text large_stringlengths 7 3.02k |
|---|
We study the large-time behavior of global energy class ($H^1$) solutions of the one-dimensional nonlinear Schrödinger equation with a general localized potential term and a defocusing nonlinear term. By using a new type of interaction Morawetz estimate localized to an exterior region, we prove that these solutions de... |
We consider a frequency localized Bernstein inequality for the fractional Laplacian operator which has wide applications in fluid dynamics such as dissipative surface quasi-geostrophic equations. We use a heat flow reformulation and prove the inequality for the full range of parameters and in all dimensions. |
This paper is to discuss a new method to quickly solve scalar $u_t+(G(u))_x=0$ numerically. |
In this paper, we finish the basic development of the discrete octonionic analysis by presenting a Weyl calculus-based approach to bounded domains in $\mathbb{R}^{8}$. In particular, we explicitly prove the discrete Stokes formula for a bounded cuboid, and then we generalise this result to arbitrary bounded domains in... |
We study a new flocking model which has the versatility to capture the physically realistic qualitative behavior of the Motsch-Tadmor model, while also retaining the entropy law, which lends to a similar 1D global well-posedness analysis to the Cucker-Smale model. This is an improvement to the situation in the Cucker-... |
We prove that, in general, given a $p$-harmonic map $F:M\to N$ and a convex function $H:N\to\mathbb{R}$, the composition $H\circ F$ is not $p$-subharmonic. By assuming some rotational symmetry on manifolds and functions, we reduce the problem to an ordinary differential inequality. |
This paper deals with the global compactness and multiplicity of positive solutions to problems of the type $$ -\Delta_{\mathbb B^N} u -\lambda u=a(x) |u|^{2^*-2}u+f(x) \quad\text{in } \mathbb B^N, \quad u\in H^1(\mathbb B^N),$$ <br>where $\mathbb B^N$ denotes the ball model of the hyperbolic space of dimension $N\geq ... |
The problem we consider is given a mixed model $F(\gamma) \rightarrow \gamma \rightarrow \tilde{\gamma}$ and a potential $\widehat{V}$ such that $\log \widehat{V} \in L^{1}$, to determine a movement $U$ such that $F(U \gamma)=F(\gamma) + \widehat{V}$. |
In this paper we propose a notion of viscosity solutions for path dependent semi-linear parabolic PDEs. This can also be viewed as viscosity solutions of non-Markovian backward SDEs, and thus extends the well-known nonlinear Feynman-Kac formula to non-Markovian case. |
The solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann-Hilbert problem whose definition involves four spectral functions $a,b,A,B$. The functions $a(k)$ and $b(k)$ are defined via a nonlinear Fourier transform of the initial data, whereas $A(k)$ and ... |
Nonlocal interactions are ubiquitous in nature and play a central role in many biological systems. In this paper, we perform a bifurcation analysis of a widely-applicable advection-diffusion model with nonlocal advection terms describing the species movements generated by inter-species interactions. |
The classical one-phase Stefan problem (without surface tension) allows for a continuum of steady state solutions, given by an arbitrary (but sufficiently smooth) domain together with zero temperature. We prove global-in-time stability of such steady states, assuming a sufficient degree of smoothness on the initial do... |
This work deals with the system $(-\Delta)^m u= a(x) v^p$, $(-\Delta)^m v=b(x) u^q$ with Dirichlet boundary condition in a domain $\Omega\subset\RR^n$, where $\Omega$ is a ball if $n\ge 3$ or a smooth perturbation of a ball when $n=2$. <br>We prove that, under appropriate conditions on the parameters ($a,b,p,q,m,n$), ... |
We prove that the knowledge of the Dirichlet-to-Neumann map, measured on a part of the boundary of a bounded domain in $\mathbb{R}^n, n\geq2$, can uniquely determine, in a nonlinear magnetic Schrödinger equation, the vector-valued magnetic potential and the scalar electric potential, both being nonlinear in the solutio... |
We consider the two-dimensional water wave problem in an infinitely long canal of finite depth both with and without surface tension. In order to describe the evolution of the envelopes of small oscillating wave packet-like solutions to this problem the Nonlinear Schrödinger equation can be derived as a formal approxi... |
By an extension of of some estimates due to Crandall and Pierre and Di Benedetto we derive consequences for fully nonlinear parabolic equations of the form $\dt v + F(t,x,D^2v)=0$, where $F$ can be both singular and degenerate elliptic and also non-homogeneous. Such equations appear in the theory of option pricing wit... |
