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We consider a class of singular Liouville equations on compact surfaces motivated by the study of Electroweak and Self-Dual Chern-Simons theories, the Gaussian curvature prescription with conical singularities and Onsager's description of turbulence. We analyse the problem of existence variationally, and show how ... |
We study periodic tessellations of the Euclidean space with unequal cells arising from the minimization of perimeter functionals. Existence results and qualitative properties of minimizers are discussed for different classes of problems, involving local and non-local perimeters. |
We prove that the results in scattering theory that involve resonances are still valid for non-analytic potentials, even if the notion of resonance is not defined in this setting. More precisely, we show that if the potential of a semiclassical Schrödinger operator is supposed to be smooth and to decrease at infinity,... |
We establish the uniqueness of the positive solution for equations of the form $-\Delta u=au-b(x)f(u)$ in $\Omega$, $u|\_{\partial\Omega}=\infty$. The special feature is to consider nonlinearities $f$ whose variation at infinity is \emph{not regular} (e.g., $\exp(u)-1$, $\sinh(u)$, $\cosh(u)-1$, $\exp(u)\log(u+1)$, $u... |
This investigation is devoted to the program to characterise continuous and variable discrete asymptotics of solutions to elliptic equations on a manifold with edge, continued in a cicle of forthcoming expositions [15], [16]. The structure of continuous and variable discrete (in general branch- ing) asymptotics is ver... |
Given a bounded domain $\Omega \subset \mathbb{R}^n$, a result by Bourgain, Brezis, and Mironescu characterizes when a function $f \in L^p(\Omega)$ is in the Sobolev space $W^{1,p}(\Omega)$ based on the limiting behavior of its Besov seminorms. We prove a direct analogue of this result which characterizes when a diffe... |
Consider the H^{1/2}-critical Schroedinger equation with a cubic nonlinearity in R^3, i \partial_t \psi + \Delta \psi + |\psi|^2 \psi = 0. It admits an eight-dimensional manifold of periodic solutions called solitons e^{i(\Gamma + vx - t|v|^2 + \alpha^2 t)} \phi(x-2tv-D, \alpha), where \phi(x, \alpha) is a positive gr... |
A notion of measure solution is formulated for a coagulation-diffusion equation, which is the natural counterpart of Smoluchowski's coagulation equation in a spatially inhomogeneous setting. Some general properties of such solutions are established. |
In this paper we investigate the zero dielectric constant limit to the non-isentropic compressible Euler-Maxwell system. We justify this singular limit rigorously in the framework of smooth solutions and obtain the non-isentropic compressible magnetohydrodynamic equations as the dielectric constant tends to zero. |
The nonlinear Schrödinger equations with nonlinearities $|u|^{2k}u$ on the $d$-dimensional torus are considered for arbitrary positive integers $k$ and $d$. The solution of the Cauchy problem is shown to be unique in the class $C_tH^s_x$ for a certain range of scale-subcritical regularities $s$, which is almost optima... |
In the language of differential geometry, the incompressible inviscid Euler equations can be written in vorticity-vector potential form as \begin{align*} \partial_t \omega + {\mathcal L}_u \omega &= 0\\ u &= \delta \tilde \eta^{-1} \Delta^{-1} \omega \end{align*} where $\omega$ is the vorticity $2$-form, ${\mat... |
This work presents an approach to the Navier-Stokes equations that is phrased in unbiased Eulerian coordinates, yet describes objects that have Lagrangian significance: particle paths, their dispersion and diffusion. The commutator between Lagrangian and Eulerian derivatives plays an important role in the Navier-Stoke... |
A linear operator in a Hilbert space defined through the form of Riesz representation naturally introduces a reproducing kernel Hilbert space structure over the range space. Such formulation, called H-HK formulation in this paper, possesses a built-in mechanism to solve some basic type problems in the formulation by u... |
We consider solutions to the full (non-isentropic) two-dimensional Euler equations that are constant in time and along rays emanating from the origin. We prove that for a polytropic equation of state, entropy admissible solutions in $L^\infty$ with non-vanishing velocity, density, and internal energy must be $BV$. |
The global regularity for the viscous Boussinesq equations is proved. |
In this paper, we consider a diffusion equation with fractional-time derivative with nonsingular Mittag-Leffler kernel in Hilbert spaces. Existence and uniqueness of solution are proved by means of a spectral argument. |
