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Local energy estimates and global solvability in a three-dimensional chemotaxis-fluid system with prescribed signal on the boundary Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2021-01-21 Yulan Wang; Michael Winkler; Zhaoyin Xiang
Abstract The chemotaxis-Stokes system { n t + u · ∇ n = Δ n − ∇ · ( n ∇ c ) , c t + u · ∇ c = Δ c − n c , u t = Δ u + ∇ P + n ∇ ϕ , ∇ · u = 0 , ( ⋆ ) is considered in a bounded domain Ω ⊂ R 3 with smooth boundary. The corresponding solution theory is quite well-developed in the case when ( ⋆ ) is accompanied by homogeneous boundary conditions of no-flux type for n and c, and of Dirichlet type for
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Distance functions with dense singular sets Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2021-01-21 Mario Santilli
Abstract We characterize the denseness of the singular set of the distance function from a C 1 -hypersurface in terms of an inner ball condition and we address the problem of the existence of viscosity solutions of the Eikonal equation whose singular set (i.e. set of non-differentiability points) is not no-where dense.
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A class of prescribed Weingarten curvature equations in Euclidean space Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2021-01-21 Li Chen; Agen Shang; Qiang Tu
Abstract In this article, we consider a class of prescribed Weingarten curvature equations. Under some sufficient conditions, we obtain an existence result by the standard degree theory based on the a priori estimates for the solutions to the prescribed Weingarten curvature equations.
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Symmetry breaking bifurcations for two overdetermined boundary value problems with non-constant Neumann condition on exterior domains in R 3 Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2021-01-21 Filippo Morabito
Abstract We study two overdetermined elliptic boundary value problems on exterior domains (the complement of a ball and the complement of a solid cylinder in R 3 respectively). The Neumann condition is non-constant and involves the mean curvature of the boundary. We show there exists a family of bifurcation branches of domains which are small deformations of the complement of a ball and of the complement
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Black hole gluing in de Sitter space Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2021-01-18 Peter Hintz
Abstract We construct dynamical many-black-hole spacetimes with well-controlled asymptotic behavior as solutions of the Einstein vacuum equation with positive cosmological constant. We accomplish this by gluing Schwarzschild–de Sitter or Kerr–de Sitter black hole metrics into neighborhoods of points on the future conformal boundary of de Sitter space, under certain balance conditions on the black hole
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Uniqueness criteria for the Oseen vortex in the 3d Navier-Stokes equations Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2021-01-18 Jacob Bedrossian; William Golding
Abstract In this paper, we consider the uniqueness of solutions to the 3d Navier-Stokes equations with initial vorticity given by ω 0 = α e z δ x = y = 0 , where δ x = y = 0 is the one dimensional Hausdorff measure of an infinite, vertical line and α ∈ R is an arbitrary circulation. This initial data corresponds to an idealized, infinite vortex filament. One smooth, mild solution is given by the self-similar
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Recovery of discontinuous Lamé parameters from exterior Cauchy data Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2021-01-13 Peter Caday; Maarten V. de Hoop; Vitaly Katsnelson; Gunther Uhlmann
Abstract Consider an isotropic elastic medium Ω ⊂ R 3 whose Lamé parameters are piecewise smooth. In the elastic wave initial value inverse problem, we are given the solution operator for the elastic wave equation, but only outside Ω and only for initial data supported outside Ω. Using the recently introduced scattering control series in the acoustic case, we prove that piecewise smooth Lamé parameters
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On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2021-01-02 Huaian Diao; Xinlin Cao; Hongyu Liu
Abstract This paper is concerned with the intrinsic geometric structures of conductive transmission eigenfunctions. The geometric properties of interior transmission eigenfunctions were first studied in Blåsten, E., Liu, H. (2017). On vanishing near corners of transmission eigenfunctions. J. Funct. Anal. 273(11):3616–3632. It is shown in two scenarios that the interior transmission eigenfunction must
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The Fuchsian approach to global existence for hyperbolic equations Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-12-30 Florian Beyer; Todd A. Oliynyk; J. Arturo Olvera-Santamaría
Abstract We analyze the Cauchy problem for symmetric hyperbolic equations with a time singularity of Fuchsian type and establish a global existence theory along with decay estimates for evolutions toward the singular time under a small initial data assumption. We, then, apply this theory to semilinear wave equations near spatial infinity on Minkowski and Schwarzschild spacetimes, and to the relativistic
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Attractors of the Einstein-Klein-Gordon system Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-11-16 David Fajman; Zoe Wyatt
Abstract It is shown that negative Einstein metrics are attractors of the Einstein-Klein-Gordon system. As an essential part of the proof we upgrade a technique that uses the continuity equation complementary to L2-estimates to control massive matter fields. In contrast to earlier applications of this idea we require a correction to the energy density to obtain sufficiently strong pointwise bounds
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Flow solutions of transport equations Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-11-01 Sholeh Karimghasemi; Siegfried Müller; Michael Westdickenberg
Abstract Under general assumptions on the velocity field, it is possible to construct a flow that is forward untangled. Once such a flow has been selected, the associated transport problem is well-posed.