In this paper we prove the existence of multiple solutions for a quasilinear elliptic boundary value problem, when the p-derivative at zero and the p-derivative at infinity of the nonlinearity are greater than the first eigenvalue of the p-Laplace operator. Our proof uses bifurcation from infinity and bifurcation from... |
We discuss variational problems on two-dimensional domains with energy densities of linear growth and with radially symmetric data. The smoothness of generalized minimizers is established under rather weak ellipticity assumptions. |
In this paper we consider a class of fractional Schrödinger equations with potentials vanishing at infinity. By using a minimization argument and a quantitative deformation Lemma, we prove the existence of a sign-changing solution. |
In this paper we prove the existence, regularity and symmetry of a ground state for a nonlinear equation in the whole space, involving a pseudo-relativistic Schrödinger operator. |
Cet article est consacre a l'etude des mesures limites associees a la solution de l'equation de Helmholtz avec un terme source se concentrant en un point. Le potentiel est suppose regulier et l'operateur non-captif. |
In this paper we study singular integrals on small (that is, measure zero and lower than full dimensional) subsets of metric groups. The main examples of the groups we have in mind are Euclidean spaces and Heisenberg groups. |
The aim of the paper is to recall the importance of the study of invertibility and monotonicity of stress-strain relations for investigating the non-uniqueness and bifurcation of homogeneous solutions of the equilibrium problem of a hyperelastic cube subjected to equiaxial tensile forces. In other words, we reconsider... |
We consider the higher order asymptotics for the mKdV equation in the Painlevé sector. We first show that the solution admits a uniform expansion to all orders in powers of $t^{-1/3}$ with coefficients that are smooth functions of $x(3t)^{-1/3}$. |
We prove a version of Bressan's mixing conjecture where the advecting field is constrained to be a shear at each time. Also, inspired by recent work of Blumenthal, Coti Zelati and Gvalani, we construct a particularly simple example of a shear flow which mixes at the optimal rate. |
We study the asymptotic behavior of the Schrödinger equation in the presence of a nonlinearity of Hartree type in the semi-classical regime. Our scaling corresponds to a weakly nonlinear regime where the nonlinearity affects the leading order amplitude of the solution without altering the rapid oscillations. |
In this paper, we consider a 3d cubic focusing nonlinear Schrödinger equation (NLS) with slowing decaying potentials. Adopting the variational method of Ibrahim-Masmoudi-Nakanishi \cite{IMN}, we obtain a condition for scattering. |
We present explicit filtration/backprojection-type formulae for the inversion of the spherical (circular) mean transform with the centers lying on the boundary of some polyhedra (or polygons, in 2D). The formulae are derived using the double layer potentials for the wave equation, for the domains with certain symmetri... |
To describe the cellular self-aggregation phenomenon, some strongly coupled PDEs named as Keller-Segel (KS) and Patlak-Keller-Segel (PKS) systems were proposed in 1970s. Since KS and PKS systems possess relatively simple structures but admit rich dynamics, plenty of scholars have studied them and obtained many signifi... |
We analyze the Vlasov equation coupled with the compressible Navier--Stokes equations with degenerate viscosities and vacuum. These two equations are coupled through the drag force which depends on the fluid density and the relative velocity between particle and fluid. |
In the article, we study variational eigenvalues on doubling metric measure spaces. We prove existence of minimizers of variational Neumann $(p,q)$-eigenvalues on metric measure spaces and on this base we obtain estimates of Neumann eigenvalues. |
It has been established that the local mass of blow-up solutions to Toda systems associated with the simple Lie algebras $\mathbf{A}_n,~\mathbf{B}_n,~\mathbf{C}_n$ and $\mathbf{G}_2$ can be represented by a finite Weyl group. In particular, at each blow-up point, after a sequence of bubbling steps (via scaling) is per... |
A rigorous proof is given for the convergence of the solutions of a viscous Cahn-Hilliard system to the solution of the regularized version of the forward-backward parabolic equation, as the coefficient of the diffusive term goes to 0. Non-homogenous Neumann boundary condition are handled for the chemical potential an... |
