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We propose a first rigorous homogenisation procedure in image-segmentation models by analysing the relative impact of (possibly random) fine-scale oscillations and phase-field regularisations for a family of elliptic functionals of Ambrosio and Tortorelli type, when the regularised volume term grows \emph{linearly} in ... |
We consider an overdetermined problem of Serrin-type with respect to an operator in divergence form with piecewise constant coefficients. We give sufficient condition for unique solvability near radially symmetric configurations by means of a perturbation argument relying on shape derivatives and the implicit function... |
We study special properties of solutions to the IVP associated to the Camassa-Holm equation on the line related to the regularity and the decay of solutions. The first aim is to show how the regularity on the initial data is transferred to the corresponding solution in a class containing the "peakon solutions"... |
This paper deals with the blow-up properties of the solutions of the semilinear heat equation |
We study the inverse problem of determining a magnetic Schrödinger operator in an unbounded closed waveguide from boundary measurements. We consider this problem with a general closed waveguide in the sense that we only require our unbounded domain to be contained into an infinite cylinder. |
We present a blow-up result for large data for relaxed compressible Navier-Stokes models avoiding the possibility of reaching the boundary of hyperbolicity. Thus a previous result is improved and further examples are given illustrating possible effects of a relaxation and contrasting the classical compressible Navier-... |
We study a class of free boundary systems with nonlocal diffusion, which are natural extensions of the corresponding free boundary problems of reaction diffusion systems. As before the free boundary represents the spreading front of the species, but here the population dispersal is described by "nonlocal diffusion... |
When two inclusions with high contrast material properties are located close to each other in a homogeneous medium, stress may become arbitrarily large in the narrow region between them. In this paper, we investigate such stress concentration in the two-dimensional Stokes flow when inclusions are the two-dimensional c... |
This survey is a slightly extended version of the lecture given by the author at the \emph{VI International Course of Mathematical Analysis in Andaluc\'ıa} (CIDAMA), in September 2014. Most results are contained (in a slightly less general setting) in the earlier paper [3] (Garrigós, Hartzstein, Signes, Torrea and... |
In this note, we characterize the sharp boundary condition such that the fractional harmonic extensions with Hölder regularity up to the boundary is globally Hölder continuous. The proofs are based on estimates of fractional harmonic measure decay and uniform fractional fatness of the complement of the domain. |
We study a non-local Cahn-Hilliard energy arising in the study of di-block copolymer melts, often referred to as the Ohta-Kawasaki energy in that context. In this model, two phases appear, which interact via a Coulombic energy. |
We consider here a problem of finding the sharp estimate for the boundedness of an arbitrary Calderón-Zygmund operator in $L^2(w)$, $w\in A_2$. We first prove that for $A_2$ weight $w$ one has that the norm a Calderon--Zygmund operator $T$ in $L^2(w)$ is bounded by the sum of its weak norm, the weak norm of its adjoin... |
We study the qualitative properties of functions belonging to the corresponding De Giorgi classes \begin{equation*} \int\limits_{B_{r(1-\sigma)}(x_{0})}\,\varPhi(x, |\nabla(u-k)_{\pm}|)\,dx \leqslant \gamma\,\int\limits_{B_{r}(x_{0})}\,\varPhi\bigg(x, \frac{(u-k)_{\pm}}{\sigma r}\bigg)\,dx, \end{equation*} where $\sigm... |
The JKO scheme is a time-discrete scheme of implicit Euler type that allows to construct weak solutions of evolution PDEs which have a Wasserstein gradient structure. The purpose of this work is to study the effect of replacing the classical quadratic optimal transport problem by the Schrödinger problem (\emph{a.k.a.}... |
In this paper, existence of solutions is established for critical exponential Kirchhoff systems on the Heisenberg group by using the variational method. The novelty of our paper is that not only the nonlinear term has critical exponential growth, but also that Kirchhoff function covers the degenerate case. |
In this paper, we prove the decay estimates of the velocity and $H^1$ scattering for the 2D linearized Euler equations around a class of monotone shear flow in a finite channel. Our result is consistent with the decay rate predicted by Case in 1960. |