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Scaling limits and homogenization of mixing Hamilton-Jacobi equations Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-10-24 Benjamin Seeger
Abstract We study the homogenization of nonlinear, first-order equations with highly oscillatory mixing spatio-temporal dependence. It is shown in a variety of settings that the homogenized equations are stochastic Hamilton-Jacobi equations with deterministic, spatially homogenous Hamiltonians driven by white noise in time. The paper also contains proofs of some general regularity and path stability
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Stability of the conical Kähler-Ricci flows on Fano manifolds Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-12-17 Jiawei Liu; Xi Zhang
Abstract In this paper, we study stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle 2 π β along the divisor, then for any β ′ sufficiently close to β, the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein metric with cone angle 2 π β ′ along the divisor. Here, we only use the condition
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Large-scale Lipschitz estimates for elliptic systems with periodic high-contrast coefficients Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-12-13 Zhongwei Shen
Abstract This article is concerned with the large-scale regularity in the homogenization of elliptic systems of elasticity with periodic high-contrast coefficients. We obtain the large-scale Lipschitz estimate that is uniform with respect to the contrast ratio δ 2 for 0 < δ < ∞ . Our study also covers the case of soft inclusions (δ = 0) as well as the case of stiff inclusions ( δ = ∞ ). The large-scale
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Density of zero sets for sums of eigenfunctions Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-12-13 Stefano Decio
Abstract We consider linear combinations of eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian manifold (M, g) and investigate a density property of their zero sets. More precisely, let f = ∑ k = 1 m a k ϕ λ j k , where − Δ g ϕ λ = λ ϕ λ . Denoting by Zf the zero-set of f, we show that for any x ∈ M , dist ( x , Z f ) ≤ C ( m ) λ j 1 − 1 / 2 . The proof is based on a new integral
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Resolvent near zero energy on Riemannian scattering (asymptotically conic) spaces, a Lagrangian approach Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-12-13 András Vasy
Abstract We use a Lagrangian regularity perspective to discuss resolvent estimates near zero energy on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. In addition to the Lagrangian perspective we introduce and use a resolved pseudodifferential algebra to deal with zero energy degeneracies in a robust manner.
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Limiting absorption principle on Riemannian scattering (asymptotically conic) spaces, a Lagrangian approach Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-12-13 András Vasy
Abstract We use a Lagrangian perspective to show the limiting absorption principle on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. More precisely we show that, for non-zero spectral parameter, the ‘on spectrum’, as well as the ‘off-spectrum’, spectral family is Fredholm in function spaces which encode the Lagrangian regularity of generalizations of ‘outgoing
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Determining the magnetic potential in the fractional magnetic Calderón problem Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-12-10 Li Li
Abstract We determine both the magnetic potential and the electric potential from the exterior partial measurements of the Dirichlet-to-Neumann map in the fractional linear magnetic Calderón problem by using an integral identity. We also determine both the magnetic potential and the nonlinearity in the fractional semi-linear magnetic Calderón problem by using a first order linearization.
