Topic overview

math.AP

5324 works7305 researchers0 institutions

Topic snapshot

What this area looks like now

5324works

7305authors

0experts visible

0communities

Next steps

Move from topic reading into action

The graph preview below keeps the nearby papers, people and communities visible in the same reading flow.

Inspect nearby papers, researchers, institutions and communities without opening a separate graph page.

Building this graph slice

BZPEER is loading the nearby papers, people, topics and institutions for this page.

preprint2013arXiv

Global Well-Posedness and Large Time Asymptotic Behavior of Classical Solutions to the Compressible Navier-Stokes Equations with Vacuum

This paper concerns the global well-posedness and large time asymptotic behavior of strong and classical solutions to the Cauchy problem of the Navier-Stokes equations for viscous compressible barotropic flows in two or three spatial dimensions with vacuum as far field density. For strong and classical solutions, some a priori decay with rates (in large time) for both the pressure and the spatial gradient of the velocity field are obtained provided that the initial total energy is suitably {small.} Moreover, by using these key decay rates and some analysis on the expansion rates of the essential support of the density, we establish the global existence and uniqueness of classical solutions (which may be of possibly large oscillations) in two spatial dimensions, provided the smooth initial data are of small total energy. In addition, the initial density can even have compact support. This, in particular, yields the global regularity and uniqueness of the re-normalized weak solutions of Lions-Feireisl to the two-dimensional compressible barotropic flows for all adiabatic number $γ>1$ provided that the initial total energy is small.

preprint2013arXiv

On a foliation-covariant elliptic operator on null hypersurfaces

We introduce a new elliptic operator on null hypersurfaces of four-dimensional Lorentzian manifolds. This operator depends on the first and second fundamental forms of the sections of a foliation of the null hypersurface and its novelty originates from its covariant transformation under change of foliation. It thus provides at any point an elliptic structure intimately connected with the geometry of the null hypersurface, independent of the choice of a specific section through that point. No analytic or algebraic symmetries or other conditions are imposed on the metric. The spectral properties of this elliptic operator are relevant to the evolution of the wave equation, and in particular, the existence of conservation laws along null hypersurfaces.

preprint2005arXiv

Analytic discs, plurisubharmonic hulls, and non-compactness of the d-bar-Neumann operator

We show that a complex manifold M in the boundary of a smooth bounded pseudoconvex domain in C^n is an obstruction to compactness of the d-bar-Neumann operator on the domain, provided that at some point of M, the Levi form has the maximal possible rank n-1-dim(M) (i.e. the domain is strictly pseudoconvex in the directions transverse to M). In particular, an analytic disc is an obstruction, provided that at some point of the disc, the Levi form has only one zero eigenvalue. We also show that a boundary point where the Levi form has only one zero eigenvalue can be picked up by the plurisubharmonic hull of a set only via an analytic disc in the boundary.

preprint2015arXiv

Transient thermal mixed boundary value problems in the half-space

The Wiener-Hopf and Cagniard-de Hoop techniques are employed in order to solve a range of transient thermal mixed boundary value problems on the half-space. The thermal field is determined via a rapidly convergent integral, which can be evaluated straightforwardly and quickly on a desktop PC.

preprint2015arXiv

Spectra and stability of spatially periodic pulse patterns: Evans function factorization via Riccati transformation

In the spectral stability analysis of localized patterns to singular perturbed evolution problems, one often encounters that the Evans function respects the scale separation. In such cases the Evans function of the full linear stability problem can be approximated by a product of a slow and a fast reduced Evans function, which correspond to properly scaled slow and fast singular limit problems. This feature has been used in several spectral stability analyses in order to reduce the complexity of the linear stability problem. In these studies the factorization of the Evans function was established via geometric arguments that need to be customized for the specific equations and solutions under consideration. In this paper we develop an alternative factorization method. In this analytic method we use the Riccati transformation and exponential dichotomies to separate slow from fast dynamics. We employ our factorization procedure to study the spectra associated with spatially periodic pulse solutions to a general class of multi-component, singularly perturbed reaction-diffusion equations. Eventually, we obtain expressions of the slow and fast reduced Evans functions, which describe the spectrum in the singular limit. The spectral stability of localized periodic patterns has so far only been investigated in specific models such as the Gierer-Meinhardt equations. Our spectral analysis significantly extends and formalizes these existing results. Moreover, it leads to explicit instability criteria.

preprint2016arXiv

On a differential equation with Caputo-Fabrizio fractional derivative of order $1<β\leq 2$ and application to mass-spring-damper system

In this work, we investigate a linear differential equation involving Caputo-Fabrizio fractional derivative of order $1<β\leq 2$. Under some assumptions the considered equation is reduced to an integer order differential equation and solutions for different cases are obtained in explicit forms. We also prove a uniqueness of a solution of an initial value problem with a nonlinear differential equation containing the Caputo-Fabrizio derivative. Application of our result to the mass-spring-damper motion is also presented.