From the solution of the heat and mass diffusion equations that describe the exothermic physical absorption of a gas into a liquid, compact formulae for the sums of two infinite series are derived. The method is general and should be applicable to other systems of linear, parabolic partial differential equations. |
The Adimurthi-Druet [1] inequality is an improvement of the standard Moser-Trudinger inequality by adding a $L^2$-type perturbation, quantified by $\alpha\in [0,\lambda\_1)$, where $\lambda\_1$ is the first Dirichlet eigenvalue of $\Delta$ on a smooth bounded domain. It is known [3,9,13,18] that this inequality admits... |
We consider the nonlinear Schrödinger equation (NLS) on a torus of arbitrary dimension. The equation is studied in presence of an external potential field whose time-dependent amplitude is taken as control. |
In this paper, we are concerned with the existence of least energy solutions for the following biharmonic equations: $$\Delta^2 u+(\lambda V(x)-\delta)u=|u|^{p-2}u \quad in\quad \mathbb{R}^N$$ where $N\geq 5, 2<p\leq\frac{2N}{N-4}, \lambda>0$ is a parameter, $V(x)$ is a nonnegative potential function with nonempt... |
We study the existence and long-time asymptotics of weak solutions to a system of two nonlinear drift-diffusion equations that has a gradient flow structure in the Wasserstein distance. The two equations are coupled through a cross-diffusion term that is scaled by a parameter $\varepsilon\ge0$. |
In this paper, we study a strongly coupled two-prey one-predator system. We first prove the unique positive equilibrium solution is globally asymptotically stable for the corresponding kinetic system (the system without diffusion) and remains locally linearly stable for the reaction-diffusion system without cross-diff... |
We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer. A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index $2$. |
We consider the porous medium equation with a power-like reaction term, posed on Riemannian manifolds. Under certain assumptions on $p$ and $m$ in (1.1), and for small enough nonnegative initial data, we prove existence of global in time solutions, provided that the Sobolev inequality holds on the manifold. |
We investigate the location of the maximal gradient of the torsion function on some non-symmetric planar domains. First, by establishing uniform estimates for narrow domains, we prove that as a planar domain bounded by two graphs of functions becomes increasingly narrow, the location of the maximal gradient of its tor... |
In this work we deal with a scalar spectral mixed boundary value problem in a spacial junction of thin rods and a plate. Constructing asymptotics of the eigenvalues, we employ two equipollent asymptotic models posed on the skeleton of the junction, that is, a hybrid domain. |
We study the semiclassical quantization of Poincaré maps arising in scattering problems with fractal hyperbolic trapped sets. The main application is the proof of a fractal Weyl upper bound for the number of resonances/scattering poles in small domains near the real axis. |
We study the consistency of Lipschitz learning on graphs in the limit of infinite unlabeled data and finite labeled data. Previous work has conjectured that Lipschitz learning is well-posed in this limit, but is insensitive to the distribution of the unlabeled data, which is undesirable for semi-supervised learning. |
We study the null-controllability of parabolic equations associated to non-autonomous Ornstein-Uhlenbeck operators. When a Kalman type condition holds for some positive time $T>0$, these parabolic equations are shown to enjoy a Gevrey regularizing effect at time $T>0$. |
We identify the short time asymptotics of the sub-Riemannian heat content for a smoothly bounded domain in the first Heisenberg group. Our asymptotic formula generalizes prior work by van den Berg-Le Gall and van den Berg-Gilkey to the sub-Riemannian context, and identifies the first few coefficients in the sub-Rieman... |
In this paper, we study the existence of nodal solutions for the non-autonomous Schrödinger--Poisson system: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+u+\lambda K(x) \phi u=f(x) |u|^{p-2}u & \text{ in }\mathbb{R}^{3}, \\ -\Delta \phi =K(x)u^{2} & \text{ in }\mathbb{R}^{3},% \end{array}% \right. \en... |
In this paper, the large time behavior of solutions of 1-D isentropic Navier-Stokes system is investigated. It is shown that a composite wave consisting of two viscous shock waves is stable for the Cauchy problem provided that the two waves are initially far away from each other. |