We consider the large time behaviour of solutions to the porous medium equation with a Fisher-KPP type reaction term and nonnegative, compactly supported initial function in $L^\infty(\mathbb{R}^N)\setminus\{0\}$: \begin{equation} \label{eq:abstract} \tag{$\star$}u_t=\Delta u^m+u-u^2\quad\text{in }Q:=\mathbb{R}^N\times... |
Starting from coupled fluid-kinetic equations for the modeling of laden flows, we derive relevant viscous corrections to be added to asymptotic hydrodynamic systems, by means of Chapman-Enskog expansions and analyse the shock profile structure for such limiting systems. Our main findings can be summarized as follows. |
In this work, we study the existence and nonexistence of solution for strongly coupled elliptic systems to m-parameters. |
In this paper, we consider the semi-linear wave systems with power-nonlinearities and a large class of space-dependent damping and potential. We obtain the same blow-up regions and the lifespan estimates for three types wave systems, compared with the systems without damping and potential. |
The energy of a type II superconductor submitted to an external magnetic field of intensity close to the second critical field is given by the celebrated Abrikosov energy. If the external magnetic field is comparable to and below the second critical field, the energy is given by a reference function obtained as a spec... |
In this paper, we provide modulated interaction energy estimates for the kernel $K(x) = |x|^{-\alpha}$ with $\alpha \in (0,d)$, and its applications to quantified asymptotic analyses for kinetic equations. The proof relies on a dimension extension argument for an elliptic operator and its commutator estimates. |
This paper concerns the structural stability of smooth cylindrically symmetric transonic flows in a concentric cylinder. Both cylindrical and axi-symmetric perturbations are considered. |
We demonstrate the existence of an open set of data which exhibits \textit{reversal} and \textit{recirculation} for the stationary Prandtl equations (data is taken in an appropriately defined product space due to the simultaneous forward and backward causality in the problem). Reversal describes the development of the... |
In this paper the existence of an absolute minimizer for a functional \[ F(u,\Omega) = \underset{x \in \Omega}{\text{ess sup}} \, f (x, u(x), Du(x)) \] is proved by using Perron's method. The function is assumed to be quasiconvex and uniformly coercive. |
We study the existence and nonexistence of positive (super) solutions to the nonlinear $p$-Laplace equation $$-\Delta_p u-\frac{\mu}{|x|^p}u^{p-1}=\frac{C}{|x|^{\sigma}}u^q$$ in exterior domains of ${\R}^N$ ($N\ge 2$). Here $p\in(1,+\infty)$ and $\mu\le C_H$, where $C_H$ is the critical Hardy constant. |
For the nonlinear wave equation $u_{tt} - c(u)\big(c(u) u_x\big)_x~=~0$, it is well known that solutions can develop singularities in finite time. For an open dense set of initial data, the present paper provides a detailed asymptotic description of the solution in a neighborhood of each singular point, where $|u_x|\t... |
We consider a class of fully nonlinear nonlocal degenerate elliptic operators which are modeled on the fractional Laplacian and converge to the truncated Laplacians. We investigate the validity of (strong) maximum and minimum principles, and their relation with suitably defined principal eigenvalues. |
In this paper, we study the long-time behavior of global solutions to the Schrödinger-Choquard equation $$i\partial_tu+\Delta u=-(I_\alpha\ast|\cdot|^b|u|^{p})|\cdot|^b|u|^{p-2}u. $$ <br>Inspired by Murphy, who gave a simple proof of scattering for the non-radial inhomogeneous NLS, we prove scattering theory below the ... |
We focus on the bilinear Strichartz estimates for free solutions to the Schrödinger equation on rescaled waveguide manifolds $\mathbb{R} \times \mathbb{T}_\lambda^n$, $\mathbb{T}_\lambda^n=(\lambda\mathbb{T})^n$ with $n\geq 1$ and their applications. First, we utilize a decoupling-type estimate originally from Fan-Sta... |
We prove the uniqueness of solutions to Dirichlet problem for p-harmonic maps with images in a small geodesic ball of the target manifold. As a consequence, we show that such maps have Hoelder continuous derivatives. |
We continue our investigation of Hardy-type inequalities involving combinations of cylindrical and spherical weights. Compared to [Cora-Musina-Nazarov, Ann. Sc. Norm. Sup., 2024], where the quasi-spherical case was considered, we handle the full range of allowed parameters. |
We consider the three-dimensional incompressible Navier-Stokes equation on the whole space. We observe that this system admits a $L^\infty$ family of global spatial plane wave solutions, which are connected with the two-dimensional equation. |
