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Variational and Quasi-variational solutions to thick flows

Jos\é Francisco Rodrigues, Lisa Santos

TL;DR

The paper develops a rigorous framework for thick, incompressible fluid flows under gradient-type constraints by casting the dynamics as evolution variational and quasi-variational inequalities with a variable deformation-rate threshold $|\mathbf D\boldsymbol u|\le \psi$. It establishes the existence and, for viscous flows, the uniqueness of strong and weak solutions, including inviscid limits, and introduces generalized Lagrange multipliers to represent constraint effects. A central contribution is the extension to quasi-variational inequalities with solution-dependent thresholds $\Psi[\boldsymbol u]$, using fixed-point methods to prove existence (and, in contraction regimes, uniqueness) of both weak and strong quasi-variational solutions, along with associated multiplier formulations. The work provides concrete mechanisms (nonlocal and parabolic-coupled thresholds) to realize $\Psi$ and offers explicit continuous dependence estimates that undergird well-posedness and potential numerical schemes for thick-fluid models with threshold-induced viscosity changes.

Abstract

We formulate the flow of thick fluids as evolution variational and quasi-variational inequalities, with a variable threshold on the absolute value of the deformation rate tensor. In the variational case, we show the existence and uniqueness of strong and weak solutions in the viscous case and also the existence of strong and weak solutions in the inviscid case. These problems correspond to solve, respectively, the Navier-Stokes and the Euler equations with an additional generalised Lagrange multiplier associated with the threshold on the deformation rate tensor. Applying the continuous dependence of strong and weak solutions to the variational inequalities for the Navier-Stokes with constraints on the derivatives, and on their respective generalised Lagrange multipliers, we can solve the case of the variable threshold depending on the solution itself that correspond to quasi-variational problems. \vspace{2mm} $$ \text{Dedicated to Vsevolod Alekseevich Solonnikov, {\em in memoriam}}$$

Variational and Quasi-variational solutions to thick flows

TL;DR

The paper develops a rigorous framework for thick, incompressible fluid flows under gradient-type constraints by casting the dynamics as evolution variational and quasi-variational inequalities with a variable deformation-rate threshold . It establishes the existence and, for viscous flows, the uniqueness of strong and weak solutions, including inviscid limits, and introduces generalized Lagrange multipliers to represent constraint effects. A central contribution is the extension to quasi-variational inequalities with solution-dependent thresholds , using fixed-point methods to prove existence (and, in contraction regimes, uniqueness) of both weak and strong quasi-variational solutions, along with associated multiplier formulations. The work provides concrete mechanisms (nonlocal and parabolic-coupled thresholds) to realize and offers explicit continuous dependence estimates that undergird well-posedness and potential numerical schemes for thick-fluid models with threshold-induced viscosity changes.

Abstract

We formulate the flow of thick fluids as evolution variational and quasi-variational inequalities, with a variable threshold on the absolute value of the deformation rate tensor. In the variational case, we show the existence and uniqueness of strong and weak solutions in the viscous case and also the existence of strong and weak solutions in the inviscid case. These problems correspond to solve, respectively, the Navier-Stokes and the Euler equations with an additional generalised Lagrange multiplier associated with the threshold on the deformation rate tensor. Applying the continuous dependence of strong and weak solutions to the variational inequalities for the Navier-Stokes with constraints on the derivatives, and on their respective generalised Lagrange multipliers, we can solve the case of the variable threshold depending on the solution itself that correspond to quasi-variational problems. \vspace{2mm}
Paper Structure (5 sections, 10 theorems, 166 equations)

This paper contains 5 sections, 10 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Strong solutions to variational inequalities
  3. Weak solutions to variational inequalities
  4. Existence of generalised Lagrange multipliers
  5. Solutions of the quasi-variational inequality with $\nu>0$

Key Result

Theorem 2.1

Assuming *, psi, assumptions and for each $\nu\ge0$, there exists a strong solution $\boldsymbol u\in H^1(0,T;\mathbb H)\cap \mathscr V_\infty^\cap$ of the evolution variational inequality Besides, the solution is unique when $\nu>0$.

Theorems & Definitions (26)

  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • ...and 16 more