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The Willmore flow with prescribed isoperimetric ratio Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2024-02-06 Fabian Rupp
We introduce a non-local L2-gradient flow for the Willmore energy of immersed surfaces which preserves the isoperimetric ratio. For spherical initial data with energy below an explicit threshold, w...
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Stationary equilibria and their stability in a Kuramoto MFG with strong interaction Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2024-02-06 Annalisa Cesaroni, Marco Cirant
Recently, R. Carmona, Q. Cormier, and M. Soner proposed a Mean Field Game (MFG) version of the classical Kuramoto model, which describes synchronization phenomena in a large population of “rational...
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Quasilinear wave equations on Schwarzschild–de Sitter Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2024-02-06 Georgios Mavrogiannis
We give an elementary new argument for global existence and exponential decay of solutions of quasilinear wave equations on Schwarzschild–de Sitter black hole backgrounds, for appropriately small i...
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Logarithmic Gross-Pitaevskii equation Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2024-02-06 Rémi Carles, Guillaume Ferriere
We consider the Schrödinger equation with a logarithmic nonlinearty and non-trivial boundary conditions at infinity. We prove that the Cauchy problem is globally well posed in the energy space, whi...
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Magnetic Schrödinger operators and landscape functions Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2024-02-06 Jeremy G. Hoskins, Hadrian Quan, Stefan Steinerberger
We study localization properties of low-lying eigenfunctions of magnetic Schrödinger operators (−i∇−A(x))2ϕ+V(x)ϕ=λϕ, where V:Ω→R≥0 is a given potential and A:Ω→Rd induces a magnetic field. We e...
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Boundary regularity for anisotropic minimal Lipschitz graphs Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2024-02-06 Antonio De Rosa, Reinaldo Resende
We prove that m-dimensional Lipschitz graphs in any codimension with C1,α boundary and anisotropic mean curvature bounded in Lp, p > m, are regular at every boundary point with density bounded abov...
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A weakly turbulent solution to the cubic nonlinear harmonic oscillator on ℝ2 perturbed by a real smooth potential decaying to zero at infinity Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2024-01-27 Ambre Chabert
We build a smooth real potential V(t, x) on (t0,+∞)×R2 decaying to zero as t→∞ and a smooth solution to the associated perturbed cubic noninear harmonic oscillator whose Sobolev norms blow up log...
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The convergence rate of p-harmonic to infinity-harmonic functions Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2024-01-09 Leon Bungert
The purpose of this paper is to prove a uniform convergence rate of the solutions of the p-Laplace equation Δpu=0 with Dirichlet boundary conditions to the solution of the infinity-Laplace equation...
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Nonlinear enhanced dissipation in viscous Burgers type equations II Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2024-01-09 Tej-Eddine Ghoul, Nader Masmoudi, Eliot Pacherie
In this follow up but self contained paper, we focus on the viscous Burgers equation. There, using the Hopf-Cole transformation, we compute the long time behavior of solutions for some classes of i...
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Monotone solutions for mean field games master equations: continuous state space and common noise Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2024-01-09 Charles Bertucci
We present the notion of monotone solution of mean field games master equations in the case of a continuous state space. We establish the existence, uniqueness and stability of such solutions under...
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A Globally Stable Self-Similar Blowup Profile in Energy Supercritical Yang-Mills Theory Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-10-30 Roland Donninger, Matthias Ostermann
This paper is concerned with the Cauchy problem for an energy-supercritical nonlinear wave equation in odd space dimensions that arises in equivariant Yang-Mills theory. In each dimension, there is...
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Synchronization in a Kuramoto mean field game Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-10-30 Rene Carmona, Quentin Cormier, H. Mete Soner
The classical Kuramoto model is studied in the setting of an infinite horizon mean field game. The system is shown to exhibit both synchronization and phase transition. Incoherence below a critical...
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Asymptotics and scattering for wave Klein-Gordon systems Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-10-30 Xuantao Chen, Hans Lindblad
We study the coupled wave-Klein-Gordon systems, introduced by LeFloch-Ma and then Ionescu-Pausader, to model the nonlinear effects from the Einstein-Klein-Gordon equation in harmonic coordinates. W...