This work focuses on the Riemann problem of Euler equations with global constant initial conditions and a single-point heating source, which comes from the physical problem of heating one-dimensional inviscid compressible constant flow. In order to deal with the source of Dirac delta-function, we propose an analytical... |
We derive new, localized geometric integral identities for solutions to the $3D$ compressible Euler equations under an arbitrary equation of state when the sound speed is positive. The identities are coercive in the first derivatives of the specific vorticity and the second derivatives of the entropy, and the error te... |
We study a Keller-Segel type chemotaxis model with a modified sensitivity function in a bounded domain $\Omega\subset \mathbb{R}^N$, $N\geq2$. The global existence of classical solutions to the fully parabolic system is established provided that the ratio of the chemotactic coefficient to the motility of cells is not ... |
In this paper we study Love waves in a layered, elastic half-space. We first address the direct problem and we characterize the existence of Love waves through the dispersion relation. |
Shortest path graph distances are widely used in data science and machine learning, since they can approximate the underlying geodesic distance on the data manifold. However, the shortest path distance is highly sensitive to the addition of corrupted edges in the graph, either through noise or an adversarial perturbat... |
In the article we study the Dirichlet $(p,q)$-eigenvalue problem for subelliptic non-commutative operators on nilpotent Lie groups. We prove solvability of this eigenvalue problem and existence of the minimizer of the corresponding variational problem. |
For two-dimensional steady pure-gravity water waves with a unidirectional flow of constant favourable vorticity, we prove an explicit bound on the amplitude of the wave, which decays to zero as the vorticity tends to infinity. Notably, our result holds true for arbitrary water waves, that is, we do not have to restric... |
We establish an explicit formula for the Half-Wave maps equation for rational functions with simple poles. The Lax pair provides a description of the evolution of the poles. |
We prove existence, non-degeneracy, and exponential decay at infinity of a non-trivial solution to Emden's equation $-\Delta u = | u |^3$ on an unbounded $L$-shaped domain, subject to Dirichlet boundary conditions. Besides the direct value of this result, we also regard this solution as a building block for soluti... |
In this note we consider boundary point principles for partial differential inequalities of elliptic type. Firstly, we highlight the difference between conditions required to establish classical strong maximum principles and classical boundary point lemmas for second order linear elliptic partial differential inequali... |
We consider minimizers of <br>\[ <br>F(\lambda_1(\Omega),\ldots,\lambda_N(\Omega)) + |\Omega|, <br>\] where $F$ is a function strictly increasing in each parameter, and $\lambda_k(\Omega)$ is the $k$-th Dirichlet eigenvalue of $\Omega$. Our main result is that the reduced boundary of the minimizer is composed of $C^{1... |
This article studies the equations of adiabatic thermoelasticity endowed with an internal energy satisfying an appropriate quasiconvexity assumption which is associated to the symmetrisability condition for the system. A Gårding-type inequality for these quasiconvex functions is proved and used to establish a weak-str... |
We introduce and study a new class of partial differential equations (PDEs) with hybrid fuzzy-stochastic parameters, coined fuzzy-stochastic PDEs. Compared to purely stochastic PDEs or purely fuzzy PDEs, fuzzy-stochastic PDEs offer powerful models for accurate representation and propagation of hybrid aleatoric-epistem... |
We develop an approach for the analysis of fundamental solutions to Hamilton-Jacobi equations of contact type based on a generalized variational principle proposed by Gustav Herglotz. We also give a quantitative Lipschitz estimate on the associated minimizers. |
In this work, the existence of solutions (in a suitable sense) to a family of inclusion systems involving fractional, possibly competing, elliptic operators, fractional convection, and homogeneous Dirichlet boundary conditions is established. The technical approach exploits Galerkin's method and a surjective resul... |
We present a general framework to study edge states for second order elliptic operators in a half channel. We associate an integer valued index to some bulk materials, and we prove that for any junction between two such materials, localised states must appear at the boundary whenever the indices differ. |