This paper investigates the existence, nonexistence, and qualitative properties of p-harmonic functions in the upper half-space $\mathbb{R}^N_+ \, (N \geq 3)$ satisfying nonlinear boundary conditions for $1<p<N$. Moreover, the symmetry of positive solutions is shown by using the method of moving planes. |
In this paper, we will solve the Leray's problem for the stationary Navier-Stokes system in a 2D infinite distorted strip with the Navier-slip boundary condition. The existence, uniqueness, regularity and asymptotic behavior of the solution will be investigated. |
Using the inverse scattering method, we construct global solutions to the Novikov-Veselov equation for real-valued decaying initial data q with the property that the associated Schrodinger operator with potential q is nonnegative. Such initial data are either critical (an arbitrarily small perturbation of the potentia... |
We prove the local Hölder continuity of strong local minimizers of the stored energy functional \[E(u)=\int_{\om}\lambda |\nabla u|^{2}+h(\det \nabla u) \,dx\] subject to a condition of `positive twist'. The latter turns out to be equivalent to requiring that $u$ maps circles to suitably star-shaped sets. |
We analyze the spectral stability of small-amplitude, periodic, traveling-wave solutions of a Boussinesq-Whitham system. These solutions are shown numerically to exhibit high-frequency instabilities when subject to bounded perturbations on the real line. |
Analog to the classical result of Kazdan-Warner for the existence of solutions to the prescribed Gaussian curvature equation on compact 2-manifolds without boundary, it is widely known that if $(M,g_0)$ is a closed 4-manifold with zero $Q$-curvature and if $f$ is any non-constant, smooth, sign-changing function with $\... |
We prove a general clustering result for the fractional Sobolev space $W^{s,p}$: whenever the positivity set of a function $u$ in a square has measure bounded from below by a multiple of the cube's volume, and the $W^{s,p}$-seminorm of $u$ is bounded from above by a convenient power of the cube's side, then $u$... |
Steady vortices for the three-dimensional Euler equation for inviscid incompressible flows and for the shallow water equation are constructed and showed to tend asymptotically to singular vortex filaments. The construction is based on the asymptotic study of solutions to a semilinear elliptic problem. |
In this paper we consider a one-dimensional Mindlin model describing linear elastic behaviour of isotropic materials with micro-structural effects. After introducing the kinetic and the potential energy, we derive a system of equations of motion by means of the Euler-Lagrange equations. |
This paper concerns the bubbling phenomena for the $L^2$-critical half-wave equation in dimension one. Given arbitrarily finitely many distinct singularities, we construct blow-up solutions concentrating exactly at these singularities. |
We classify all the blow-up solutions in self-similar form to the following reaction-diffusion equation $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,T)$, with $m>1$, $1\leq p<m$ and $-2(p-1)/(m-1)<\sigma<\infty$. We prove that there are several types of self-similar solu... |
We consider time discretizations of the two-dimensional Euler equation written in vorticity form. The discretization method uses a Crouch-Grossman integrator that proceeds in two stages: first freezing the velocity vector field at the beginning of the time step, and then solve the resulting elementary transport equati... |
In this paper, we study the $a$-contraction property of small extremal shocks for 1-d systems of hyperbolic conservation laws endowed with a single convex entropy, when subjected to large perturbations. We show that the weight coefficient $a$ can be chosen with amplitude proportional to the size of the shock. |
We investigate the properties of the operator \Delta(\sigma\Delta. ):H^2_0(\Omega)-> H^{-2}(\Omega), where \sigma is a given parameter whose sign can change on the bounded domain \Omega. |
We investigate the barotropic compressible Navier-Stokes equations with slip boundary conditions in a three-dimensional (3D) simply connected bounded domain, whose smooth boundary has a finite number of two-dimensional connected components. For any adiabatic exponent bigger than one, after discovering some new estimat... |
We consider the homogenization of the Navier-Stokes equation, set in a channel with a rough boundary, of small amplitude and wavelength $\epsilon$. It was shown recently that, for any non-degenerate roughness pattern, and for any reasonable condition imposed at the rough boundary, the homogenized boundary condition in... |