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A dynamical system approach to a class of radial weighted fully nonlinear equations Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-12-10 Liliane Maia; Gabrielle Nornberg; Filomena Pacella
Abstract In this paper we study existence, nonexistence and classification of radial positive solutions of some weighted fully nonlinear equations involving Pucci extremal operators. Our results are entirely based on the analysis of the dynamics induced by an autonomous quadratic system which is obtained after a suitable transformation. This method allows to treat both regular and singular solutions
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Concavity of solutions to degenerate elliptic equations on the sphere Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-12-10 Mat Langford; Julian Scheuer
Abstract We prove the concavity of classical solutions to a wide class of degenerate elliptic differential equations on strictly convex domains of the unit sphere. The proof employs a suitable two-point maximum principle, a technique which originates in works of Korevaar, Kawohl and Kennington for equations on Euclidean domains. We emphasize that no differentiability of the differential operator is
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Short-time existence of the α-Dirac-harmonic map flow and applications Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-12-10 Jürgen Jost; Jingyong Zhu
Abstract In this paper, we discuss the general existence theory of Dirac-harmonic maps from closed surfaces via the heat flow for α-Dirac-harmonic maps and blow-up analysis. More precisely, given any initial map along which the Dirac operator has nontrivial minimal kernel, we first prove the short time existence of the heat flow for α-Dirac-harmonic maps. The obstacle to the global existence is the
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Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-12-10 Yannick Sire; Susanna Terracini; Stefano Vita
Abstract We consider a class of equations in divergence form with a singular/degenerate weight − div ( | y | a A ( x , y ) ∇ u ) = | y | a f ( x , y ) or div ( | y | a F ( x , y ) ) . Under suitable regularity assumptions for the matrix A and f (resp. F) we prove Hölder continuity of solutions which are even in y ∈ R , and possibly of their derivatives up to order two or more (Schauder estimates)
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Asymptotic behaviour of the Steklov spectrum on dumbbell domains Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-12-06 Dorin Bucur; Antoine Henrot; Marco Michetti
Abstract We analyse the asymptotic behaviour of the eigenvalues and eigenvectors of a Steklov problem in a dumbbell domain consisting of two Lipschitz sets connected by a thin tube with vanishing width. All the eigenvalues are collapsing to zero, the speed being driven by some power of the width which multiplies the eigenvalues of a one dimensional problem. In two dimensions of the space, the behaviour
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Global representation and multiscale expansion for the Dirichlet problem in a domain with a small hole close to the boundary Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-12-06 Virginie Bonnaillie-Noël; Matteo Dalla Riva; Marc Dambrine; Paolo Musolino
Abstract For each pair ε = ( ε 1 , ε 2 ) of positive parameters, we define a perforated domain Ω ε by making a small hole of size ε 1 ε 2 in an open regular subset Ω of ℝ n ( n ≥ 3 ). The hole is situated at distance ε 1 from the outer boundary ∂ Ω of the domain. Thus, when ε → ( 0 , 0 ) both the size of the hole and its distance from ∂ Ω tend to zero, but the size shrinks faster than the distance
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Mild assumptions for the derivation of Einstein’s effective viscosity formula Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-12-01 David Gérard-Varet; Richard M. Höfer
Abstract We provide a rigorous derivation of Einstein’s formula for the effective viscosity of dilute suspensions of n rigid balls, n ≫ 1 , set in a volume of size 1. So far, most justifications were carried under a strong assumption on the minimal distance between the balls: d min ≥ c n − 1 3 , c > 0. We relax this assumption into a set of two much weaker conditions: one expresses essentially that
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Large-time behavior of unbounded solutions of viscous Hamilton-Jacobi equations in R N Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-11-19 Guy Barles; Alexander Quaas; Andrei Rodríguez-Paredes