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Qualitative properties of solutions to a mass-conserving free boundary problem modeling cell polarization Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-08-30 Anna Logioti, Barbara Niethammer, Matthias Röger, Juan J. L. Velázquez
We consider a parabolic non-local free boundary problem that has been derived as a limit of a bulk-surface reaction-diffusion system which models cell polarization. We have justified the well-posed...
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Slow periodic homogenization for Hamilton–jacobi equations Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-08-24 William Cooperman
Capuzzo-Dolcetta–Ishii proved that the rate of periodic homogenization for coercive Hamilton–Jacobi equations is O(ε1/3) . We complement this result by constructing examples of coercive nonconvex H...
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A microscopic derivation of Gibbs measures for the 1D focusing cubic nonlinear Schrödinger equation Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-08-08 Andrew Rout, Vedran Sohinger
Abstract In this paper, we give a microscopic derivation of Gibbs measures for the focusing cubic nonlinear Schrödinger equation on the one-dimensional torus from many-body quantum Gibbs states. Since we are not making any positivity assumptions on the interaction, it is necessary to introduce a truncation of the mass in the classical setting and of the rescaled particle number in the quantum setting
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Recovery of a spatially-dependent coefficient from the NLS scattering map Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-08-06 Jason Murphy
We follow up on work of Strauss, Weder, and Watanabe concerning scattering and inverse scattering for nonlinear Schrödinger equations with nonlinearities of the form α(x)|u|pu.
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Homogenization of some periodic Hamilton-Jacobi equations with defects Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-08-04 Yves Achdou, Claude Le Bris
We study homogenization for a class of stationary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the l...
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Correction Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-07-13
Published in Communications in Partial Differential Equations (Vol. 48, No. 6, 2023)
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Hyperbolic–parabolic normal form and local classical solutions for cross-diffusion systems with incomplete diffusion Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-06-20 Pierre-Étienne Druet, Katharina Hopf, Ansgar Jüngel
We investigate degenerate cross-diffusion equations, with a rank-deficient diffusion-matrix, modelling multispecies population dynamics driven by partial pressure gradients. These equations have re...
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Gradient estimates for the Schrödinger potentials: convergence to the Brenier map and quantitative stability Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-06-14 Alberto Chiarini, Giovanni Conforti, Giacomo Greco, Luca Tamanini
Abstract We show convergence of the gradients of the Schrödinger potentials to the (uniquely determined) gradient of Kantorovich potentials in the small-time limit under general assumptions on the marginals, which allow for unbounded densities and supports. Furthermore, we provide novel quantitative stability estimates for the optimal values and optimal couplings for the Schrödinger problem (SP), that
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Correction to: Remarks on local regularity of axisymmetric solutions to the 3D Navier–Stokes equations Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-05-31 Hui Chen, Tai-Peng Tsai, Ting Zhang
Published in Communications in Partial Differential Equations (Vol. 48, No. 6, 2023)
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Threshold solutions for the 3d cubic-quintic NLS Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-05-25 Alex H. Ardila, Jason Murphy
Abstract We study the cubic-quintic NLS in three space dimensions. It is known that scattering holds for solutions with mass-energy in a region corresponding to positive virial, the boundary of which is delineated both by ground state solitons and by certain rescalings thereof. We classify the possible behaviors of solutions on the part of the boundary attained solely by solitons. In particular, we
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Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-05-12 Biagio Cassano, Valentina Franceschi, David Krejčiřík, Dario Prandi
Abstract In this article, we introduce a notion of magnetic field in the Heisenberg group and we study its influence on spectral properties of the corresponding magnetic (sub-elliptic) Laplacian. We show that uniform magnetic fields uplift the bottom of the spectrum. For magnetic fields vanishing at infinity, including Aharonov–Bohm potentials, we derive magnetic improvements to a variety of Hardy-type
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Phenotypic heterogeneity in a model of tumour growth: existence of solutions and incompressible limit Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-05-09 Noemi David
Abstract We consider a (degenerate) cross-diffusion model of tumour growth structured by phenotypic trait. We prove the existence of weak solutions and the incompressible limit as the pressure becomes stiff extending methods recently introduced in the context of two-species cross-diffusion systems. In the stiff-pressure limit, the compressible model generates a free boundary problem of Hele-Shaw type