In this paper we study the following nonlinear Choquard equation $$ -\Delta u+u=\left(\ln\frac{1}{|x|}\ast F(u)\right)f(u),\quad\text{ in }\,\mathbb{R}^2, $$ where $f\in C^1(\mathbb{R})$ and $F$ is the primitive of the nonlinearity $f$ vanishing at zero. We use an asymptotic approximation approach to establish the exi... |
Given $A,B\in M_n(\mathbb R)$, we consider the Cauchy problem for partially dissipative hyperbolic systems having the form \begin{equation*} <br>\partial_{t}u+A\partial_{x}u+Bu=0, \end{equation*} with the aim of providing a detailed description of the large-time behavior. Sharp $L^p$-$L^q$ estimates are established fo... |
We characterize the well known self-similar Blasius profiles, $[\bar{u}, \bar{v}]$, as downstream attractors to solutions $[u,v]$ to the 2D, stationary Prandtl system. It was established in \cite{Serrin} that $\| u - \bar{u}\|_{L^\infty_y} \rightarrow 0$ as $x \rightarrow \infty$. |
Let $n\geq 3$, $\alpha$, $\beta\in\mathbb{R}$, and let $v$ be a solution $\Delta v+\alpha e^v+\beta x\cdot\nabla e^v=0$ in $\mathbb{R}^n$, which satisfies the conditions $\lim_{R\to\infty}\frac{1}{\log R}\int_{1}^{R}\rho^{1-n} (\int_{B_{\rho}}e^v\,dx)d\rho\in (0,\infty)$ and $|x|^2e^{v(x)}\le A_1$ in $\R^n$. We prove ... |
We consider the stochastic electrokinetic flow in a smooth bounded domain $\mathcal{D}$, modelled by a Nernst-Planck-Navier-Stokes system with a blocking boundary conditions for ionic species concentrations, perturbed by multiplicative noise. Several results are established in this paper. |
This paper deals with the two-species chemotaxis-competition system. About the problem, Bai--Winkler first obtained asymptotic stability under some conditions. |
We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. |
We consider the relativistic Schrödinger equation with a time dependent vector and scalar potential on a bounded cylindrical domain. Using a Geometric Optics Ansatz we establish a logarithmic stability estimate for the recovery of the vector and scalar potentials. |
Discrete Ginzburg-Landau (DGL) equations with non-local nonlinearities have been established as significant inherently discrete models in numerous physical contexts, similar to their counterparts with local nonlinear terms. We study two prototypical examples of non-local and local DGLs on the one-dimensional infinite ... |
We study the vanishing dissipation limit of the three-dimensional (3D) compressible Navier-Stokes-Fourier equations to the corresponding 3D full Euler equations. Our results are twofold. |
Two mathematical models are developed within the theoretical framework of large strain elasticity for the determination of upper and lower bounds on the total strain energy of a finitely deformed hyperelastic body in unilateral contact with a rigid surface or with an elastic substrate. The model problems take the form... |
Results of Struwe, Grillakis, Struwe-Shatah, Kapitanski, Bahouri-Shatah, Bahouri-Gérard and Nakanishi have established global wellposedness, regularity, and scattering in the energy class for the energy-critical nonlinear wave equation $\Box u = u^5$ in $\R^{1+3}$, together with a spacetime bound $$ \| u \|_{L^4_t L^{1... |
The aim of this work is to give a broad panorama of the control properties of fractional diffusive models from a numerical analysis and simulation perspective. We do this by surveying several research results we obtained in the last years, focusing in particular on the numerical computation of controls. |
A singularly perturbed free boundary problem arising from a real problem associated with a Radiographic Integrated Test Stand concerns a solution of the equation $\Delta u = f(u)$ in a domain $\Omega$ subject to constant boundary data, where the function $f$ in general is not monotone. When the domain $\Omega$ is a pe... |
We consider inverse boundary value problems for general real principal type differential operators. The first results state that the Cauchy data set uniquely determines the scattering relation of the operator and bicharacteristic ray transforms of lower order coefficients. |
We prove global well-posedness of the $ 3d $ Yang-Mills equation in the temporal gauge in $ H^{\sigma} $ for $ \sigma > \frac{5}{6} $. <br>Unlike related equations, Yang-Mills is not directly amenable to the method of almost conservation laws (I-method) in its Fourier and global version. |
We investigate the problem of dimension reduction for plates in nonlinear magnetoelasticity. The model features a mixed Eulerian-Lagrangian formulation, as magnetizations are defined on the deformed set in the actual space. |
The main topic of this note is a discussion of applicability conditions of the Nehari manifold method depending on the value of parameters of equations. As the main tool, we apply the nonlinear generalized Rayleigh quotient method. |