We consider imaging in a scattering medium where the illumination goes through this medium but there is also an auxiliary, passive receiver array that is near the object to be imaged. Instead of imaging with the source-receiver array on the far side of the object we image with the data of the passive array on the near... |
Assuming a lower bound on the dimension, we prove a long standing conjecture concerning the classification of global solutions of the obstacle problem with unbounded coincidence sets. |
We study a homogenization question for stochastic divergence type operator |
The present paper commences the study of higher order differential equations in composition form. Specifically, we consider the equation Lu=\Div B^*\nabla(a\Div A\nabla u)=0, where A and B are elliptic matrices with complex-valued bounded measurable coefficients and a is an accretive function. |
There are a few examples of solutions to the incompressible Euler equations which are piecewise smooth with a discontinuity of the tangential velocity across a hypersurface evolving in time: the so-called vortex sheets. An important open problem is to determine whether or not these solutions can be obtained as zero vi... |
We study the regularity of the interface between the disjoint supports of a pair of nonnegative subharmonic functions. The portion of the interface where the Alt-Caffarelli-Friedman (ACF) monotonicity formula is asymptotically positive forms an $\mathcal{H}^{n-1}$-rectifiable set. |
This article completes the study of the influence of the intensity parameter $\alpha$ in the boundary condition $\varepsilon \partial_{\boldsymbol{\nu}_\varepsilon} u_\varepsilon - u_\varepsilon \, \overrightarrow{V_\varepsilon}\boldsymbol{\cdot}\boldsymbol{\nu}_\varepsilon = \varepsilon^{\alpha} \varphi_\varepsilon $ ... |
We study the weak universality of the two-dimensional fractional nonlinear wave equation. For a sequence of Hamiltonians of high-degree potentials scaling to the fractional $\Phi_2^4$, we first establish a \emph{sufficient and almost necessary} criteria for the convergence of invariant measures to the fractional $\Phi... |
In this paper, curved fronts are constructed for spatially periodic bistable reaction-diffusion equations under the a priori assumption that there exist pulsating fronts in every direction. Some sufficient and some necessary conditions of the existence of curved fronts are given. |
This article deals with the uniqueness and stability issues in the inverse problem of determining the unbounded potential of the Schrödinger operator in a bounded domain of dimension 3 or greater, endowed with Robin boundary condition, from knowledge of its boundary spectral data. These data are defined by the pairs f... |
We present Chapman--Enskog and Hilbert expansions applied to the $\BigO(v/c)$ Boltzmann equation for the radiative transfer of neutrinos in core-collapse supernovae. Based on the Legendre expansion of the scattering kernel for the collision integral truncated after the second term, we derive the diffusion limit for th... |
We study the Stokes-transport system in a two-dimensional channel with horizontally moving boundaries, which serves as a reduced model for oceanography and sedimentation. The density is transported by the velocity field, satisfying the momentum balance between viscosity, pressure, and gravity effects, described by the... |
In this manuscript we consider a porous medium equation with non-local diffusion effects given by a fractional heat operator $\partial_t + (-\Delta)^s$ in two space dimensions. Global in time existence of weak solutions is shown by employing a time semi-discretization of the equations, an energy inequality, a higher o... |
This paper examines systems of poly-harmonic equations of the Hardy--Sobolev type and the closely related weighted systems of integral equations involving Riesz potentials. Namely, it is shown that the two systems are equivalent under some appropriate conditions. |
In this paper, we deal with the following singular perturbed fractional elliptic problem $ \epsilon^{} (-\Delta)^{1/2}{u}+V(z)u=f(u)\,\,\, \mbox{in} \,\,\, \mathbb{R}, $ where $ (-\Delta)^{1/2}u$ is the square root of the Laplacian and $f(s)$ has exponential critical growth. Under suitable conditions on $f(s)$, we con... |
The aim of this article is to show a local-in-time existence of a strong solution to the generalized compressible Navier-Stokes equation for arbitrarily large initial data. The goal is reached by $L^p$-theory for linearized equations which are obtained with help of the Weis multiplier theorem and can be seen as genera... |
We provide a natural simple argument using anistropic flows to prove the existence of weak solutions to Lutwak's $L^p$-Minkowski problem on $S^n$ which were obtained by other methods. |