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On the motion of a small rigid body in a viscous compressible fluid Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-05-09 Eduard Feireisl, Arnab Roy, Arghir Zarnescu
Abstract We consider the motion of a small rigid object immersed in a viscous compressible fluid in the 3-dimensional Eucleidean space. Assuming the object is a ball of a small radius ε we show that the behavior of the fluid is not influenced by the object in the asymptotic limit ε→0.ε→0. The result holds for the isentropic pressure law p(ϱ)=aϱγp(ϱ)=aϱγ for any γ>32 under mild assumptions concerning
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On the vanishing discount approximation for compactly supported perturbations of periodic Hamiltonians: the 1d case Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-05-05 Italo Capuzzo Dolcetta, Andrea Davini
Abstract We study the asymptotic behavior of the viscosity solutions uGλ of the Hamilton-Jacobi (HJ) equation λu(x)+G(x,u′)=c(G) in R as the positive discount factor λ tends to 0, where G(x,p):=H(x,p)−V(x) is the perturbation of a Hamiltonian H∈ C(R×R), Z–periodic in the space variable and convex and coercive in the momentum, by a compactly supported potential V∈Cc(R). The constant c(G) appearing above
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Erratum to “optimal relaxation of bump-like solutions of the one-dimensional Cahn–Hilliard equation” Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-05-05 Sarah Biesenbach, Richard Schubert, Maria G. Westdickenberg
Abstract A dissipation estimate from the original article is corrected. The main result carries through unchanged, as explained below.
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Quantum entanglement and the growth of Laplacian eigenfunctions Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-04-20 Stefan Steinerberger
Abstract We study the growth of Laplacian eigenfunctions −Δϕk=λkϕk on compact manifolds (M, g). Hörmander proved sharp polynomial bounds on ||ϕk||L∞ which are attained on the sphere. On a “generic” manifold, the behavior seems to be different: both numerics and Berry’s random wave model suggest ||ϕk||L∞≲ log λk as the typical behavior. We propose a mechanism, centered around an L1− analog of the spectral
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Sticky particle Cucker–Smale dynamics and the entropic selection principle for the 1D Euler-alignment system Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-04-20 Trevor M. Leslie, Changhui Tan
Abstract We develop a global wellposedness theory for weak solutions to the 1D Euler-alignment system with measure-valued density, bounded velocity, and locally integrable communication protocol. A satisfactory understanding of the low-regularity theory is an issue of pressing interest, as smooth solutions may lose regularity in finite time. However, no such theory currently exists except for a very
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Cost of observability inequalities for elliptic equations in 2-d with potentials and applications to control theory Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-04-10 Sylvain Ervedoza, Kévin Le Balc’h
Abstract The goal of this article is to obtain observability estimates for non-homogeneous elliptic equations in the presence of a potential, posed on a smooth bounded domain Ω in R2 and observed from a non-empty open subset ω⊂Ω. More precisely, for V∈L∞(Ω;R), our main result shows that, when Ω⊂R2 has a finite number of holes, the observability constant of the elliptic operator −Δ+V, with domain H2(Ω)∩H01(Ω)
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Stable discontinuous stationary solutions to reaction-diffusion-ODE systems Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-04-10 Szymon Cygan, Anna Marciniak-Czochra, Grzegorz Karch, Kanako Suzuki
Abstract A general system of n ordinary differential equations coupled with one reaction-diffusion equation, considered in a bounded N-dimensional domain, with no-flux boundary condition is studied in a context of pattern formation. Such initial boundary value problems may have different types of stationary solutions. In our parallel work [Instability of all regular stationary solutions to reaction-diffusion-ODE
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On the growth of generalized Fourier coefficients of restricted eigenfunctions Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-03-31 Madelyne M. Brown
Abstract Let (M, g) be a smooth, compact, Riemannian manifold and {ϕh} a sequence of L2-normalized Laplace eigenfunctions on M. For a smooth submanifold H⊂M, we consider the growth of the restricted eigenfunctions ϕh|H by testing them against a sequence of functions {ψh} on H whose wavefront set avoids S*H. That is, we study what we call the generalized Fourier coefficients: 〈ϕh,ψh〉L2(H). We give an
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Analysis of the generalized Aw-Rascle model Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-03-28 Nilasis Chaudhuri, Piotr Gwiazda, Ewelina Zatorska
Abstract We consider the multi-dimensional generalization of the Aw-Rascle system for vehicular traffic. For arbitrary large initial data and the periodic boundary conditions, we prove the existence of global-in-time measure-valued solutions. We also show, using the relative energy technique, that the measure-valued solutions coincide with the classical solutions as long as the latter exist.
