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Anti-selfdual Lagrangians II: Unbounded non self-adjoint operators and evolution equations

Nassif Ghoussoub, Leo Tzou

TL;DR

The paper advances the anti-selfdual Lagrangian framework by enabling variational treatment of unbounded, non-self-adjoint operators, notably transport and transport-Diffusion PDEs, via compositions with skew-adjoint (modulo boundary) operators and Yosida-type regularization. It develops Lax-Milgram-type results, a general evolution principle yielding semigroups, and parabolic initial-value resolutions, establishing existence, regularity, and boundary-compatibility for nonlinear PDEs outside classical Euler–Lagrange settings. Key contributions include the construction of ASD Lagrangians under unbounded operators, variational resolutions for stationary and evolution problems, and applications to nonlinear boundary-value and parabolic problems driven by transport-type operators. The results offer a versatile variational pathway to handle dissipative dynamics with transport components, with potential impact on nonlinear PDE theory and boundary-value problems exhibiting strong non-self-adjointness.

Abstract

This paper is a continuation of [13], where new variational principles were introduced based on the concept of anti-selfdual (ASD) Lagrangians. We continue here the program of using these Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary equations of the form $ -Au\in \partial \phi (u)$ as well as i dissipative evolutions of the form $-\dot{u}(t)-A_t u(t)+\omega u(t) \in \partial \phi (t, u(t))$ were $\phi$ is a convex potential on an infinite dimensional space. In this paper, the emphasis is on the cases where the differential operators involved are not necessarily bounded, hence completing the results established in [13] for bounded linear operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by the transport operator with or without a diffusion term.

Anti-selfdual Lagrangians II: Unbounded non self-adjoint operators and evolution equations

TL;DR

The paper advances the anti-selfdual Lagrangian framework by enabling variational treatment of unbounded, non-self-adjoint operators, notably transport and transport-Diffusion PDEs, via compositions with skew-adjoint (modulo boundary) operators and Yosida-type regularization. It develops Lax-Milgram-type results, a general evolution principle yielding semigroups, and parabolic initial-value resolutions, establishing existence, regularity, and boundary-compatibility for nonlinear PDEs outside classical Euler–Lagrange settings. Key contributions include the construction of ASD Lagrangians under unbounded operators, variational resolutions for stationary and evolution problems, and applications to nonlinear boundary-value and parabolic problems driven by transport-type operators. The results offer a versatile variational pathway to handle dissipative dynamics with transport components, with potential impact on nonlinear PDE theory and boundary-value problems exhibiting strong non-self-adjointness.

Abstract

This paper is a continuation of [13], where new variational principles were introduced based on the concept of anti-selfdual (ASD) Lagrangians. We continue here the program of using these Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary equations of the form as well as i dissipative evolutions of the form were is a convex potential on an infinite dimensional space. In this paper, the emphasis is on the cases where the differential operators involved are not necessarily bounded, hence completing the results established in [13] for bounded linear operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by the transport operator with or without a diffusion term.

Paper Structure

This paper contains 13 sections, 24 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. ASD Lagrangians and unbounded operators
  3. ASD Lagrangians and unbounded skew-adjoint operators
  4. The transport operator $A =\vec{a}\cdot\vec{\nabla} u+\frac{(\vec{\nabla} \cdot \vec{a})}{2} u$
  5. Anti-symmetric operators and ASD Lagrangians
  6. Variational resolution of equations of the form $-Au\in \partial {\varphi} (u)$
  7. A Lax-Milgram type result for unbounded operators
  8. Applications to PDE involving the transport operator
  9. ASD Lagrangians and maximal monotone operators
  10. A variational principle for general evolution equations
  11. Variational resolution of parabolic initial-value problems
  12. Parabolic equations involving a diffusion term
  13. Parabolic equations driven by first-order operators

Key Result

Proposition 2.1

Let L:X\times X^*\to \R be an ASD Lagrangian on a reflexive Banach space X such that for ever p\in X^*, the function x\to L(x,p) is bounded on the bounded sets of X. Let A:D(A) \to X^* be skew-adjoint modulo the boundary operators (b_1,b_2):X\to H_1\times H_2. Then the Lagrangian defined by is anti-self dual on X.

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1
  • Lemma 2.4
  • Lemma 2.5
  • Proposition 3.1
  • Proposition 3.2
  • Theorem 3.1
  • Theorem 3.2
  • ...and 17 more