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Dissipative solutions to a Beris-Edwards type model for compressible active nematic liquid crystals

Kuntal Bhandari, Apala Majumdar, Šárka Nečasová

TL;DR

This work proves the global existence of dissipative solutions for a compressible active nematic system of Beris–Edwards type in a bounded 3D domain, incorporating nonhomogeneous boundary data and inflow/outflow conditions. The authors develop a three-level approximation scheme—artificial viscosity in the continuity equation, convex regularization of the elastic potential, and Galerkin discretization of the momentum equation—and derive uniform energy bounds. They introduce Reynolds stress and energy defect measures to formulate dissipative solutions and carefully pass to the limits $oldsymbol extdelta o0$, $oldsymbol extvarepsilon o0$, and $n o ty$, obtaining weak formulations and an energy inequality that persists in the limit. The analysis accommodates a general pressure law $p( ho)$ with convexity properties and non-Newtonian stresses, enabling $ ho$-dependent pressures with $ ho>0$ and admitting the physically relevant range $oldsymbol o1< oldsymbol o$ in the dissipative setting. The results provide a robust mathematical foundation for active nematic hydrodynamics and offer a framework compatible with numerical schemes and potential extensions to more general rheologies and boundary conditions.

Abstract

We study the hydrodynamics of compressible active nematic liquid crystals in a three-dimensional and bounded domain, with a nonlinear viscosity tensor and nonhomogeneous boundary data, in a Landau-de Gennes framework. We prove the existence of dissipative solutions within a Beris-Edwards type model for active nematodynamics, which are weak solutions satisfying the underlying equations modulo a defect measure. The proof follows from a three level approximation scheme -- the Galerkin approximation, the classical parabolic regularization of the continuity equation, and the convex regularization of the potential generating the viscous stress. New techniques are required to deal with non-Newtonian stress tensor, larger classes of admissible pressure potentials and nonhomogeneous boundary conditions.

Dissipative solutions to a Beris-Edwards type model for compressible active nematic liquid crystals

TL;DR

This work proves the global existence of dissipative solutions for a compressible active nematic system of Beris–Edwards type in a bounded 3D domain, incorporating nonhomogeneous boundary data and inflow/outflow conditions. The authors develop a three-level approximation scheme—artificial viscosity in the continuity equation, convex regularization of the elastic potential, and Galerkin discretization of the momentum equation—and derive uniform energy bounds. They introduce Reynolds stress and energy defect measures to formulate dissipative solutions and carefully pass to the limits , , and , obtaining weak formulations and an energy inequality that persists in the limit. The analysis accommodates a general pressure law with convexity properties and non-Newtonian stresses, enabling -dependent pressures with and admitting the physically relevant range in the dissipative setting. The results provide a robust mathematical foundation for active nematic hydrodynamics and offer a framework compatible with numerical schemes and potential extensions to more general rheologies and boundary conditions.

Abstract

We study the hydrodynamics of compressible active nematic liquid crystals in a three-dimensional and bounded domain, with a nonlinear viscosity tensor and nonhomogeneous boundary data, in a Landau-de Gennes framework. We prove the existence of dissipative solutions within a Beris-Edwards type model for active nematodynamics, which are weak solutions satisfying the underlying equations modulo a defect measure. The proof follows from a three level approximation scheme -- the Galerkin approximation, the classical parabolic regularization of the continuity equation, and the convex regularization of the potential generating the viscous stress. New techniques are required to deal with non-Newtonian stress tensor, larger classes of admissible pressure potentials and nonhomogeneous boundary conditions.
Paper Structure (28 sections, 14 theorems, 191 equations)

This paper contains 28 sections, 14 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Concept of dissipative solution and main result
  3. Pressure--density equation of state
  4. Continuity equation -- mass conservation
  5. Momentum balance
  6. Concentration of active particles
  7. Nematic tensor order parameter
  8. Energy inequality and defect compatibility condition
  9. A-priori estimates
  10. Existence of approximate solutions
  11. Artificial viscosity approximation of the equation of continuity
  12. Galerkin approximation of the momentum balance
  13. Limit with respect to $\delta \to 0$
  14. Limits in the continuity, concentration and tensor parameter equations
  15. Limit in the momentum balance
  16. ...and 13 more sections

Key Result

Theorem 2.2

Let $T>0$ and $\Omega\subset \mathbb R^3$ be a bounded Lipschitz domain. Suppose that $p=p(\varrho)$ is given by S2 and $F=F(\mathbb D_x \mathbf{u})$ is given by F-convex--Coercivity-F. Then the compressible active nematic liquid crystal system dens-eq-1--nematic-tensor-eq-1, with initial conditions

Theorems & Definitions (16)

  • Definition 2.1: Dissipative solution
  • Theorem 2.2: Global existence of dissipative solution
  • Lemma 4.1: Crippa et al. Crippa-et-al
  • Lemma 4.2: Maximum principle; Crippa et al. Crippa-et-al
  • Lemma 4.3: Renormalization
  • Corollary 4.4: Abbatiello et al. Dissipative-Feireisl
  • Lemma 4.5
  • Remark 4.6
  • Proposition 4.7: Solution to approximate system
  • Proposition 5.1
  • ...and 6 more