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Propagation of velocity moments and uniqueness for the magnetized Vlasov–Poisson system Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-03-23 Alexandre Rege
Abstract We present two results regarding the three-dimensional Vlasov–Poisson system in the full space with an external magnetic field. First, we investigate the propagation of velocity moments for solutions to the system when the magnetic field is uniform and time-dependent. We combine the classical moment approach with an induction procedure depending on the cyclotron period Tc=‖B‖∞−1. This allows
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A transmission problem with (p, q)-Laplacian Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-03-23 Maria Colombo, Sunghan Kim, Henrik Shahgholian
Abstract In this paper we consider the so-called double-phase problem where the phase transition takes place across the interface of the positive and negative phase of minimizers of the functional J(v,Ω)=∫Ω(|Dv+|p+|Dv−|q)dx. We prove that minimizers exist, are Hölder regular and verify (q−1)|Du−|q=(p−1)|Du+|p on ∂{u>0}, in a weak sense. We also prove that their free boundary is C1,α a.e. with respect
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Asymptotic estimates for the wave functions of the Dirac-Coulomb operator and applications Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-03-22 Federico Cacciafesta, Éric Séré, Junyong Zhang
Abstract In this paper we prove some uniform asymptotic estimates for confluent hypergeometric functions making use of the steepest-descent method. As an application, we obtain Strichartz estimates that are L2-averaged over angular direction for the massless Dirac-Coulomb equation in 3D.
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The Yang-Mills heat flow with random distributional initial data Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-03-10 Sky Cao, Sourav Chatterjee
Abstract We construct local solutions to the Yang–Mills heat flow (in the DeTurck gauge) for a certain class of random distributional initial data, which includes the 3D Gaussian free field. The main idea, which goes back to work of Bourgain as well as work of Da Prato–Debussche, is to decompose the solution into a rougher linear part and a smoother nonlinear part, and to control the latter by probabilistic
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Quantitative unique continuation for the elasticity system with application to the kinematic inverse rupture problem Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-03-08 Maarten V. de Hoop, Matti Lassas, Jinpeng Lu, Lauri Oksanen
Abstract We obtain explicit estimates on the stability of the unique continuation for a linear system of hyperbolic equations. In particular, our result applies to the elasticity system and also the Maxwell system. As an application, we study the kinematic inverse rupture problem of determining the jump in displacement and the friction force at the rupture surface, and we obtain new features on the
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Lotka–Volterra competition-diffusion system: the critical competition case Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-03-08 Matthieu Alfaro, Dongyuan Xiao
Abstract We consider the reaction-diffusion competition system in the so-called critical competition case. The associated ODE system then admits infinitely many equilibria, which makes the analysis intricate. We first prove the nonexistence of ultimately monotone traveling waves by applying the phase plane analysis. Next, we study the large time behavior of the solution of the Cauchy problem with a
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Scaling asymptotics of Wigner distributions of harmonic oscillator orbital coherent states Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-02-22 Nicholas Lohr
Abstract The main result of this article gives scaling asymptotics of the Wigner distributions WφNγ,φNγ of isotropic harmonic oscillator orbital coherent states φNγ concentrating along Hamiltonian orbits γ in shrinking tubes around γ in phase space. In particular, these Wigner distributions exhibit a hybrid semi-classical scaling. That is, simultaneously, we have an Airy scaling when the tube has radius
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Erratum: the Poisson equation involving surface measures Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2023-01-11 Marius Müller
Abstract This erratum points out an error in “The Poisson equation involving surface measures” (Vol. 47 of Communications in Partial Differential Equations, (2022)) and provides a counterexample and discussion of the erroneous theorem.
