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.Comment: 29p

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.Comment: 31 page

Regularity for Shape Optimizers: The Degenerate Case

We consider minimizers of $$F(\lambda_1(\Omega),\ldots,\lambda_N(\Omega)) + |\Omega|,$$ where $F$ is a function nondecreasing in each parameter, and $\lambda_k(\Omega)$ is the $k$-th Dirichlet eigenvalue of $\Omega$. This includes, in particular, functions $F$ which depend on just some of the first $N$ eigenvalues, such as the often studied $F=\lambda_N$. 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 $\Omega$ 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.Comment: Minor typos fixe

Superfluids Passing an Obstacle and Vortex Nucleation

We consider a superfluid described by the Gross-Pitaevskii equation passing an obstacle \epsilon^2 \Delta u+ u(1-|u|^2)=0 \ \mbox{in} \ {\mathbb R}^d \backslash \Omega, \ \ \frac{\partial u}{\partial \nu}=0 \ \mbox{on}\ \partial \Omega where $\Omega$ is a smooth bounded domain in ${\mathbb R}^d$ ($d\geq 2$), which is referred as the obstacle and $\epsilon>0$ is sufficiently small. We first construct a vortex free solution of the form $u= \rho_\epsilon (x) e^{i \frac{\Phi_\epsilon}{\epsilon}}$ with $\rho_\epsilon (x) \to 1-|\nabla \Phi^\delta(x)|^2, \Phi_\epsilon (x) \to \Phi^\delta (x)$ where $\Phi^\delta (x)$ is the unique solution for the subsonic irrotational flow equation \nabla ( (1-|\nabla \Phi|^2)\nabla \Phi )=0 \ \mbox{in} \ {\mathbb R}^d \backslash \Omega, \ \frac{\partial \Phi}{\partial \nu} =0 \ \mbox{on} \ \partial \Omega, \ \nabla \Phi (x) \to \delta \vec{e}_d \ \mbox{as} \ |x| \to +\infty and $|\delta | <\delta_{*}$ (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 $|\nabla \Phi^\delta (x)|^2$ (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.Comment: 21 pages; comments are very welcom

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.Comment: Update the wording and reference

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 \tau=\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

Global solutions of two dimensional incompressible viscoelastic flows with discontinuous initial data

The global existence of weak solutions of the incompressible viscoelastic flows in two spatial dimensions has been a long standing open problem, and it is studied in this paper. We show the global existence if the initial deformation gradient is close to the identity matrix in $L^2\cap L^\infty$, and the initial velocity is small in $L^2$ and bounded in $L^p$, for some $p>2$. While the assumption on the initial deformation gradient is automatically satisfied for the classical Oldroyd-B model, the additional assumption on the initial velocity being bounded in $L^p$ for some $p>2$ may due to techniques we employed. The smallness assumption on the $L^2$ norm of the initial velocity is, however, natural for the global well-posedness . One of the key observations in the paper is that the velocity and the \textquotedblleft effective viscous flux\textquotedblright $\mathcal{G}$ are sufficiently regular for positive time. The regularity of $\mathcal{G}$ leads to a new approach for the pointwise estimate for the deformation gradient without using $L^\infty$ bounds on the velocity gradients in spatial variables

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.Comment: 12 page

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.Comment: Phil. Trans. R. Soc., A, to appea

Systematic Low-Thrust Trajectory Optimization for a Multi-Rendezvous Mission using Adjoint Scaling

A deep-space exploration mission with low-thrust propulsion to rendezvous with multiple asteroids is investigated. Indirect methods, based on the optimal control theory, are implemented to optimize the fuel consumption. The application of indirect methods for optimizing low-thrust trajectories between two asteroids is briefly given. An effective method is proposed to provide initial guesses for transfers between close near-circular near-coplanar orbits. The conditions for optimality of a multi-asteroid rendezvous mission are determined. The intuitive method of splitting the trajectories into several legs that are solved sequentially is applied first. Then the results are patched together by a scaling method to provide a tentative guess for optimizing the whole trajectory. Numerical examples of optimizing three probe exploration sequences that contain a dozen asteroids each demonstrate the validity and efficiency of these methods