preprint2007arXiv

Sobolev regularity of solutions of the cohomological equation

We refine the theory of the cohomological equation for translation flows on higher genus surfaces with the goal of proving optimal results on the Sobolev regularity of solutions and of distributional obstructions. For typical translation surfaces our results are sharp and we find the expected relation between the regularity of the distributional obstructions and the Lyapunov exponents of the Kontsevich-Zorich renormalization cocycle. As a consequence we exactly determine the dimension of the space of obstructions in each Sobolev regularity class in terms of the Kontsevich-Zorich exponents. For a fixed arbitrary translation surface and a typical direction, our results are probably not optimal but are the best which can be achieved with the available harmonic analysis techniques we have introduced in an earlier paper.

preprint2014arXiv

Functional central limit theorem for Brownian particles in domains with Robin boundary condition

We rigorously derive non-equilibrium space-time fluctuation for the particle density of a system of reflected diffusions in bounded Lipschitz domains in $\mathbb R^d$. The particles are independent and are killed by a time-dependent potential which is asymptotically proportional to the boundary local time. We generalize the functional analytic framework introduced by Kotelenez [19, 20] to deal with time-dependent perturbations. Our proof relies on Dirichlet form method rather than the machineries derived from Kotelenez's sub-martingale inequality. Our result holds for any symmetric reflected diffusion, for any bounded Lipschitz domain and for any dimension $d\geq 1$.

preprint2016arXiv

Parabolic BMO and the forward-in-time maximal operator

We study if the parabolic forward-in-time maximal operator is bounded on parabolic BMO. It turns out that for non-negative functions the answer is positive, but the behaviour of sign changing functions is more delicate. The class parabolic BMO and the forward-in-time maximal operator originate from the regularity theory of nonlinear parabolic partial differential equations. In addition to that context, we also study the question in dimension one.

preprint2014arXiv

Nonzero radial solutions for a class of elliptic systems with nonlocal BCs on annular domains

We provide new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for systems of Hammerstein integral equations. Some of the criteria involve a comparison with the spectral radii of some associated linear operators. We apply our results to prove the existence of multiple nonzero radial solutions for some systems of elliptic boundary value problems subject to nonlocal boundary conditions. Our approach is topological and relies on the classical fixed point index. We present an example to illustrate our theory.

preprint2014arXiv

Positive solutions of a nonlocal Caputo fractional BVP

We discuss the existence of multiple positive solutions for a nonlocal fractional problem recently considered by Nieto and Pimental. Our approach relies on classical fixed point index.

preprint2016arXiv

Multi- to one-dimensional transportation

Fix probability densities $f$ and $g$ on open sets $X \subset \mathbf{R}^m$ and $Y \subset \mathbf{R}^n$ with $m\ge n\ge1$. Consider transporting $f$ onto $g$ so as to minimize the cost $-s(x,y)$. We give a non-degeneracy condition (a) on $s \in C^{1,1}$ which ensures the set of $x$ paired with [$g$-a.e.] $y\in Y$ lie in a codimension $n$ submanifold of $X$. Specializing to the case $m>n=1$, we discover a nestedness criteria relating $s$ to $(f,g)$ which allows us to construct a unique optimal solution in the form of a map $F:X \longrightarrow \overline Y$. When $s\in C^2 \cap W^{3,1}$ and $\log f$ and $\log g$ are bounded, the Kantorovich dual potentials $(u,v)$ satisfy $v \in C^{1,1}_{loc}(Y)$, and the normal velocity $V$ of $F^{-1}(y)$ with respect to changes in $y$ is given by $V(x) = v"(f(x))-s_{yy}(x,f(x))$. Positivity (b) of $V$ locally implies a Lipschitz bound on $f$; moreover, $v \in C^2$ if ${F^{-1}(y)}$ intersects $\partial X \in C^1$ transversally (c). On subsets where (a)-(c) can be be quantified, for each integer $r \ge1$ the norms of $u,v \in C^{r+1,1}$ and $F \in C^{r,1}$ are controlled by these bounds, $||\log f,\log g, \partial X ||_{C^{r-1,1}}, ||\partial X||_{C^{1,1}}$, $||s||_{C^{r+1,1}}$, and the smallness of $F^{-1}(y)$. We give examples showing regularity extends from $X$ to part of $\bar X$, but not from $Y$ to $\bar Y$. We also show that when $s$ remains nested for all $(f,g)$, the problem in $\mathbf{R}^m \times \mathbf{R}$ reduces to a supermodular problem in $\mathbf{R} \times \mathbf{R}$.