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The Hele–Shaw flow as the sharp interface limit of the Cahn–Hilliard equation with disparate mobilities Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-12-28 Milan Kroemer, Tim Laux
Abstract In this paper, we study the sharp interface limit for solutions of the Cahn–Hilliard equation with disparate mobilities. This means that the mobility function degenerates in one of the two energetically favorable configurations, suppressing the diffusion in that phase. First, we construct suitable weak solutions to this Cahn–Hilliard equation. Second, we prove precompactness of these solutions
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Zvonkin’s transform and the regularity of solutions to double divergence form elliptic equations Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-11-12 Vladimir I. Bogachev, Michael Röckner, Stanislav V. Shaposhnikov
Abstract We study qualitative properties of solutions to double divergence form elliptic equations (or stationary Kolmogorov equations) on Rd. It is shown that the Harnack inequality holds for nonnegative solutions if the diffusion matrix A is nondegenerate and satisfies the Dini mean oscillation condition and the drift coefficient b is locally integrable to some power p > d. We establish new estimates
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On the blow-up analysis at collapsing poles for solutions of singular Liouville-type equations Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-11-11 Gabriella Tarantello
Abstract We analyze a blow-up sequence of solutions for Liouville-type equations involving Dirac measures with “collapsing” poles. We consider the case where blow-up occurs exactly at a point where the poles coalesce. After proving that a” quantization” property still holds for the” blow-up mass,” we obtain precise pointwise estimates when blow-up occurs with the least blow-up mass. Interestingly,
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A Hamilton-Jacobi approach to evolution of dispersal Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-11-10 King-Yeung Lam, Yuan Lou, Benoît Perthame
Abstract The evolution of dispersal is a classical question in evolutionary biology, and it has been studied in a wide range of mathematical models. A selection-mutation model, in which the population is structured by space and a phenotypic trait, with the trait acting directly on the dispersal (diffusion) rate, was formulated by Perthame and Souganidis [Math. Model. Nat. Phenom. 11:154–166, 2016]
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Spectral asymptotics for the vectorial damped wave equation Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-11-09 Guillaume Klein
Abstract The eigenfrequencies associated to a scalar damped wave equation are known to belong to a band parallel to the real axis. Sjöstrand showed that up to a set of density 0, the eigenfrequencies are confined in a thinner band determined by the Birkhoff limits of the damping term. In this article we show that this result is still true for a vectorial damped wave equation. In this setting the Lyapunov
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Filamentation near Hill’s vortex Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-11-09 Kyudong Choi, In-Jee Jeong
Abstract For the axi-symmetric incompressible Euler equations, we prove linear in time filamentation near Hill’s vortex: there exists an arbitrary small outward perturbation growing linearly for all times. This is based on combining the recent nonlinear orbital stability obtained by the first author with a dynamical bootstrapping scheme for particle trajectories. These results rigorously confirm numerical
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Construction of high regularity invariant measures for the 2D Euler equations and remarks on the growth of the solutions Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-10-29 Mickaël Latocca
Abstract We consider the Euler equations on the two-dimensional torus and construct invariant measures for the dynamics of these equations, concentrated on sufficiently regular Sobolev spaces so that strong solutions are also known to exist. The proof follows the method of Kuksin (J. Stat. Phys. 115(1/2):469–492) and we obtain in particular that these measures do not have atoms, excluding trivial invariant
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Toward uniform existence and convergence theorems for three-scale systems of hyperbolic PDEs with general initial data Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-10-25 Steve Schochet, Xin Xu
Abstract Uniform existence of solutions to initial-value problems and convergence of appropriately filtered solutions are proven for a special class of three-scale singular limit equations, without any restriction on the initial data. The uniform existence is proven using a novel system of energy estimates. The convergence result is based on a detailed analysis of the fastest-scale oscillations, which
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Taxis-driven persistent localization in a degenerate Keller-Segel system Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-10-18 Angela Stevens, Michael Winkler
Abstract The degenerate Keller-Segel type system {∂tu=∇·(um−1∇u)−∇·(u∇v),x∈Ω, t>0,0=Δv−μ+u, ∫Ωv=0, μ=1|Ω|∫Ωu, x∈Ω, t>0, is considered in balls Ω=BR(0)⊂ℝn with n≥1, R > 0 and m > 1. Our main results reveal that throughout the entire degeneracy range m∈(1,∞), the interplay between degenerate diffusion and cross-diffusive attraction herein can enforce persistent localization of solutions inside a compact
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The Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-09-19 Peter Hintz, Gunther Uhlmann, Jian Zhai
Abstract We consider the semilinear wave equation □gu+au4=0, a≠0, on a Lorentzian manifold (M, g) with timelike boundary. We show that from the knowledge of the Dirichlet-to-Neumann map one can recover the metric g and the coefficient a up to natural obstructions. Our proof rests on the analysis of the interaction of distorted plane waves together with a scattering control argument, as well as Gaussian
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Quantitative mixing and dissipation enhancement property of Ornstein–Uhlenbeck flow Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-09-15 Umberto Pappalettera
Abstract This work deals with mixing and dissipation enhancement for the solution of advection-diffusion equation driven by a Ornstein–Uhlenbeck velocity field. We are able to prove a quantitative mixing result, uniform in the diffusion parameter, and enhancement of dissipation over a finite time horizon.