Hyperbolic partial differential equations on a one-dimensional spatial domain are studied. This class of systems includes models of beams and waves as well as the transport equation and networks of non-homogeneous transmission lines. |
In this paper, we consider the sharp interface limit of a matrix-valued Allen-Cahn equation, which takes the form: $$\partial_t A=\Delta A-\varepsilon^{-2}( A A^{\mathrm{T}}A- A)~~\text{with}~~A:\Omega\subset\mathbb{R}^m\to\mathbb{R}^{n\times n}. $$ We show that the sharp interface system is a two-phases flow system: t... |
This paper is dedicated to the study of both viscous compressible barotropic fluids and Navier-Stokes equation with dependent density, when the viscosity coefficients are variable, in dimension $d\geq2$. We aim at proving the local and global well-posedness for respectively {\it large} and \textit{small} initial data ... |
In the present paper, we prove the existence of solutions $(\lambda, u)\in \R\times H^1(\R^N)$ to the following elliptic equations with potential $\displaystyle -\Delta u+(V(x)+\lambda)u=g(u)\;\hbox{in}\;\R^N, $ satisfying the normalization constraint $\displaystyle \int_{\R^N}u^2=a>0,$ which is deduced by searching... |
Let $Q$ be a first-order differential operator on a compact, smooth oriented Riemannian manifold with smooth boundary. Then, Friedrichs' extension lemma states that the minimal closed extension $Q_{min}$ (the closure of the graph) and the maximal closed extension $Q_{max}$ (in the sense of distributions) of $Q$ in... |
We give a global bilateral estimate on the maximal solution $\bar u_F$ of $ \prt_tu-\Delta u+u^q=0$ in $\BBR^N\times (0,\infty)$, $q>1$, $N\geq 1$, which vanishes at $t=0 $ on the complement of a closed subset $F\subset \BBR^N$. This estimate is expressed by a Wiener test involving the Bessel capacity $C_{2/q,q'... |
We study the Boltzmann equation near a global Maxwellian. We prove the global existence of a unique mild solution with initial data which belong to the $L^r_v L^\infty_t L^\infty_x $ spaces where $r \in (1,\infty]$ by using the excess conservation laws and entropy inequality introduced in [5]. |
In this article we consider cloaking for a quasi-linear elliptic partial differential equation of divergence type defined on a bounded domain in $\mathbb{R}^N$ for $N=2,3$. We show that a perfect cloak can be obtained via a singular change of variables scheme and an approximate cloak can be achieved via a regular chan... |
The goal of this paper is to present a generic multi-region nonlinear age-size structured fish population model, and to assess its mathematical well-posedness. An initial-boundary-value problem is formulated. |
We study the localization of the interior transmission eigenvalues (ITEs) in the case when the domain is the unit ball $\{x \in {\mathbb R}^d:\: |x| \leq 1\}, \: d\geq 2,$ and the coefficients $c_j(x), \: j =1,2,$ and the indices of refraction $n_j(x), \: j =1,2,$ are constants near the boundary $|x| = 1$. We prove th... |
In this paper, we study the well-posedness of Poisson-Nernst-Planck system with no-flux boundary condition and singular permanent charges in two dimension. The main difficulty comes from the lack of integrability of singular permanent charges. |
We consider a parabolic partial differential equation with Dirichlet boundary conditions and measure or $L^1$ data. The key difficulty consists in a presence of a monotone operator~$A$ subjected to a non-standard growth condition, controlled by the exponent $p$ depending on the time and the spatial variable. |
In this work, we derive a result of exponential stability for a coupled system of partial differential equations (PDEs) which governs a certain fluid-structure interaction. In particular, a three-dimensional Stokes flow interacts across a boundary interface with a two-dimensional mechanical plate equation. |
The existence of singular solutions of the incompressible Navier-Stokes system with singular external forces, the existence of regular solutions for more regular forces as well as the asymptotic stability of small solutions (including stationary ones), and a pointwise loss of smoothness for solutions are proved in the ... |
In this paper we consider the model of phase relaxation introduced in [22], where an asymptotic analysis is performed toward an integral formulation of the Stefan problem when the relaxation parameter approaches zero. Assuming the natural physical assumption that the initial condition of the phase is constrained, but ... |
In this work we study oceanic waves in a shallow water flow subject to strong wind forcing and rotation, and linearized around a inhomogeneous (non zonal) stationary profile. This extends the study \cite{CGPS}, where the profile was assumed to be zonal only and where explicit calculations were made possible due to the... |