Abstract We study the large-time behavior of bounded from below solutions of parabolic viscous Hamilton-Jacobi Equations in the whole space R N in the case of superquadratic Hamiltonians. Existence and uniqueness of such solutions are shown in a very general framework, namely when the source term and the initial data are only bounded from below with an arbitrary growth at infinity. Our main result
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Regularity of nonvanishing – at infinity or at the boundary – solutions of the defocusing nonlinear Shrödinger equation Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-11-19 Nikolaos Gialelis; Nikos I. Karachalios; Ioannis G. Stratis
Abstract Considering the defocusing nonlinear Schrödinger equation (NLSE) in generic (bounded or unbounded) open sets U ⊆ R n for n = 1, 2, and 3, we prove the regularity of weak, non-vanishing solutions at infinity or at the boundary of U. Our approach is based on suitably defined extension operators, along with a priori estimates for regular functions, under certain assumptions on the smoothness
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The Hartree and Vlasov equations at positive density Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-09-10 Mathieu Lewin; Julien Sabin
Abstract We consider the nonlinear Hartree and Vlasov equations around a translation-invariant (homogeneous) stationary state in infinite volume, for a short range interaction potential. For both models, we consider time-dependent solutions which have a finite relative energy with respect to the reference translation-invariant state. We prove the convergence of the Hartree solutions to the Vlasov ones
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The dirichlet-conormal problem with homogeneous and inhomogeneous boundary conditions Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-11-16 Hongjie Dong; Zongyuan Li
Abstract We consider the mixed Dirichlet-conormal problem on irregular domains in R d . Two types of regularity results will be discussed: the W 1 , p regularity and a non-tangential maximal function estimate. The domain is assumed to be Reifenberg-flat, and the interfacial boundary is either Reifenberg-flat of co-dimension 2 or is locally sufficiently close to a Lipschitz function of m variables,
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Localisation of the first eigenfunction of a convex domain Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-11-04 Thomas Beck
Abstract We study the first Dirichlet eigenfunction of the Laplacian in a n-dimensional convex domain. For domains of a fixed inner radius, estimates of Chiti [1 Chiti, G. (1982). A reverse Hölder inequality for the eigenfunctions of linear second order elliptic operators. Z. angew. Math. Phys . 33(1):143–148.[Crossref], [Web of Science ®] , [Google Scholar],2 Chiti, G. (1982). An isoperimetric inequality
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Gaussian fluctuations from random Schrödinger equation Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-10-24 Yu Gu; Tomasz Komorowski
Abstract We study the Schrödinger equation driven by a weak Brownian forcing, and derive Gaussian fluctuations in the form of a time-inhomogeneous Ornstein-Uhlenbeck process. As a result, when evaluated at a fixed frequency, the intensity of the incoherent wave is of exponential distribution.
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Stochastic PDEs via convex minimization Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-10-14 Luca Scarpa; Ulisse Stefanelli
Abstract We prove the applicability of the Weighted Energy-Dissipation (WED) variational principle to nonlinear parabolic stochastic partial differential equations in abstract form. The WED principle consists in the minimization of a parameter-dependent convex functional on entire trajectories. Its unique minimizers correspond to elliptic-in-time regularizations of the stochastic differential problem
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Semiclassical estimates for pseudodifferential operators and the Muskat problem in the unstable regime Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-10-14 Víctor Arnaiz; Ángel Castro; Daniel Faraco
Abstract We obtain new semiclassical estimates for pseudodifferential operators with low regular symbols. Such symbols appear naturally in a Cauchy Problem related to recent weak solutions to the unstable Muskat problem constructed via convex integration. In particular, our new estimates reveal the tight relation between the speed of opening of the mixing zone and the regularity of the interphase.