preprint2014arXiv

A topological approach to the existence and multiplicity of positive solutions of $(p,q)$-Laplacian systems

In this paper we develop a new theory for the existence, localization and multiplicity of positive solutions for a class of non-variational,quasilinear, elliptic systems. In order to do this, we provide a fairly general abstract framework for the existence of fixed points of nonlinear operators acting on cones that satisfy an inequality of Harnack type. Our methodology relies on fixed point index theory. We also provide a non-existence result and an example to illustrate the theory.

preprint2015arXiv

Relativistic Chern--Simons--Higgs Vortex Equations

An existence theorem is established for the solutions to the non-Abelian relativistic Chern--Simons--Higgs vortex equations over a doubly periodic domain when the gauge group $G$ assumes the most general and important prototype form, $G=SU(N)$

preprint2015arXiv

Schrödinger Operators With $A_\infty$ Potentials

We study the heat kernel $p(x,y,t)$ associated to the real Schrödinger operator $H = -Δ+ V$ on $L^2(\mathbb{R}^n)$, $n \geq 1$. Our main result is a pointwise upper bound on $p$ when the potential $V \in A_\infty$. In the case that $V\in RH_\infty$, we also prove a lower bound. Additionally, we compute $p$ explicitly when $V$ is a quadratic polynomial.

preprint2013arXiv

Chern--Simons Vortices in the Gudnason Model

We present a series of existence theorems for multiple vortex solutions in the Gudnason model of the ${\cal N}=2$ supersymmetric field theory where non-Abelian gauge fields are governed by the pure Chern--Simons dynamics at dual levels and realized as the solutions of a system of elliptic equations with exponential nonlinearity over two-dimensional domains. In the full plane situation, our method utilizes a minimization approach, and in the doubly periodic situation, we employ an-inequality constrained minimization approach. In the latter case, we also obtain sufficient conditions under which we show that there exist at least two gauge-distinct solutions for any prescribed distribution of vortices. In other words, there are distinct solutions with identical vortex distribution, energy, and electric and magnetic charges.

preprint2013arXiv

Horizon Instability of Extremal Black Holes

We show that axisymmetric extremal horizons are unstable under linear scalar perturbations. Specifically, we show that translation invariant derivatives of generic solutions to the wave equation do not decay along such horizons as advanced time tends to infinity, and in fact, higher order derivatives blow up. This result holds in particular for extremal Kerr-Newman and Majumdar-Papapetrou spacetimes and is in stark contrast with the subextremal case for which decay is known for all derivatives along the event horizon.

preprint2015arXiv

Invariant distributions and X-ray transform for Anosov flows

For Anosov flows preserving a smooth measure on a closed manifold $\mathcal{M}$, we define a natural self-adjoint operator $Π$ which maps into the space of invariant distributions in $\cap_{u<0} H^{u}(\mathcal{M})$ and whose kernel is made of coboundaries in $\cup_{s>0} H^{s}(\mathcal{M})$. We describe relations to Livsic theorem and recover regularity properties of cohomological equations using this operator. For Anosov geodesic flows on the unit tangent bundle $\mathcal{M}=SM$ of a compact manifold, we apply this theory to study questions related to $X$-ray transform on symmetric tensors on $M$: in particular we prove that injectivity implies surjectivity of X-ray transform, and we show injectivity for surfaces.

preprint2013arXiv

The characteristic gluing problem and conservation laws for the wave equation on null hypersurfaces

We obtain necessary and sufficient conditions for the existence of "conservation laws" on null hypersurfaces for the wave equation on general four-dimensional Lorentzian manifolds. Examples of null hypersurfaces exhibiting such conservation laws include the standard null cones of Minkowski spacetime and the degenerate horizons of extremal black holes. Another (limiting) example of such a conservation law is that which gives rise to the well-known Newman-Penrose constants along the null infinity of asymptotically flat spacetimes. The existence of such conservation laws can be viewed as an obstruction to a certain gluing construction for characteristic initial data for the wave equation. We initiate the general study of the latter gluing problem and show that the existence of conservation laws is in fact the only obstruction. Our method relies on a novel elliptic structure associated to a foliation with 2-spheres of a null hypersurface.

preprint2017arXiv

Nodal intersections and Geometric Control

This article contains a generalization of the authors' results on numbers of nodal points of eigenfunctions on "good curves" in analytic plane domains (arXiv:0710.0101). The term `good' means that the $L^2$ norms of restrictions of eigenfunctions of eigenvalue $λ^2$ to the curve are bounded below by $e^{- C λ}$. In this article, the result is generalized to all real analytic Riemannian manifolds $(M, g)$ of any dimension $m$ without boundary. Moreover, a similar lower bound is given for the Hausdorff $m-2$ measure of the intersection of the nodal set with a good real analytic hypersurface. Most of the article is devoted to giving a dynamical or geometric control condition for `goodness' of a hypersurface. The conditions are that the hypersurface $H$ be asymmetric with respect to geodesics and that the flowout of the unit vectors with footpoint on $H$ have full measure in $S^*M. $ This gives a partial answer to a question of Bourgain-Rudnick of characterizing hypersurfaces $H$ on which a sequence of eigenfunctions vanishes. We show that under our conditions, a positive density sequence cannot vanish on $H$ or even have smaller $L^2$ norms than $e^{- C λ}$

preprint2017arXiv

Non-local initial problem for second order time-fractional and space-singular equation

