Search arXiv⌕ Search

arXiv subjects

Yuning Liu

Publications and source records attributed to Yuning Liu.

16 recordsLinked to original sources
arXiv2026

Contact set of solutions to gradient flow of Landau-de Gennes energy with a singular entropy potential

This paper investigates the convexity structure and contact set of the $Q$-tensor flow with anisotropic Landau--de Gennes elastic energy and a singular entropy potential $f(Q)$. The flow naturally exhibits a coercivity property, yielding the $H^1$ regularity of $f(Q)$ for almost every time. Combining this $H^1$ regularity with spherical averaging and capacity theory, the contact set $$\mathcal{C}(Q)\triangleq \{x\mid f(Q(x))=\infty\}$$ is characterized. In particular, it is shown to be empty in two spatial dimensions and to have Hausdorff dimension at most one in three spatial dimensions.

math.AP↗
arXiv2025

Social Media Harm Abatement: Mechanisms for Transparent Public Health Assessment

Social media platforms have been accused of causing a range of harms, resulting in dozens of lawsuits across jurisdictions. These lawsuits are situated within the context of a long history of American product safety litigation, suggesting opportunities for remediation outside of financial compensation. Anticipating that at least some of these cases may be successful and/or lead to settlements, this article outlines an implementable mechanism for an abatement and/or settlement plan capable of mitigating abuse. The paper describes the requirements of such a mechanism, implications for privacy and oversight, and tradeoffs that such a procedure would entail. The mechanism is framed to operate at the intersection of legal procedure, standards for transparent public health assessment, and the practical requirements of modern technology products.

cs.CY↗
arXiv2024

Existence of curvature flow with forcing in a critical Sobolev space

Suppose that a closed 1-rectifiable set $\Gamma_0\subset\mathbb R^2$ of finite 1-dimensional Hausdorff measure and a vector field $u$ in a dimensionally critical Sobolev space are given. It is proved that, starting from $\Gamma_0$, there exists a non-trivial flow of curves with the normal velocity given by the sum of the curvature and the given vector field $u$. The motion law is satisfied in the sense of Brakke and the flow exists through singularities.

math.AP↗
arXiv2022

Phase Transition of Parabolic Ginzburg--Landau Equation with Potentials of High-Dimensional Wells

In this work, we study the co-dimensional one interface limit and geometric motions of parabolic Ginzburg--Landau systems with potentials of high-dimensional wells. The main result generalizes the one by Lin et al. (Comm. Pure Appl. Math., 65(6):833-888, 2012) to a dynamical case. In particular combining modulated energy methods and weak convergence methods, we derive the limiting harmonic heat flows in the inner and outer bulk regions segregated by the sharp interface, and a non-standard boundary condition for them. These results are valid provided that the initial datum of the system is well-prepared under natural energy assumptions.

math.AP↗
arXiv2022

The sharp interface limit of a Navier--Stokes/Allen--Cahn system with constant mobility: Convergence rates by a relative energy approach

We investigate the sharp interface limit of a diffuse interface system that couples the Allen--Cahn equation with the instationary Navier--Stokes system in a bounded domain in $\mathbb{R}^d$ with $d \in \{2,3\}$. This model is used to describe a propagating front in a viscous incompressible flow with the width of the transition layer being characterized by a small parameter $\varepsilon>0$. We show that the solutions converge to a limit two-phase fluid system with surface tension that couples the mean curvature flow and the Navier--Stokes system. The main assumptions are that the evolution of the limit system is sufficiently regular and that the associated evolving interface does not intersect the boundary of the container. For quantitatively well-prepared initial data, we even establish an optimal convergence rate. This is the first rigorous result of this kind which is valid in all physically relevant ambient dimensions.