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On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-09-25 Raphaël Danchin; Jin Tan
Abstract We investigate the existence and uniqueness issues of the 3D incompressible Hall-magnetohydrodynamic system supplemented with initial velocity u0 and magnetic field B0 in critical regularity spaces. In the case where u0, B0 and the current J0:=∇×B0 belong to the homogeneous Besov space Ḃp,13p−1, 1≤p<∞, and are small enough, we establish a global result and the conservation of higher regularity
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Globally bounded trajectories for the barotropic Navier–Stokes system with general boundary conditions Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-09-08 Jan Březina; Eduard Feireisl; Antonín Novotný
Abstract We consider the barotropic Navier–Stokes system describing the motion of a viscous compressible fluid interacting with the outer world through general in/out flux boundary conditions. We consider a hard-sphere type pressure EOS and show that all trajectories eventually enter a bounded absorbing set. In particular, the associated ω − limit sets are compact and support a stationary statistical
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Existence of weak solutions to a continuity equation with space time nonlocal Darcy law Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-09-08 Luis Caffarelli; Maria Gualdani; Nicola Zamponi
Abstract In this manuscript, we consider a non-local porous medium equation with non-local diffusion effects given by a fractional heat operator { ∂ t u = div ( u ∇ p ) , ∂ t p = − ( − Δ ) s p + u β , in two space dimensions for β > 1 , 1 β < s < 1 . Global in time existence of weak solutions is shown by employing a time semi-discretization of the equations, an energy inequality and the Div-Curl lemma
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Linear and fully nonlinear elliptic equations with Ld -drift Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-08-17 Nicolai V. Krylov
In subdomains of R d , we consider uniformly elliptic equations H ( v ( x ) , D v ( x ) , D 2 v ( x ) , x ) = 0 with the growth of H with respect to | D v | controlled by the product of a function from Ld and | D v | . The dependence of H on x is assumed to be of BMO type. Among other things we prove that there exists d 0 ∈ ( d / 2 , d ) such that for any p ∈ ( d 0 , d ) the equation with prescribed
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Linear and fully nonlinear elliptic equations with Ld -drift Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-08-17 Nicolai V. Krylov
Abstract In subdomains of R d , we consider uniformly elliptic equations H ( v ( x ) , D v ( x ) , D 2 v ( x ) , x ) = 0 with the growth of H with respect to | D v | controlled by the product of a function from Ld and | D v | . The dependence of H on x is assumed to be of BMO type. Among other things we prove that there exists d 0 ∈ ( d / 2 , d ) such that for any p ∈ ( d 0 , d ) the equation with
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Nonlocal dissipation measure and L1 kinetic theory for fractional conservation laws Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-06-17 Nathaël Alibaud; Boris Andreianov; Adama Ouédraogo
We introduce a kinetic formulation for scalar conservation laws with nonlocal and nonlinear diffusion terms. We deal with merely L1 initial data, general self-adjoint pure jump Lévy operators, and locally Lipschitz nonlinearities of porous medium kind possibly strongly degenerate. The cornerstone of the formulation and the uniqueness proof is an adequate explicit representation of the dissipation measure
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Stable blowup for the focusing energy critical nonlinear wave equation under random perturbations Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-08-11 Bjoern Bringmann
We consider the radial focusing energy critical nonlinear wave equation in three spatial dimensions. We establish the stability of the ODE-blowup under random perturbations below the energy space. The argument relies on probabilistic Strichartz estimates in similarity coordinates.
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Stable blowup for the focusing energy critical nonlinear wave equation under random perturbations Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-08-11 Bjoern Bringmann
Abstract We consider the radial focusing energy critical nonlinear wave equation in three spatial dimensions. We establish the stability of the ODE-blowup under random perturbations below the energy space. The argument relies on probabilistic Strichartz estimates in similarity coordinates.
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Enhanced dissipation for the 2D couette flow in critical space Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-07-21 Nader Masmoudi; Weiren Zhao
Abstract We consider the 2 D incompressible Navier-Stokes equations on T × R , with initial vorticity that is δ close in H x log L y 2 to −1(the vorticity of the Couette flow ( y , 0 ) ). We prove that if δ ≪ ν 1 / 2 , where ν denotes the viscosity, then the solution of the Navier-Stokes equation approaches some shear flow which is also close to Couette flow for time t ≫ ν − 1 / 3 by a mixing-enhanced
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Homogenization for the cubic nonlinear Schrödinger equation on R2 Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-07-07 Maria Ntekoume
Abstract We study the defocusing inhomogeneous mass-critical nonlinear Schrödinger equation on R2 i∂tun+Δun=g(nx)|un|2un for initial data in L2(R2). We obtain sufficient conditions on g to ensure existence and uniqueness of global solutions for n sufficiently large, as well as homogenization.
