arXiv subjects

Fanghua Lin

Publications and source records attributed to Fanghua Lin.

At least 37 records · Page 2Linked to original sources

arXiv2020

Upper bounds of nodal sets for eigenfunctions of eigenvalue problems

The aim of this article is to provide a simple and unified way to obtain the sharp upper bounds of nodal sets of eigenfunctions for different types of eigenvalue problems on real analytic domains. The examples include biharmonic Steklov eigenvalue problems, buckling eigenvalue problems and champed-plate eigenvalue problems. The geometric measure of nodal sets are derived from doubling inequalities and growth estimates for eigenfunctions. It is done through analytic estimates of Morrey-Nirenberg and Carleman estimates.

math.AP↗

arXiv2020

Isotropic-Nematic Phase Transition and Liquid Crystal Droplets

Liquid crystal droplets are of great interest from physics and applications. Rigorous mathematical analysis is challenging as the problem involves harmonic maps (and in general the Oseen-Frank model), free interfaces and topological defects which could be either inside the droplet or on its surface along with some intriguing boundary anchoring conditions for the orientation configurations. In this paper, through a study of the phase transition between the isotropic and nematic states of liquid crystal based on the Ericksen model, we can show, when the size of droplet is much larger in comparison with the ratio of the Frank constants to the surface tension, a $Γ$-convergence theorem for minimizers. This $Γ$-limit is in fact the sharp interface limit for the phase transition between the isotropic and nematic regions when the small parameter $\varepsilon$, corresponding to the transition layer width, goes to zero. This limiting process not only provides a geometric description of the shape of the droplet as one would expect, and surprisingly it also gives the anchoring conditions for the orientations of liquid crystals on the surface of the droplet depending on material constants. In particular, homeotropic, tangential, and even free boundary conditions as assumed in earlier phenomenological modelings arise naturally provided that the surface tension, Frank and Ericksen constants are in suitable ranges.

math.AP↗

arXiv2019

Finite time blow-up for the nematic liquid crystal flow in dimension two

We consider the initial-boundary value problem of a simplified nematic liquid crystal flow in a bounded, smooth domain $Ω\subset \mathbb R^2$. Given any $k$ distinct points in the domain, we develop a new {\em inner--outer gluing method} to construct solutions which blow up exactly at those $k$ points as $t$ goes to a finite time $T$. Moreover, we obtain a precise description of the blow-up.

math.AP↗

arXiv2019

Vanishing Viscosity Limit for Incompressible Viscoelasticity in Two Dimensions

This paper studies the inviscid limit of the two-dimensional incompressible viscoelasticity, which is a system coupling a Navier-Stokes equation with a transport equation for the deformation tensor. The existence of global smooth solutions near the equilibrium with a fixed positive viscosity was known since the work of F. H. Lin, C. Liu, and P. Zhang in "On hydrodynamics of viscoelastic fluids". The inviscid case was solved recently by the second author Z. Lei. in "Global well-posedness of incompressible elastodynamics in two dimensions". While the latter was solely based on the techniques from the studies of hyperbolic equations, and hence the 2D problem is in general more challenge than that in higher dimensions, the former was relied crucially upon a dissipative mechanism. Indeed, after a symmetrization and a linearization around the equilibrium, the system of the incompressible viscoelasticity reduces to an incompressible system of damped wave equations for both the fluid velocity and the deformation tensor. These two approaches are not compatible. In this paper, we prove global existence of solutions, uniformly in both time $t \in [0, \infty)$ and viscosity $μ\geq 0$. This allows us to justify in particular the vanishing viscosity limit for all time. In order to overcome difficulties coming from the incompatibility between the purely hyperbolic limiting system and the systems with additional parabolic viscous perturbations, we introduce in this paper a rather robust method which may apply to a wide class of physical systems of similar nature. Roughly speaking, the method works in two dimensional case whenever the hyperbolic system satisfies intrinsically a "Strong Null Condition". For dimensions not less than three, the usual null condition is sufficient for this method to work.

math.AP↗

arXiv2018

Harmonic maps in connection of phase transitions with higher dimensional potential wells

This is in the sequel of authors' paper \cite{LPW} in which we had set up a program to verify rigorously some formal statements associated with the multiple component phase transitions with higher dimensional wells. The main goal here is to establish a regularity theory for minimizing maps with a rather non-standard boundary condition at the sharp interface of the transition. We also present a proof, under simplified geometric assumptions, of existence of local smooth gradient flows under such constraints on interfaces which are in the motion by the mean-curvature. In a forthcoming paper, a general theory for such gradient flows and its relation to Keller-Rubinstein-Sternberg's work \cite{KRS1, KRS2} on the fast reaction, slow diffusion and motion by the mean curvature would be addressed.