In this paper we show the existence of global minimizers for the geometrically exact, non-linear equations of elastic plates, in the framework of the general 6-parametric shell theory. A characteristic feature of this model for shells is the appearance of two independent kinematic fields: the translation vector field ... |
In this paper, by providing the uniform gradient estimates for a sequence of the approximating equations, we prove the existence, uniqueness and regularity of the conical parabolic complex Monge-Ampère equation with weak initial data. As an application, we prove a regularity estimates, that is, any $L^{\infty}$-soluti... |
We consider the quadratic Zakharov-Kuznetsov equation $$ \partial_t u + \partial_x \Delta u + \partial_x u^2 =0 $$ on $\mathbb{R}^3$. A solitary wave solution is given by $Q(x-t,y,z)$, where $Q$ is the ground state solution to $-Q + \Delta Q + Q^2 =0$. |
Eikonal equations in metric spaces have strong connections with the local slope operator (or the De Giorgi slope). In this manuscript, we explore and delve into an analogous model based on the global slope operator, expressed as $\lambda u + G[u] = \ell$, where $\lambda \geq 0$. |
For given total mass $m>0$ we show unique solvability of the stationary chemotaxis-consumption model \[ <br>\begin{cases} <br>0= \Delta u - \chi \nabla \cdot (\frac{u}{v} \nabla v) \\ <br>0= \Delta v - uv \\ <br>\int_\Omega u = m <br>\end{cases} \] under no-flux-Dirichlet boundary conditions in bounded smooth domain... |
In this paper, we consider the 1D compressible Euler equation with the damping coefficient $\lambda/(1+t)^{\mu}$. Under the assumption that $0\leq \mu <1$ and $\lambda >0$ or $\mu=1$ and $\lambda > 2$, we prove that solutions exist globally in time, if initial data are small $C^1$ perturbation near constant s... |
We consider a size-structured model for cell division and address the question of determining the division (birth) rate from the measured stable size distribution of the population. We formulate such question as an inverse problem for an integro-differential equation posed on the half line. |
We study the system $$ \left\{ -\Delta u+u+K(x) \phi |u|^{q-2}u&=(I_\alpha*|u|^p)|u|^{p-2}u &&\mbox{ in }{\mathbb R}^N, -\Delta \phi&=K(x)|u|^q&&\mbox{ in }{\mathbb R}^N, \right. $$ where $N\geq 3$, $\alpha\in (0,N)$, $p,q>1$ and $K\geq 0$. |
In a recent work, Paycha and Scott establish formulas for all the Laurent coefficients of Tr(AP^{-s}) at the possible poles. In particular, they show a formula for the zero'th coefficient at s=0, in terms of two functions generalizing, respectively, the Kontsevich-Vishik canonical trace density, and the Wodzicki-G... |
In [19-20], we have established the existence and singularity structures of low regularity solutions to the semilinear generalized Tricomi equations in the degenerate hyperbolic regions and to the higher order degenerate hyperbolic equations, respectively. In the present paper, we shall be concerned with the low regul... |
In this paper, we study the existence of solutions for the new fractinal Robin equations with variable exponents. Moreover, we deal with the logarithm-type nonlinearity. |
We consider the asymptotic behavior of small global-in-time solutions to a 1D Klein-Gordon equation with a spatially localized, variable coefficient quadratic nonlinearity and a non-generic linear potential. The purpose of this work is to continue the investigation of the occurrence of a novel modified scattering beha... |
Results regarding off-diagonal Gaussian upper heat kernel bounds on discrete weighted graphs with possibly unbounded geometry are summarized and related. <br>After reviewing uniform upper heat kernel bounds obtained by Carlen, Kusuoka, and Stroock, the universal Gaussian term on graphs found by Davies is addressed and... |
This paper is devoted to Bresse systems, a robust model for circular beams, given by a set of three coupled wave equations. The main objective is to establish the existence of global attractors for dynamics of semilinear problems with localized damping. |
The study of the uniqueness and nondegeneracy of ground state solutions to semilinear elliptic equations is of great importance because of the resulting energy landscape and its implications for the various dynamics. In [AIKN3], semilinear elliptic equations with combined power-type nonlinearities involving the Sobole... |
A rigorous mathematical theory is developed to explain the super-resolution phenomenon observed in the experiment by <a href="http://F.Lemoult" rel="external noopener nofollow" class="link-external link-http">this http URL</a>, <a href="http://M.Fink" rel="external noopener nofollow" class="link-external link-http">thi... |