math.AP↗
arXiv2021

Phase transition of anisotropic Ginzburg--Landau equation

We study the effective geometric motions of an anisotropic Ginzburg--Landau equation with a small parameter $\varepsilon>0$ which characterizes the width of the transition layer. For well-prepared initial datum, we show that as $\varepsilon$ tends to zero the solutions will develop a sharp interface limit which evolves under mean curvature flow. The bulk limits of the solutions correspond to a vector field $\mathbf{u}(x,t)$ which is of unit length on one side of the interface, and is zero on the other side. The proof combines the modulated energy method and weak convergence methods. In particular, by a (boundary) blow-up argument we show that $\mathbf{u}$ must be tangent to the sharp interface. Moreover, it solves a geometric evolution equation for the Oseen--Frank model in liquid crystals.

math.AP↗
arXiv2020

Regularity of a gradient flow generated by the anisotropic Landau-de Gennes energy with a singular potential

In this paper we study a gradient flow generated by the Landau-de Gennes free energy that describes nematic liquid crystal configurations in the space of $Q$-tensors. This free energy density functional is composed of three quadratic terms as the elastic energy density part, and a singular potential in the bulk part that is considered as a natural enforcement of a physical constraint on the eigenvalues of $Q$. The system is a non-diagonal parabolic system with a singular potential which trends to infinity logarithmically when the eigenvalues of $Q$ approaches the physical boundary. We give a rigorous proof that for rather general initial data with possibly infinite free energy, the system has a unique strong solution after any positive time $t_0$. Furthermore, this unique strong solution detaches from the physical boundary after a sufficiently large time $T_0$. We also give estimate of the Hausdorff measure of the set where the solution touches the physical boundary and thus prove a partial regularity result of the solution in the intermediate stage $(0,T_0)$.

math.AP↗
arXiv2020

Nematic-Isotropic phase transition in Liquid crystals: a variational derivation of effective geometric motions

In this work, we study the nematic-isotropic phase transition based on the dynamics of the Landau--De Gennes theory of liquid crystals. At the critical temperature, the Landau--De Gennes bulk potential favors the isotropic phase and nematic phase equally. When the elastic coefficient is much smaller than that of the bulk potential, a scaling limit can be derived by formal asymptotic expansions: the solution gradient concentrates on a closed surface evolving by mean curvature flow. Moreover, on one side of the surface the solution tends to the nematic phase which is governed by the harmonic map heat flow into the sphere while on the other side, it tends to the isotropic phase. To rigorously justify such a scaling limit, we prove a convergence result by combining weak convergence methods and the modulated energy method. Our proof applies as long as the limiting mean curvature flow remains smooth.

math.AP↗
arXiv2019

Phase-field approximation of the Willmore flow

We investigate the phase-field approximation of the Willmore flow. This is a fourth-order diffusion equation with a parameter $\epsilon>0$ that is proportional to the thickness of the diffuse interface. We show rigorously that for well-prepared initial data, as $\epsilon$ trends to zero the level-set of solution will converge to motion by Willmore flow before the singularity of the later occurs. This is done by constructing an approximate solution from the limiting flow via matched asymptotic expansions, and then estimating its difference with the real solution. The crucial step and also the major contribution of this work is to show a spectrum condition of the linearized operator at the optimal profile. This is a fourth-order operator written as the sum of the squared Allen-Cahn operator and a singular linear perturbation.

math.AP↗
arXiv2018

Global Well-posedness of the Two Dimensional Beris-Edwards System with General Laudau-de Gennes Free Energy

In this paper, we consider the Beris-Edwards system for incompressible nematic liquid crystal flows. The system under investigation consists of the Navier-Stokes equations for the fluid velocity $\mathbf{u}$ coupled with an evolution equation for the order parameter $Q$-tensor. One important feature of the system is that its elastic free energy takes a general form and in particular, it contains a cubic term that possibly makes it unbounded from below. In the two dimensional periodic setting, we prove that if the initial $L^\infty$-norm of the $Q$-tensor is properly small, then the system admits a unique global weak solution. The proof is based on the construction of a specific approximating system that preserves the $L^\infty$-norm of the $Q$-tensor along the time evolution.