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An analytic construction of singular solutions related to a critical Yamabe problem Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-07-06 Hardy Chan; Azahara DelaTorre
Abstract We answer affirmatively a question posed by Aviles in 1983, concerning the construction of singular solutions of semilinear equations without using phase-plane analysis. Fully exploiting the semilinearity and the stability of the linearized operator in any dimension, our techniques involve a careful gluing in weighted L∞ spaces that handles multiple occurrences of criticality, without the
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Asymptotic behavior at the isolated singularities of solutions of some equations on singular manifolds with conical metrics Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-07-06 Zongming Guo; Jiayu Li; Fangshu Wan
Abstract We present the sharp characterization of the behavior at the isolated singularities of positive solutions of some equations on singular manifolds with conical metrics. It is seen that the equations on singular manifolds with conical metrics are equivalent to weighted elliptic equations in B \ { 0 } , where B ⊂ R N is the unit ball. The weights can be singular at x = 0. We present the sharp
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Interface asymptotics of eigenspace Wigner distributions for the harmonic oscillator Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-06-28 Boris Hanin; Steve Zelditch
Abstract Eigenspaces of the quantum isotropic Harmonic Oscillator Ĥℏ:=−ℏ22Δ+12||x||2 on Rd have extremally high multiplicites and the eigenspace projections Πℏ,EN(ℏ) have special asymptotic properties. This article gives a detailed study of their Wigner distributions Wℏ,EN(ℏ)(x,ξ). Heuristically, if EN(ℏ)=E,Wℏ,EN(ℏ)(x,ξ) is the “quantization” of the energy surface ΣE, and should be like the delta-function
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On two methods for quantitative unique continuation results for some nonlocal operators Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-06-28 María Ángeles García-Ferrero; Angkana Rüland
Abstract In this article, we present two mechanisms for deducing logarithmic quantitative unique continuation bounds for certain classes of integral operators. In our first method, expanding the corresponding integral kernels, we exploit the logarithmic stability of the moment problem. In our second method we rely on the presence of branch-cut singularities for certain Fourier multipliers. As an application
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Inverse source problem for a system of wave equations on a Lorentzian manifold Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-06-24 Hiroshi Takase
Abstract A system of wave equations on a Lorentzian manifold, the coefficients of which depend on time relates to the Einstein equation in general relativity. We consider inverse source problem for the system in this paper. Having established the Carleman estimate with a second large parameter for the Laplace–Beltrami operator on a Lorentzian manifold under assumptions independent of a choice of local
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Gradient estimates for capillary-type problems via the maximum principle, a second look Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-06-18 Gary M. Lieberman
Abstract We prove gradient bounds for solutions of a class of boundary value problems related to the capillary problem, using the maximum principle. Our results extend those of the author and those of Ma and Xu. In addition to the capillary boundary condition, we can prove results for the general boundary condition (1+|Du|2)q−1Du·γ=ψ(x) for any choice of the parameter q≥0. (The range 0
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Scattering for a particle interacting with a Bose gas Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-06-18 Tristan Léger
Abstract We study the long-time behavior of solutions to an ODE- Schrödinger type system that models the interaction of a particle with a Bose gas. We show that the particle has sonic or subsonic ballistic trajectory asymptotically, and that the wave function describing the Bose gas converges to a soliton in L∞.