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A superposition principle for the inhomogeneous continuity equation with Hellinger–Kantorovich-regular coefficients Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-09-15 Kristian Bredies, Marcello Carioni, Silvio Fanzon
Abstract We study measure-valued solutions of the inhomogeneous continuity equation ∂tρt+div (vρt)=gρt where the coefficients v and g are of low regularity. A new superposition principle is proven for positive measure solutions and coefficients for which the recently-introduced dynamic Hellinger–Kantorovich energy is finite. This principle gives a decomposition of the solution into curves t↦h(t)δγ(t)
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On a generalized Aviles-Giga functional: compactness, zero-energy states, regularity estimates and energy bounds Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-09-10 Xavier Lamy, Andrew Lorent, Guanying Peng
Abstract Given any strictly convex norm ‖·‖ on R2 that is C1 in R2∖{0}, we study the generalized Aviles-Giga functional Iϵ(m):=∫Ω(ϵ|∇m|2+1ϵ(1−‖m‖2)2) dx, for Ω⊂R2 and m:Ω→R2 satisfying ∇·m=0. Using, as in the euclidean case ‖·‖=|·|, the concept of entropies for the limit equation ‖m‖=1, ∇·m=0, we obtain the following. First, we prove compactness in Lp of sequences of bounded energy. Second, we prove
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Analysis and mean-field derivation of a porous-medium equation with fractional diffusion Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-09-09 Li Chen, Alexandra Holzinger, Ansgar Jüngel, Nicola Zamponi
Abstract A mean-field-type limit from stochastic moderately interacting many-particle systems with singular Riesz potential is performed, leading to nonlocal porous-medium equations in the whole space. The nonlocality is given by the inverse of a fractional Laplacian, and the limit equation can be interpreted as a transport equation with a fractional pressure. The proof is based on Oelschläger’s approach
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Green’s function of heat equation for heterogeneous media in 3-D Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-09-08 Ching-hsiao Arthur Cheng, Tai-Ping Liu, Shih-Hsien Yu
Abstract The purpose of the present paper is to study the structure of Green’s function for heat equation in several spatial dimensions and with rough heat conductivity coefficient. We take the heat conductivity coefficient to be of bounded variation in the x direction and study the dispersion in the (y, z) direction. The goal is to understand the coupling of dissipation across rough heat conductivity
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Segregated solutions for nonlinear Schrödinger systems with weak interspecies forces Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-09-01 Angela Pistoia, Giusi Vaira
Abstract We find positive non-radial solutions for a system of Schrödinger equations in a weak fully attractive or repulsive regime in presence of an external radial trapping potential that exhibits a maximum or a minimum at infinity.
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Boundary renormalisation of SPDEs Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-08-27 Máté Gerencsér, Martin Hairer
Abstract We consider the continuum parabolic Anderson model (PAM) and the dynamical Φ4 equation on the 3-dimensional cube with boundary conditions. While the Dirichlet solution theories are relatively standard, the case of Neumann/Robin boundary conditions gives rise to a divergent boundary renormalisation. Furthermore for Φ34 a ‘boundary triviality’ result is obtained: if one approximates the equation
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The streamlines of ∞-harmonic functions obey the inverse mean curvature flow Commun. Partial Differ. Equ. (IF 1.9) Pub Date : 2022-08-23 Roger Moser
Abstract Given an ∞-harmonic function u∞ on a domain Ω⊆R2, consider the function w=− log |∇u∞|. If u∞∈C2(Ω) with ∇u∞≠0 and ∇|∇u∞|≠0, then it is easy to check that the streamlines of u∞ are the level sets of w and w solves the level set formulation of the inverse mean curvature flow. For less regular solutions, neither statement is true in general, but even so, w is still a weak solution of the inverse