In this paper we analyse the stability of the system of partial differential equations modelling scalar image velocimetry. We first revisit a successful numerical technique to reconstruct velocity vectors ${u}$ from images of a passive scalar field $\psi$ by minimising a cost functional, that penalises the difference ... |
We show that t^{3/4}|| u(.,t) ||_{sup} --> 0 as t --> infty for all (global) Leray solutions of the incompressible Navier-Stokes equations in R3. It is also shown that t || u(.,t) - v(.,t) ||_{sup} --> 0 as t --> infty, where v(.,t) is the Stokes approximation, as well as other fundamental results. |
The original Leray's problem concerns the well-posedness of weak solutions to the steady incompressible Navier-Stokes equations in a distorted pipe, which approach to the Poiseuille flow subject to the no-slip boundary condition at spacial infinity. In this paper, the same problem with the Navier-slip boundary con... |
We give an easy proof of the fact that $C^\infty$ solutions to non-linear elliptic equations of second order $$ <br>\phi(x, u, D u, D^2 u)=0 $$ are analytic. Following ideas of Kato, the proof uses an inductive estimate for suitable weighted derivatives. |
In this article we apply local bifurcation theory to prove the existence of small-amplitude steady periodic water waves, which propagate over a flat bed with a specified fixed mean-depth, and where the underlying flow has a discontinuous vorticity distribution. |
The hot spots conjecture of J. Rauch states that the second Neumann eigenfunction of the Laplace operator on a bounded Lipschitz domain in $\mathbb{R}^n$ attains its extrema only on the boundary of the domain. We present an analogous problem for domains with mixed Dirichlet-Neumann boundary conditions. |
Via continuous deformations based on natural flow evolutions, we prove several novel monotonicity results for Riesz-type nonlocal energies on triangles and quadrilaterals. Some of these results imply new and simpler proofs for known theorems without relying on any symmetrization arguments. |
We introduce, analyze and test a new interpolation operator for use with continuous data assimilation (DA) of evolution equations that are discretized spatially with the finite element method. The interpolant is constructed as an approximation of the L2 projection operator onto piecewise constant functions on a coarse... |
We consider the asymptotic behavior of compressible isentropic flow when the initial mass is finite, which is modeled by the compressible Euler equation with frictional damping. It is shown in \cite{HUA} (resp. |
Existence and uniqueness of solutions is shown for a class of viscoelastic flows in porous media with particular attention to problems with nonsmooth porosities. The considered models are formulated in terms of the time-dependent nonlinear interaction between porosity and effective pressure, which in certain cases lea... |
In this work, we investigate the IVP for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn-Hilliard equation. |
This paper is devoted to the study of the global existence of smooth solutions for the 3+1 dimensional Einstein-Klein-Gordon systems with a $U(1) \times \mathbb{R}$ isometry group for a class of regular Cauchy data. In our first paper \cite{chen}, we reduce the Einstein equations to a 2+1 dimensional Einstein-wave-Kle... |
In this paper, we deal with the electrostatic Born-Infeld equation \begin{equation}\label{eq:BI-abs} \tag{$\mathcal{BI}$} \left\{ \begin{array}{ll} -\operatorname{div}\left(\displaystyle\frac{\nabla \phi}{\sqrt{1-|\nabla \phi|^2}}\right)= \rho, & \hbox{in } \mathbb{R}^N, \\ \displaystyle\lim_{|x|\to \infty}\phi(x)=... |
The occurrence of a finite time singularity is shown for a free boundary problem modeling microelectromechanical systems (MEMS) when the applied voltage exceeds some value. The model involves a singular nonlocal reaction term and a nonlinear curvature term accounting for large deformations. |
This paper considers the existence of weak and strong solutions to the Poisson equation on a surface with a boundary condition in co-normal direction. We apply the Lax-Milgram theorem and some properties of $H^1$-functions to show the existence of a unique weak solution to the surface Poisson equation when the exterio... |
We develop the techniques of \cite{KS1} and \cite{ES1} in order to derive dispersive estimates for a matrix Hamiltonian equation defined by linearizing about a minimal mass soliton solution of a saturated, focussing nonlinear Schrödinger equation {c} i u_t + \Delta u + \beta (|u|^2) u = 0 u(0,x) = u_0 (x), in $\reals^3... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.