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On the Bochner Laplacian operator on theta line bundle over quasi-tori Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-06-10 Ahmed Intissar; Mohammed Ziyat
Abstract In this paper, we consider the Laplcian operator on theta line bundle over the quasi-torus, which is called the Bochner Laplacian. This operator has a canonical realization as a magnetic Laplacian acting on complex valued functions satisfying a functional equation. We study the spectral properties of such Laplacian and we show that its spectrum is reduced to eigenvalues πm; m=0,1,…. Then,
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Geometric and obstacle scattering at low energy Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-06-05 Alexander Strohmaier; Alden Waters
Abstract We consider scattering theory of the Laplace Beltrami operator on differential forms on a Riemannian manifold that is Euclidean at infinity. The manifold may have several boundary components caused by obstacles at which relative boundary conditions are imposed. Scattering takes place because of the presence of these obstacles and possible non-trivial topology and geometry. Unlike in the case
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The ∂¯-Neumann operator with Sobolev estimates up to a finite order Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-06-05 Phillip Harrington; Bingyuan Liu
Abstract Let Ω⊂Cm be a bounded pseudoconvex domain with smooth boundary. For each k∈N, we give a sufficient condition to estimate the ∂¯-Neumann operator in the Sobolev space Wk(Ω). The key feature of our results is a precise formula for k in terms of the geometry of the boundary of Ω. It also shows the bound in the precise formula for Sobolev estimates of order k is comparably reciprocal of the Sobolev
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Inverse random source problems for time-harmonic acoustic and elastic waves Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-06-05 Jianliang Li; Tapio Helin; Peijun Li
Abstract This paper concerns the random source problems for the time-harmonic acoustic and elastic wave equations in two and three dimensions. The goal is to determine the compactly supported external force from the radiated wave field measured in a domain away from the source region. The source is assumed to be a microlocally isotropic generalized Gaussian random function such that its covariance
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Long time existence of solutions to an elastic flow of networks Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-06-05 Harald Garcke; Julia Menzel; Alessandra Pluda
Abstract The L2-gradient flow of the elastic energy of networks leads to a Willmore type evolution law with non-trivial nonlinear boundary conditions. We show local in time existence and uniqueness for this elastic flow of networks in a Sobolev space setting under natural boundary conditions. In addition, we show a regularisation property and geometric existence and uniqueness. The main result is a
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Global existence, regularity, and uniqueness of infinite energy solutions to the Navier-Stokes equations Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-05-14 Zachary Bradshaw; Tai-Peng Tsai
This paper addresses several problems associated to local energy solutions (in the sense of Lemarié-Rieusset) to the Navier-Stokes equations with initial data which is sufficiently small at large or small scales as measured using truncated Morrey-type quantities, namely: (1) global existence for a class of data including the critical L2-based Morrey space; (2) initial and eventual regularity of local
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On the improvement of the Hardy inequality due to singular magnetic fields Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-05-13 Luca Fanelli; David Krejčiřík; Ari Laptev; Luis Vega
We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the Aharonov-Bohm field in all dimensions and establish a sharp Hardy-type inequality that takes into account both the dimensional as well as the magnetic flux contributions. Second, in the three-dimensional Euclidean space, we derive a non-trivial magnetic
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Reconstruction of piecewise constant layered conductivities in electrical impedance tomography Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-05-13 Henrik Garde
This work presents a new constructive uniqueness proof for Calderón’s inverse problem of electrical impedance tomography, subject to local Cauchy data, for a large class of piecewise constant conductivities that we call piecewise constant layered conductivities (PCLC). The resulting reconstruction method only relies on the physically intuitive monotonicity principles of the local Neumann-to-Dirichlet
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Exponential stability for the nonlinear Schrödinger equation with locally distributed damping Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-05-07 Marcelo M. Cavalcanti; Wellington J. Corrêa; Türker Özsarı; Mauricio Sepúlveda; Rodrigo Véjar-Asem
In this paper, we study the defocusing nonlinear Schrödinger equation with a locally distributed damping on a smooth bounded domain as well as on the whole space and on an exterior domain. We first construct approximate solutions using the theory of monotone operators. We show that approximate solutions decay exponentially fast in the L2-sense by using the multiplier technique and a unique continuation
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Semi-classical propagation of singularities for the Stokes system Commun. Partial Differ. Equ. (IF 1.079) Pub Date : 2020-05-06 Chenmin Sun
We study the quasi-mode of Stokes system posed on a smooth bounded domain Ω with Dirichlet boundary condition. We prove that the semi-classical defect measure associated with a sequence of solutions concentrates on the bicharacteristics of Laplacian as a matrix-valued Radon measure. Moreover, we show that the support of the measure is invariant under the Melrose-Sjöstrand flow.