math.AP↗

arXiv2018

On ancient solutions of the heat equation

An explicit representation formula for all positive ancient solutions of the heat equation in the Euclidean case is found. In the Riemannian case with nonnegative Ricci curvature, a similar but less explicit formula is also found. Here it is proven that any positive ancient solution is the standard Laplace transform of positive solutions of the family of elliptic operator $Δ- s$ with $s>0$. Further relaxation of the curvature assumption is also possible. It is also shown that the linear space of ancient solutions of polynomial growth has finite dimension and these solutions are polynomials in time.

math.AP↗

arXiv2018

Riesz transform under perturbations via heat kernel regularity

Let $M$ be a complete non-compact Riemannian manifold. In this paper, we derive sufficient conditions on metric perturbation for stability of $L^p$-boundedness of the Riesz transform, $p\in (2,\infty)$. We also provide counter-examples regarding in-stability for $L^p$-boundedness of Riesz transform.

math.DG↗

arXiv2018

Nodal Sets and Doubling Conditions in Elliptic Homogenization

This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators $\{ \mathcal{L}_\e\}$ in divergence form with rapidly oscillating and periodic coefficients. We show that the $(d-1)$-dimensional Hausdorff measures of the nodal sets of solutions to $\mathcal{L}_\e (u_\e)=0$ in a ball in $\R^d$ are bounded uniformly in $\e>0$. The proof relies on a uniform doubling condition and approximation of $u_\e$ by solutions of the homogenized equation.

math.AP↗

arXiv2018

Superfluids Passing an Obstacle and Vortex Nucleation

We consider a superfluid described by the Gross-Pitaevskii equation passing an obstacle \[ε^2 Δu+ u(1-|u|^2)=0 \ \mbox{in} \ {\mathbb R}^d \backslash Ω, \ \ \frac{\partial u}{\partial ν}=0 \ \mbox{on}\ \partial Ω\] where $ Ω$ is a smooth bounded domain in $ {\mathbb R}^d$ ($d\geq 2$), which is referred as the obstacle and $ ε>0$ is sufficiently small. We first construct a vortex free solution of the form $ u= ρ_ε(x) e^{i \frac{Φ_ε}ε}$ with $ ρ_ε(x) \to 1-|\nabla Φ^δ(x)|^2, Φ_ε(x) \to Φ^δ(x) $ where $Φ^δ(x)$ is the unique solution for the subsonic irrotational flow equation \[ \nabla ( (1-|\nabla Φ|^2)\nabla Φ)=0 \ \mbox{in} \ {\mathbb R}^d \backslash Ω, \ \frac{\partial Φ}{\partial ν} =0 \ \mbox{on} \ \partial Ω, \ \nabla Φ(x) \to δ\vec{e}_d \ \mbox{as} \ |x| \to +\infty \] and $|δ| <δ_{*}$ (the sound speed). In dimension $d=2$, on the background of this vortex free solution we also construct solutions with single vortex close to the maximum or minimum points of the function $|\nabla Φ^δ(x)|^2$ (which are on the boundary of the obstacle). The latter verifies the vortex nucleation phenomena (for the steady states) in superfluids described by the Gross-Pitaevskii equations. Moreover, after some proper scalings, the limits of these vortex solutions are traveling wave solution of the Gross-Pitaevskii equation. These results also show rigorously the conclusions drawn from the numerical computations in \cite{huepe1, huepe2}. Extensions to Dirichlet boundary conditions, which may be more consistent with the situation in the physical experiments and numerical simulations (see \cite{ADP} and references therein) for the trapped Bose-Einstein condensates, are also discussed.

math.AP↗

arXiv2017

Regularity for Shape Optimizers: The Degenerate Case

We consider minimizers of \[ F(λ_1(Ω),\ldots,λ_N(Ω)) + |Ω|, \] where $F$ is a function nondecreasing in each parameter, and $λ_k(Ω)$ is the $k$-th Dirichlet eigenvalue of $Ω$. This includes, in particular, functions $F$ which depend on just some of the first $N$ eigenvalues, such as the often studied $F=λ_N$. The existence of a minimizer, which is also a bounded set of finite perimeter, was shown recently. Here we show that the reduced boundary of the minimizers $Ω$ is made up of smooth graphs, and examine the difficulties in classifying the singular points. Our approach is based on an approximation ("vanishing viscosity") argument, which--counterintuitively--allows us to recover an Euler-Lagrange equation for the minimizers which is not otherwise available.