math.AP↗
arXiv2018

Control of harmonic map heat flow with an external field

We investigate the control problem of harmonic map heat flow by means of an external magnetic field. In contrast to the situation of a parabolic system with internal or boundary control, the magnetic field acts as the coefficients of the lower order terms of the equation. We show that for initial data whose image stays in a hemisphere, with one control acting on a subset of the domain plus a spatial-independent control acting on the whole domain, the state of the system can be steered to any ground state, i.e. any given unit vector, within any short time. To achieve this, in the first step a spatial independent control is applied to steer the solution into a small neighborhood of the peak of the hemisphere. Then under stereographic projection, the original system is reduced to an internal parabolic control system with initial data sufficiently close to $0$ such that the existing method for local controllability can be applied. The key process in this step is to give an explicit solution of an underdetermined algebraic system such that the affine type control can be converted into an internal control.

math.AP↗
arXiv2017

The small Deborah number limit of the Doi-Onsager equation without hydrodynamics

We study the small Deborah number limit of the Doi-Onsager equation for the dynamics of nematic liquid crystals without hydrodynamics. This is a Smoluchowski-type equation that characterizes the evolution of a number density function, depending upon both particle position $x\in \mathbb{R}^d(d=2,3)$ and orientation vector $m\in\mathbb{S}^2$ (the unit sphere). We prove that, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge strongly to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat flow into $\mathbb{S}^2$. This flow is a special case of the gradient flow to the Oseen-Frank energy functional for nematic liquid crystals. The key ingredient is to show the strong compactness of the family of number density functions and the proof relies on the strong compactness of the corresponding second moment (or the $Q$-tensor), a spectral decomposition of the linearized operator near the limit local equilibrium distribution, as well as the energy dissipation estimate.

math.AP↗
arXiv2016

On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals

This work is concerned with the solvability of a Navier-Stokes/$Q$-tensor coupled system modeling the nematic liquid crystal flow on a bounded domain in three dimensional Euclidian space with strong anchoring boundary condition for the order parameter. We prove the existence of local in time strong solution to the system with the anisotropic elastic energy. The proof is based on mainly two ingredients: first, we show that the Euler-Lagrange operator corresponding to the Landau-de Gennes free energy with general elastic coefficients fulfills the strong Legendre condition. This result together with a higher order energy estimate leads to the well-posedness of the linearized system, and then a local in time solution of the original system which is regular in temporal variable follows via a fixed point argument. Secondly, the hydrodynamic part of the coupled system can be reformulated into a quasi-stationary Stokes type equation to which the regularity theory of the generalized Stokes system, and then a bootstrap argument can be applied to enhance the spatial regularity of the local in time solution.

math.AP↗
arXiv2016

The Oseen-Frank limit of Onsager's molecular theory for liquid crystals

We study the relationship between Onsager's molecular theory and the Oseen-Frank theory for nematic liquid crystals. Under the molecular setting, we consider the free energy that includes the effects of nonlocal molecular interactions. By imposing the strong anchoring boundary condition on the second moment of the number density function, we prove the existence of global minimizers for the free energy. Moreover, when the re-scaled interaction distance tends to zero, the corresponding global minimizers will converge to a uniaxial distribution whose orientation is described by a minimizer of Oseen-Frank energy.

math.AP↗
arXiv2012

Unique Continuation and Observability Estimates for 2-D Stokes Equations with the Navier Slip Boundary Condition

This paper presents a unique continuation estimate for 2-D Stokes equations with the Naiver slip boundary condition in a bounded and simply connected domain. Consequently, an observability estimate for this equation from a subset of positive measure in time follows from the aforementioned unique continuation estimate and the new strategy developed in [16]. Several applications of the above-mentioned observability estimate to control problems of the Stokes equations are given.

math.AP↗