Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1310.8474

Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy

Eduard Feireisl·Elisabetta Rocca·Giulio Schimperna·Arghir Zarnescu
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we prove the existence of global in time weak solutions for an evolutionary PDE system modelling nonisothermal Landau-de Gennes nematic liquid crystal (LC) flows in three dimensions of space. In our model, the incompressible Navier-Stokes system for the macroscopic velocity $\vu$ is coupled to a nonlinear convective parabolic equation describing the evolution of the Q-tensor $\QQ$, namely a tensor-valued variable representing the normalized second order moments of the probability distribution function of the LC molecules. The effects of the (absolute) temperature $\vt$ are prescribed in the form of an energy balance identity complemented with a global entropy production inequality. Compared to previous contributions, we can consider here the physically realistic singular configuration potential $f$ introduced by Ball and Majumdar. This potential gives rise to severe mathematical difficulties since it introduces, in the Q-tensor equation, a term which is at the same time singular in $\QQ$ and degenerate in $\vt$. To treat it a careful analysis of the properties of $f$, particularly of its blow-up rate, is carried out.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. 2013-10-31. Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy. https://arxiv.org/abs/1310.8474

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Blow-Up Dynamics for the $L^2$ critical case of the $2$D Zakharov-Kuznetsov equation

We study blow-up dynamics for the $L^2$-critical cubic Zakharov--Kuznetsov equation in two dimensions, \[ \partial_tu+\partial_{x_1}(Δu+u^3)=0 \qquad\text{on }\mathbb R^2. \] For a class of localized $H^1$ perturbations of the ground state $Q$, we establish a trichotomy near the soliton manifold: exit from a small $L^2$-tube, global asymptotic stability, or finite-time blow-up. In the stable blow-up regime, the solution concentrates a single bubble and \[ λ(t)\sim \ell_0(T-t)^{1/(3-c)}, \] where $\ell_0>0$ depends on the initial datum and $c\in(1,2)$ is an explicit constant determined by the transverse tail of the first-order approximate profile. Consequently, \[ \|\nabla u(t)\|_{L^2} \sim \frac{\|\nabla Q\|_{L^2}} {\ell_0(T-t)^{1/(3-c)}}. \] After subtraction of the concentrating soliton, the radiation converges strongly in $L^p(\mathbb R^2)$ for every $2\leq p<\infty$ to a common nonzero profile $u^*$, while \[ u^*\notin H^s(\mathbb R^2) \qquad\text{for every }s\geq\frac c2. \] The stable blow-up branch is open in the relative $H^1$ topology of the localized class. Finally, every non-soliton datum in this class with non-positive energy blows up in finite time. Interval-arithmetic computer-assisted proofs certify the numerical inputs to the virial coercivity argument. They also yield a rigorous enclosure of $c$, justifying the polynomial moment of order $21$ imposed on the initial data.

math.AP↗
arXiv2026

Propagation of wave packets close to conical intersections

In this paper, we study the propagation of wave packets close to conical intersections with respect to a system of two Schr{ö}dinger equations presenting a codimension 2 crossing. We focus on the dynamics that occur when the wave packets pass through an area close to the crossing, and our main results provide an explicit formula for the outgoing wave packet in terms of the incoming one, with a complete description of its phase and of the classical trajectories it follows, including a drift.

math.AP↗
arXiv2026

A Volterra Calculus for Lie Groupoids

A pseudodifferential Volterra calculus for inverting parabolic differential equations on Lie groupoids is introduced. This enables the study of fundamental solutions of various cases of heat flows on singular manifolds with corners with non-resonant boundary indicial symbols, such as the $b$-manifolds, as well as other geometric bisection covariant heat flows. We also establish the short time asymptotic expansion for the heat kernel of a positive, elliptic differential operator on a Lie groupoid that acts on suitable Sobolev Hilbert modules and is positive definite with respect to the appropriate $L^2$ inner product.

math.AP↗