math.AP↗

arXiv2017

Regularity for Shape Optimizers: The Nondegenerate Case

We consider minimizers of \[ F(λ_1(Ω),\ldots,λ_N(Ω)) + |Ω|, \] where $F$ is a function strictly increasing in each parameter, and $λ_k(Ω)$ is the $k$-th Dirichlet eigenvalue of $Ω$. Our main result is that the reduced boundary of the minimizer is composed of $C^{1,α}$ graphs, and exhausts the topological boundary except for a set of Hausdorff dimension at most $n-3$. We also obtain a new regularity result for vector-valued Bernoulli type free boundary problems.

math.AP↗

arXiv2016

On the Two-Dimensional Muskat Problem with Monotone Large Initial Data

We consider the evolution of two incompressible, immiscible fluids with different densities in porous media, known as the Muskat problem [21], which in two dimensions is analogous to the Hele-Shaw cell [26]. We establish, for a class of large and monotone initial data, the global existence of weak solutions. The proof is based on a local well-posedness result for the initial data with certain specific asymptotics at spatial infinity and a new maximum principle for the first derivative of the graph function.

math.AP↗

arXiv2016

On the Cauchy problem for two dimensional incompressible viscoelastic flows

We study the large-data Cauchy problem for two dimensional Oldroyd model of incompressible viscoelastic fluids. We prove the global-in-time existence of the Leray-Hopf type weak solutions in the physical energy space. Our method relies on a new $\textit{a priori}$ estimate on the space-time norm in $L^{\f32}_{loc}$ of the Cauchy-Green strain tensor $τ=\F\F^\top$, or equivalently the $L^3_{loc}$ norm of the Jacobian of the flow map $\F$. It allows us to rule out possible concentrations of the energy due to deformations associated with the flow maps. Following the general compactness arguments due to DiPerna and Lions (\cite{DL}, \cite{FNP}, \cite{PL}), and using the so-called \textit{effective viscous flux}, $\mathcal{G}$, which was introduced in our previous work \cite{HL}, we are able to control the possible oscillations of deformation gradients as well.

math.AP↗

arXiv2015

Finite time singularity of the nematic liquid crystal flow in dimension three

In this paper, we consider the initial and boundary value problem of a simplified nematic liquid crystal flow in dimension three and construct two examples of finite time singularity. The first example is constructed within the class of axisymmetric solutions, while the second example is constructed for any generic initial data $(u_0,d_0)$ that has sufficiently small energy, and $d_0$ has a nontrivial topology

math.AP↗

arXiv2014

Global existence of weak solutions of the nematic liquid crystal flow in dimensions three

For any bounded smooth domain $Ω\subset\mathbb R^3$, we establish the global existence of a weak solution $u:Ω\times (0,+\infty)\to\mathbb R^3\times\mathbb S^2$ of the initial-boundary value (or the Cauchy) problem of the simplified Ericksen-Leslie system (1.1) modeling the hydrodynamic flow of nematic liquid crystals for any initial and boundary (or Cauchy) data $(u_0. d_0)\in {\bf H}\times H^1(Ω,\mathbb S^2$), with $d_0(Ω)\subset\mathbb S^2_+$ (the upper hemisphere). Furthermore, ($u,d$) satisfies the global energy inequality (1.4).

math.AP↗

arXiv2014

Recent developments of analysis for hydrodynamic flow of nematic liquid crystals

The study of hydrodynamics of liquid crystal leads to many fasci- nating mathematical problems, which has prompted various interesting works recently. This article reviews the static Oseen-Frank theory and surveys some recent progress on the existence, regularity, uniqueness, and large time asymp- totic of the hydrodynamic flow of nematic liquid crystals. We will also propose a few interesting questions for future investigations.

math.AP↗

arXiv2014

Global Existence for Two Dimensional Incompressible Magnetohydrodynamic Flows with Zero Magnetic Diffusivity

The existence of global-in-time classical solutions to the Cauchy problem of incompressible Magnetohydrodynamic flows with zero magnetic diffusivity is considered in two dimensions. The linearization of equations is a degenerated parabolic-hyperbolic system. The solution is constructed as a small perturbation of a constant background in critical spaces. The deformation gradient has been introduced to decouple the subtle coupling between the flow and the magnetic field. The $L^1$ dissipation of the velocity is obtained.

math.AP↗

arXiv2014

Global Small Solutions to a Complex Fluid Model in 3D

In this paper, we provide a much simplified proof of the main result in [Lin and Zhang, Comm. Pure Appl. Math.,67(2014), 531--580] concerning the global existence and uniqueness of smooth solutions to the Cauchy problem for a 3D incompressible complex fluid model under the assumption that the initial data are close to some equilibrium states. Beside the classical energy method, the interpolating inequalities and the algebraic structure of the equations coming from the incompressibility of the fluid are crucial in our arguments. We combine the energy estimates with the $L^\infty$ estimates for time slices to deduce the key $L^1$ in time estimates. The latter is responsible for the global in time existence.

math.AP↗