In the case of scalar conservation laws $$ u_{t} + f(u)_{x}~=~0,\qquad t\geq 0, x\in\mathbb{R}, $$ with uniformly strictly convex flux $f$, quantitative compactness estimates - in terms of Kolmogorov entropy in ${\bf L}^{1}_{loc}$ - were established in~\cite{DLG,AON1} for sets of entropy weak solutions evaluated at a f... |
We consider a discrete-continuum model of a biomembrane with embedded particles. While the membrane is represented by a continuous surface, embedded particles are described by rigid discrete objects which are free to move and rotate in lateral direction. |
This paper explores the phenomena of enhanced dissipation and Taylor dispersion in solutions to the passive scalar equations subject to time-dependent shear flows. The hypocoercivity functionals with carefully tuned time weights are applied in the analysis. |
We establish uniqueness of ground states $u(x) \geq 0$ for the $L^2$-critical boson star equation $\sqrt{-\Delta} u - (|x|^{-1} \ast |u|^2) u = -u$ in $\R^3$. The proof blends variational arguments with the harmonic extension to the halfspace $\R^4_+$. |
A method is proposed to deal with some multivalued semiflows with weak continuity properties. An application to the reaction-diffusion problems with nonmonotone multivalued semilinear boundary condition and nonmonotone multivalued semilinear source term is presented. |
This article represents the second installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on global solutions for small and localized data. |
A class of vector-valued elliptic operators with unbounded coefficients, coupled up to the second-order is investigated in the Lebesgue space $L^p(\mathbb R^d;\mathbb R^m)$ with $p \in (1,\infty)$, providing sufficient conditions for the generation of an analytic $C_0$-semigroup $T(t)$. Under further assumptions, a ch... |
Following a recent paper by N. Mandache (Inverse Problems 17 (2001), pp. 1435-1444), we establish a general procedure for determining the instability character of inverse problems. We apply this procedure to many elliptic inverse problems concerning the determination of defects of various types by different kinds of b... |
We prove several scattering results for dispersion-managed nonlinear Schrödinger equations. In particular, we establish small-data scattering for both `intercritical' and `mass-subcritical' powers by suitable modifications of the standard approach via Strichartz estimates. |
In this paper, we study the stationary magnetohydrodynamics system in $\mathbb{R}^2\times\mathbb{T}$. We prove trivialness of D-solutions (the velocity field $u$ and the magnetic field $h$) when they are swirl-free. |
This paper examines the Maxwell system of electrodynamics within the framework of distributions. A primary objective is to establish boundary conditions for fields at interfaces when the charge and current densities are measures localized on the interface. |
We give blow-up analysis for a Brezis-Merle's problem on the boundary. Also we give a proof of a compactness result with Lipschitz condition and weaker assumption on the regularity of the domain (smooth domain or $ C^{2,\alpha} $ domain). |
Consider a finite energy radial solution to the focusing energy critical semilinear wave equation in 1+4 dimensions. Assume that this solution exhibits type-II behavior, by which we mean that the critical Sobolev norm of the evolution stays bounded on the maximal interval of existence. |
In this article, we give a unified theory for constructing boundary layer expansions for dis-cretized transport equations with homogeneous Dirichlet boundary conditions. We exhibit a natural assumption on the discretization under which the numerical solution can be written approximately as a two-scale boundary layer e... |
We study the radial-hedgehog solution in a three-dimensional spherical droplet, with homeotropic boundary conditions, within the Landau-de Gennes theory for nematic liquid crystals. The radial-hedgehog solution is a candidate for a globally stable configuration in this model framework and is also a prototype configura... |
In this article we introduce a new Riemann solver for traffic flow on networks. The Priority Riemann solver (PRS) provides a solution at junctions by taking into consideration priorities for the incoming roads and maximization of through flux. |
We study various qualitative and quantitative (global) unique continuation properties for the fractional discrete Laplacian. We show that while the fractional Laplacian enjoys striking rigidity properties in the form of (global) unique continuation properties, the fractional discrete Laplacian does not enjoy these in ... |
For $0<s<1$, we consider the Dirichlet problem for the fractional nonlocal Ornstein--Uhlenbeck equation $$\begin{cases} (-\Delta+x\cdot\nabla)^su=f&\hbox{in}~\Omega\\ u=0&\hbox{on}~\partial\Omega, \end{cases}$$ where $\Omega$ is a possibly unbounded open subset of $\mathbb{R}^n$, $n\geq2$. The appropriat... |
