) to the target equation. It further analyzes the viscous limit $\rho\to0$ via Gamma-convergence and shows convergence to the parabolic limit, thereby linking hyperbolic and parabolic dynamics within a unified variational framework. The results provide a rigorous, convex-analytic pathway to approximate and analyze nonlinear damped wave equations with nonquadratic dissipation, with potential applications to nonlinear viscoelasticity and related materials models.

Abstract

We discuss a variational approach to doubly nonlinear wave equations of the form $ρu_{tt} + g (u_t) - Δu + f (u)=0$. This approach hinges on the minimization of a parameter-dependent family of uniformly convex functionals over entire trajectories, namely the so-called Weighted Inertia-Dissipation-Energy (WIDE) functionals. We prove that the WIDE functionals admit minimizers and that the corresponding Euler-Lagrange system is solvable in the strong sense. Moreover, we check that the parameter-dependent minimizers converge, up to subsequences, to a solution of the target doubly nonlinear wave equation as the parameter goes to $0$. The analysis relies on specific estimates on the WIDE minimizers, on the decomposition of the subdifferential of the WIDE functional, and on the identification of the nonlinearities in the limit. Eventually, we investigate the viscous limit $ρ\to 0$, both at the functional level and on that of the equation.

Weighted Inertia-Dissipation-Energy approach to doubly nonlinear wave equations

TL;DR

The paper extends the variational WIDE framework to genuinely doubly nonlinear hyperbolic equations of the form by employing parameter-dependent convex functionals with an exponential time weight. It develops a full approximation scheme using Moreau–Yosida regularizations, proves existence and strong solvability of the Euler–Lagrange system, and establishes convergence as (causal limit) to the target equation. It further analyzes the viscous limit via Gamma-convergence and shows convergence to the parabolic limit, thereby linking hyperbolic and parabolic dynamics within a unified variational framework. The results provide a rigorous, convex-analytic pathway to approximate and analyze nonlinear damped wave equations with nonquadratic dissipation, with potential applications to nonlinear viscoelasticity and related materials models.

Abstract

We discuss a variational approach to doubly nonlinear wave equations of the form . This approach hinges on the minimization of a parameter-dependent family of uniformly convex functionals over entire trajectories, namely the so-called Weighted Inertia-Dissipation-Energy (WIDE) functionals. We prove that the WIDE functionals admit minimizers and that the corresponding Euler-Lagrange system is solvable in the strong sense. Moreover, we check that the parameter-dependent minimizers converge, up to subsequences, to a solution of the target doubly nonlinear wave equation as the parameter goes to . The analysis relies on specific estimates on the WIDE minimizers, on the decomposition of the subdifferential of the WIDE functional, and on the identification of the nonlinearities in the limit. Eventually, we investigate the viscous limit , both at the functional level and on that of the equation.
Paper Structure (22 sections, 3 theorems, 148 equations, 1 figure)

This paper contains 22 sections, 3 theorems, 148 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Assumptions and main results
  3. Preliminary materials
  4. Existence of solutions to the Euler-Lagrange problem
  5. Approximating functional $I_{\rho\varepsilon}^{\lambda\mu}$
  6. Representation of subdifferentials
  7. Euler-Lagrange equation for $I_{\rho\varepsilon}^{\lambda \mu}$
  8. A priori estimates
  9. Passage to the limit as $\mu \rightarrow 0$
  10. Identification of the nonlinearities
  11. Identification of $\eta_{\varepsilon}$
  12. Identification of $\xi_{\varepsilon\lambda}$
  13. Passage to the limit as $\lambda \rightarrow 0$
  14. Identification of the nonlinearities
  15. Identification of $\eta_{\varepsilon}$
  16. ...and 7 more sections

Key Result

Theorem 1

Assume def_psi -(growth condition hespsi2), and eq.growthF. Then,

Figures (1)

  • Figure 1: Theorems \ref{['main_thm']}-\ref{['viscous_thm']}: illustration of the various limits with respect to $\varepsilon$ and $\rho$.

Theorems & Definitions (8)

  • Theorem 1: WIDE variational approach
  • Theorem 2: Viscous limit
  • Lemma 3: Weighted limsup tool
  • Remark 4
  • proof : Proof of Lemma \ref{['lemma:doublelimsup']}
  • proof : Proof of Theorem \ref{['viscous_thm']}.i
  • proof : Proof of Theorem \ref{['viscous_thm']}.ii
  • Remark 5