This study is concerned with a family of parabolic system with general singular nonlinearities, which is a generalization of MEMS system. To some extent, the classification of global existence and quenching according to parameters and initial data is given. |
We give a natural convexity proof of an elementary inequality used by Ozawa and Rogers in proving their bilinear estimate for the one-dimensional Klein-Gordon equation. This robust approach also enables us to derive the optimality of Ozawa-Rogers estimate and establish a new bilinear estimate. |
In this paper we investigate existence and characterization of non-radial pseudo-radial (or separable) solutions of some semi-linear elliptic equations on symmetric 2-dimensional domains. The problem reduces to the phase plane analysis of a dynamical system. |
We describe a method to show short time uniqueness results for viscosity solutions of general nonlocal and non-monotone second-order geometric equations arising in front propagation problems. Our method is based on some lower gradient bounds for the solution. |
A classical result of Wente, motivated by the study of sessile capillarity droplets, demonstrates the axial symmetry of every hypersurface which meets a hyperplane at a constant angle and has mean curvature dependent only on the distance from that hyperplane. An analogous result is proven here for the fractional mean ... |
It has been proved by the authors that the (extended) Sadowsky functional can be deduced as the Gamma-limit of the Kirchhoff energy on a rectangular strip of height $\varepsilon$, as $\varepsilon$ tends to 0. In this paper we show that this Gamma-convergence result is stable when affine boundary conditions are prescri... |
In this short note we explore the validity of Wente-type estimates for Neumann boundary problems involving Jacobians. We show in particular that such estimates do not in general hold under the same hypotheses on the data for Dirichlet boundary problems. |
The main theorem has been slightly generalized to include a larger class of symbols. |
Uniform large deviations for the laws of the paths of the solutions of the stochastic nonlinear Schrodinger equation when the noise converges to zero are presented. The noise is a real multiplicative Gaussian noise. |
We introduce many families of explicit solutions to the three dimensional incompressible Euler equations for nonviscous fluid flows using the Lagrangian framework. Almost no exact Lagrangian solutions exist in the literature prior to this study. |
In this article, we apply the viscosity solutions theory for integro-differential equations to the \emph{one-phase} Muskat equation (also known as the Hele-Shaw problem with gravity). We prove global well-posedness for the corresponding Hamilton-Jacobi-Bellmann equation with bounded, uniformly continuous initial data,... |
In this paper, we consider the inverse problem of recovering a sound soft scatterer from the measured scattered field. The scattered field is assumed to be induced by a point source on a curve/surface that is known. |
We consider the optimization problem of minimizing $\int_{\Omega}|\nabla u|^{p(x)}+ \lambda \chi_{\{u>0\}} dx$ in the class of functions $W^{1,p(\cdot)}(\Omega)$ with $u-\phi_0\in W_0^{1,p(\cdot)}(\Omega)$, for a given $\phi_0\geq 0$ and bounded. $W^{1,p(\cdot)}(\Omega)$ is the class of weakly differentiable functi... |
Predicting the evolution of expanding population is critical to control biological threats such as invasive species and virus explosion. In this paper, we consider a two species chemotaxis system of parabolic-parabolic-elliptic type with Lotka-Volterra type competition terms and a free boundary. |
The paper is devoted to investigating long time behavior of smooth small data solutions to 3-D quasilinear wave equations outside of compact convex obstacles with Neumann boundary conditions. |
We present a classification of strict limits of planar BV homeomorphisms. The authors and S. Hencl showed in a previous work \cite{CHKR} that such mappings allow for cavitations and fractures singularities but fulfill a suitable generalization of the INV condition. |
In this paper, we show the existence and uniqueness of local regular solutions to the initial-Neumann boundary value problem of Schrödinger flow from a smooth bounded domain $\Omega\subset\mathbb{R}^3$ into $\mathbb{S}^2$(namely Landau-Lifshitz equation without dissipation). The proof is built on a parabolic perturbat... |
We study the problem of existence and uniqueness of strong solutions to a degenerate quasilinear parabolic non-Newtonian thin-film equation. Originating from a non-Newtonian Navier--Stokes system the equation is derived by lubrication theory and under the assumption that capillarity is the only driving force. |