Here, we prove the existence of smooth solutions for mean-field games with a singular mean-field coupling; that is, a coupling in the Hamilton-Jacobi equation of the form $g(m)=-m^{-\alpha}$. We consider stationary and time-dependent settings. |
Let $u_k$ be a solution of the Helmholtz equation with the wave number $k$, $\Delta u_k+k^2 u_k=0$, on a small ball in either $\mathbb{R}^n$, $\mathbb{S}^n$, or $\mathbb{H}^n$. For a fixed point $p$, we define $M_{u_k}(r)=\max_{d(x,p)\le r}|u_k(x)|. |
We are concerned with the inverse boundary problem of determining anomalies associated with a semilinear elliptic equation of the form $-\Delta u+a(\mathbf x, u)=0$, where $a(\mathbf x, u)$ is a general nonlinear term that belongs to a Hölder class. It is assumed that the inhomogeneity of $f(\mathbf x, u)$ is containe... |
We study the existence of singular separable solutions to a class of quasilinear equations with reaction term. In the 2-dim case, we use a dynamical system approach to construct our solutions. |
In this work, we study the Brinkman-Forchheimer equations driven under slip boundary conditions of friction type. We prove the existence and uniqueness of weak solutions by means of regularization combined with the Faedo-Galerkin approach. |
A pointwise bound for local weak solutions to the p-Laplace system is established in terms of data on the right-hand side in divergence form. The relevant bound involves a Havin-Maz'ya- Wulff potential of the datum, and is a counterpart for data in divergence form of a classical result of [KiMa], that has recently... |
In this paper for all $p>1$ we prove that the pull-in voltage of the $p$-MEMS (micro-electro mechanical systems) problems on a smooth bounded domain of $\mathbb R^{d}, d\geq1,$ is minimized by symmetrizing the domain and the permittivity profile. The proofs rely on some suitable version of Talenti's comparison ... |
We propose a level-set method for a mean curvature flow whose boundary is prescribed by interpreting the boundary as an obstacle. Since the corresponding obstacle problem is globally solvable, our method gives a global-in-time level-set mean curvature flow under a prescribed boundary with no restriction of the profile... |
We consider the singularly perturbed problem $F_\varepsilon (u,\Omega):=\int_\Omega \varepsilon |\nabla^2u| + \varepsilon^{-1}|1-|\nabla u|^2|^2$ on bounded domains $\Omega \subset\mathbb{R}^2$. Under appropriate boundary conditions, we prove that if $\Omega$ is an ellipse then the minimizers of $F_\varepsilon(\cdot,\... |
We consider the Langevin equation describing a stochastically perturbed by uniform noise non-viscous Burgers fluid and introduce a deterministic function that corresponds to the mean of the velocity when we keep the value of position fixed. We study interrelations between this function and the solution of the non-pert... |
We investigate a general nonhomogeneous parabolic initial-boundary value problem in some anisotropic Hörmander inner product spaces. We prove that the operators corresponding to this problem are isomorphisms between appropriate Hörmander spaces. |
In this paper, we investigate the existence and uniqueness of solutions to a stationary mean field game model introduced by J. -M. |
This work concerns resonant scattering by a perfectly conducting slab with periodically arranged subwavelength slits, with two slits per period. There are two classes of resonances, corresponding to poles of a scattering problem. |
This paper studies the pressureless Euler-Poisson system and its fully non-linear counterpart, the Euler-Monge-Ampère system, where the fully non-linear Monge-Ampère equation substitutes for the linear Poisson equation. While the first is a model of plasma physics, the second is derived as a geometric approximation to... |
This paper is a first step in the study of the so-called Taylor model, introduced by J.B. Taylor in \cite{Taylor}. This system of nonlinear PDE's is derived from the viscous incompressible MHD equations, through an asymptotics relevant to the Earth's magnetic field. |
We show the global well-posedness for the two-dimensional Zakharov-Kuznetsov equation in $H^{s}({\mathbb{R}^2})$ when $\frac{11}{13}<s<1$ via the I-method. Additionally, local well-posedness for the symmetrized ZK equation in $ B^\frac{1}{2}_{2,1}(\mathbb{R}^2)$ is established by using atomic spaces. |
In this paper, we establish the existence and uniqueness of solutions to the two-dimensional Burgers equation using the framework of infinite-dimensional dynamical systems. The two-dimensional Burgers equation, which models the interplay between nonlinear advection and viscous dissipation, is given by: $$ u_{t} + u \c... |