In this work, we consider an initial problem for second order partial differential equations with Caputo fractional derivatives in the time-variable and Bessel operator in the space-variable. For non-local boundary conditions, we present a solution of this problem in an explicit form representing it by the Fourier-Bessel series. The obtained solution is written in terms of multinomial Mittag-Leffler functions and first kind Bessel functions.

preprint2018arXiv

ODE and PDE based modeling of biological transportation networks

We study the global existence of solutions of a discrete (ODE based) model on a graph describing the formation of biological transportation networks, introduced by Hu and Cai. We propose an adaptation of this model so that a macroscopic (PDE based) system can be obtained as its formal continuum limit. We prove the global existence of weak solutions of the macroscopic PDE model. Finally, we present results of numerical simulations of the discrete model, illustrating the convergence to steady states, their non-uniqueness as well as their dependence on initial data and model parameters.

preprint2017arXiv

On concavity of the monopolist's problem facing consumers with nonlinear price preferences

A monopolist wishes to maximize her profits by finding an optimal price policy. After she announces a menu of products and prices, each agent $x$ will choose to buy that product $y(x)$ which maximizes his own utility, if positive. The principal's profits are the sum of the net earnings produced by each product sold. These are determined by the costs of production and the distribution of products sold, which in turn are based on the distribution of anonymous agents and the choices they make in response to the principal's price menu. In this paper, we provide a necessary and sufficient condition for the convexity or concavity of the principal's (bilevel) optimization problem, assuming each agent's disutility is a strictly increasing but not necessarily affine (i.e. quasilinear) function of the price paid. Concavity when present, makes the problem more amenable to computational and theoretical analysis; it is key to obtaining uniqueness and stability results for the principal's strategy in particular. Even in the quasilinear case, our analysis goes beyond previous work by addressing convexity as well as concavity, by establishing conditions which are not only sufficient but necessary, and by requiring fewer hypotheses on the agents' preferences.

Researcher

Wei Chen contributes to research discovery and scholarly infrastructure.

Open to collaborate

math.AP

What this area looks like now

Move from topic reading into action

See the topic as a live network

Building this graph slice

24 featured work(s)

Global Well-Posedness and Large Time Asymptotic Behavior of Classical Solutions to the Compressible Navier-Stokes Equations with Vacuum

On a foliation-covariant elliptic operator on null hypersurfaces

Analytic discs, plurisubharmonic hulls, and non-compactness of the d-bar-Neumann operator

Transient thermal mixed boundary value problems in the half-space

Spectra and stability of spatially periodic pulse patterns: Evans function factorization via Riccati transformation

On a differential equation with Caputo-Fabrizio fractional derivative of order $1<β\leq 2$ and application to mass-spring-damper system

Sobolev regularity of solutions of the cohomological equation

Functional central limit theorem for Brownian particles in domains with Robin boundary condition

Parabolic BMO and the forward-in-time maximal operator

Nonzero radial solutions for a class of elliptic systems with nonlocal BCs on annular domains

Positive solutions of a nonlocal Caputo fractional BVP

Multi- to one-dimensional transportation

A topological approach to the existence and multiplicity of positive solutions of $(p,q)$-Laplacian systems

Relativistic Chern--Simons--Higgs Vortex Equations

Schrödinger Operators With $A_\infty$ Potentials

Chern--Simons Vortices in the Gudnason Model

Horizon Instability of Extremal Black Holes

Invariant distributions and X-ray transform for Anosov flows

The characteristic gluing problem and conservation laws for the wave equation on null hypersurfaces

Nodal intersections and Geometric Control

Non-local initial problem for second order time-fractional and space-singular equation

ODE and PDE based modeling of biological transportation networks

On concavity of the monopolist&#39;s problem facing consumers with nonlinear price preferences

New explicit solutions to the $p$-Laplace equation based on isoparametric foliations

12 visible researcher(s)

Yang Liu

Lei Zhang

Wei Wang

Hao Wang

Yang Yang

Yang Zhang

Yang Li

Chang Liu

Wei Li

Jian Wang

Chao Zhang

Wei Chen

On concavity of the monopolist's problem facing consumers with nonlinear price preferences