We give an instability estimate for the Gel'fand inverse boundary value problem at high energies. Our instability estimate shows an optimality of several important preceeding stability results on inverse problems of such a type. |
This paper focuses on parabolic final value problems, and well-posedness is proved for a large class of these. The clarification is obtained from Hilbert spaces that characterise data that give existence, uniqueness and stability of the solutions. |
We prove in this short report the existence of a fundamental solution (F.S.) for the Cauchy initial boundary problem on the whole space for the parabolic differential equation having at origin the point of non-integrable unbounded discontinuity for coefficient before a first order derivative. <br>We give also the non-... |
In this article, we consider a simplified model of time dynamics for a mosquito population subject to the artificial introduction of {\itshape Wolbachia}-infected mosquitoes, in order to fight arboviruses <a href="http://transmission.Indeed" rel="external noopener nofollow" class="link-external link-http">this http URL... |
We study the asymptotic stability of a planar rarefaction wave (in the $ x_1 $- direction) for the 3-d isentropic Navier-Stokes equations, where the initial perturbation is periodic on the torus $ \mathbb{T}^3 $ with zero average. To solve this Cauchy problem in which the initial data is periodic with respect to only ... |
The dynamics of short intense electromagnetic pulses propagating in a relativistic pair plasma with small temperature asymmetry is of current interest. Such pulses were shown to be governed by a nonlinear Schrodinger equation with a new type of focusing-defocusing nonlinearity. |
We consider the following nonlinear Schrödinger equations with critical growth: \begin{equation} - \Delta u + V(|y|)u=u^{\frac{N+2}{N-2}},\quad u>0 \ \ \mbox{in} \ \mathbb {R}^N, \end{equation} where $V(|y|)$ is a bounded positive radial function in $C^1$, $N\ge 5$. By using a finite reduction argument, we show tha... |
We derive bounds on the volume of an inclusion in a body in two or three dimensions when the conductivities of the inclusion and the surrounding body are complex and assumed to be known. The bounds are derived in terms of average values of the electric field, current, and certain products of the electric field and cur... |
Using a standard linearization technique and previously obtained microlocal properties for pseudodifferential operators with smooth coefficients, the authors state results of microlocal regularity in generalized Besov spaces for solutions to non linear PDE. |
We consider the isentropic compressible Euler system in 2 space dimensions with pressure law $p({\rho}) = {\rho}^2$ and we show the existence of classical Riemann data, i.e. pure jump discontinuities across a line, for which there are infinitely many admissible bounded weak solutions (bounded away from the void). We a... |
We obtain results for the following question where $m\ge 1$ and $n\ge 2$ are integers. {\bf Question. |
Consider a class of chemotaxis-fluid model incorporating a volume-filling effect in the sense of Painter and Hillen (Can. Appl. |
We show the existence of local and global in time weak martingale solutions for a stochastic version of the Othmer-Dunbar-Alt kinetic model of chemotaxis under suitable assumptions on the turning kernel and stochastic drift coefficients, using dispersion and stochastic Strichartz estimates. The analysis is based on ne... |
We consider a class of $1D$ NLS perturbed with a steplike potential. We prove that the nonlinear solutions satisfy the double scattering channels in the energy space. |
We propose a systematic approach to analysing the complex structure of time-periodic solutions to the cubic wave equation on an interval with Dirichlet boundary conditions first reported in <a href="https://arxiv.org/abs/2407.16507" data-arxiv-id="2407.16507" class="link-https">arXiv:2407.16507</a>. The analysis we pr... |
This paper considers the supercritical Navier-Stokes equations posed in the whole space $\R^d$, with suitably randomized initial data, in the weak solution setting. The global weak solutions are constructed for a large set of initial data in $H^{-s}(\R^d)$ for some $s>0$ via a probabilistic argument, and this in tu... |
We obtain an estimate for the Hölder continuity exponent for weak solutions to the following elliptic equation in divergence form: \[ \mathrm{div}(A(x)\nabla u)=0 \qquad\mathrm{in\}\Omega, \] where $\Omega$ is a bounded open subset of $\R^2$ and, for every $x\in\Omega$, $A(x)$ is a matrix with bounded measurable coeffi... |
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