In this paper, the problem of continuous data assimilation of three dimensional primitive equations with magnetic field in thin domain is studied. We establish the well-posedness of the assimilation system and prove that the $H^2$-strong solution of the assimilation system converges exponentially to the reference solu... |
We study some zero-flux attraction-repulsion chemotaxis models, with nonlinear production rates for the chemorepellent and the chemoattractant. This paper partially improves some known results in the literature and moreover solves an open question. |
Systems consisting of a single ordinary differential equation coupled with one reaction-diffusion equation in a bounded domain and with the Neumann boundary conditions are studied in the case of particular nonlinearities from the Brusselator model, the Gray-Scott model, the Oregonator model and a certain predator-prey ... |
In this paper, we study local regularity properties of minimizers of nonlocal variational functionals with variable exponents and weak solutions to the corresponding Euler--Lagrange equations. We show that weak solutions are locally bounded when the variable exponent $p$ is only assumed to be continuous and bounded. |
This paper is concerned with the asymptotic behavior of the spreading speeds of Fisher-KPP equations in rapidly and slowly varying almost periodic settings respectively. Thanks to the variational formula for the spreading speeds involving generalized principal eigenvalue, we are able to concentrate on the homogenizati... |
We consider strongly coupled competitive elliptic systems of Gross-Pitaevskii type that arise in the study of two-component Bose-Einstein condensates, in general smooth bounded domains of $\mathbb{R}^N$, $N\geq 1$. As the coupling parameter tends to infinity, solutions that remain uniformly bounded are known to conver... |
In this paper, we study the amount of information contained in the Steklov spectrum of some compact manifolds with connected boundary equipped with a warped product metric. Examples of such manifolds can be thought of as deformed balls in R^d. |
We study singularity formation for the pressureless Euler-Poisson system of cold ion dynamics. In contrast to the Euler-Poisson system with pressure, when its smooth solutions experience $C^1$ blow-up, the $L^\infty$ norm of the density becomes unbounded, which is often referred to as a delta-shock. |
We study the existence and multiplicity of sign changing solutions of the following equation $ \begin{cases} -\Delta u = \mu |u|^{2^{\star}-2}u+\frac{|u|^{2^{*}(t)-2}u}{|x|^t}+a(x)u \quad\text{in}\quad \Omega, u=0 \quad\text{on}\quad\partial\Omega, \end{cases} $where $\Omega$ is a bounded domain in $R^N$, $0\in\partial... |
We consider a reaction-diffusion system including discontinuous hysteretic relay operators in reaction terms. This system is motivated by an epigenetic model that describes the evolution of a population of organisms which can switch their phenotype in response to changes of the state of the environment. |
In dimension three, the existence of global weak solutions to the axisymmetric simplified Ericksen-Leslie system without swirl is established. This is achieved by analyzing weak convergence of solutions of the axisymmetric Ginzburg-Landau approximated solutions as the penalization parameter $\varepsilon$ tends to zero... |
We develop local elliptic regularity for operators having coefficients in a range of Sobolev-type function spaces (Bessel potential, Sobolev-Slobodeckij, Triebel-Lizorkin, Besov) where the coefficients have a regularity structure typical of operators in geometric analysis. The proofs rely on a nonstandard technique us... |
We consider the Bean's critical state model for anisotropic superconductors. A variational problem solved by the quasi--static evolution of the internal magnetic field is obtained as the $\Gamma$-limit of functionals arising from the Maxwell's equations combined with a power law for the dissipation. |
For a large class of semiclassical operators $P(h)-z$ which includes Schrödinger operators on manifolds with boundary, we construct the Quantum Monodromy operator $M(z)$ associated to a periodic orbit $\gamma$ of the classical flow. Using estimates relating $M(z)$ and $P(h)-z$, we prove semiclassical estimates for sma... |
We prove global well-posedness for 3D Dirac equation with a concentrated nonlinearity. |
We study the regularity and the growth rates of solutions to two-dimensional Monge-Ampère equations with the right-hand side exhibiting polynomial growth. Utilizing this analysis, we demonstrate that the translators for the flow by sub-affine-critical powers of the Gauss curvature are smooth, strictly convex entire gr... |
The paper is concerned with the regularity of weak solutions to the Navier-Stokes equations. The aim is to investigate on a relaxed Prodi-Serrin condition in order to obtain regularity for t > 0. |
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