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Schauder estimates for equations with cone metrics, II Anal. PDE (IF 2.2) Pub Date : 2024-04-24 Bin Guo, Jian Song
We continue our work on the linear theory for equations with conical singularities. We derive interior Schauder estimates for linear elliptic and parabolic equations with a background Kähler metric of conical singularities along a divisor of simple normal crossings. As an application, we prove the short-time existence of the conical Kähler–Ricci flow with conical singularities along a divisor with
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The stability of simple plane-symmetric shock formation for three-dimensional compressible Euler flow with vorticity and entropy Anal. PDE (IF 2.2) Pub Date : 2024-04-24 Jonathan Luk, Jared Speck
Consider a one-dimensional simple small-amplitude solution (ϱ(bkg ),v(bkg )1) to the isentropic compressible Euler equations which has smooth initial data, coincides with a constant state outside a compact set, and forms a shock in finite time. Viewing (ϱ(bkg ),v(bkg )1) as a plane-symmetric solution to the full compressible Euler equations in three dimensions, we prove that the shock-formation
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Families of functionals representing Sobolev norms Anal. PDE (IF 2.2) Pub Date : 2024-04-24 Haïm Brezis, Andreas Seeger, Jean Van Schaftingen, Po-Lam Yung
We obtain new characterizations of the Sobolev spaces Ẇ1,p(ℝN) and the bounded variation space BV ˙(ℝN). The characterizations are in terms of the functionals νγ(Eλ,γ∕p[u]), where Eλ,γ∕p[u] ={(x,y) ∈ ℝN × ℝN : x≠y, |u(x) − u(y)| |x − y|1+γ∕p > λ} and the measure νγ is given by d νγ(x,y) = |x − y|γ−N d xd y. We provide characterizations which involve the Lp,∞-quasinorms sup λ>0 λνγ(Eλ,γ∕p[u])1∕p
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Schwarz–Pick lemma for harmonic maps which are conformal at a point Anal. PDE (IF 2.2) Pub Date : 2024-04-24 Franc Forstnerič, David Kalaj
We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc 𝔻 in ℂ into the unit ball 𝔹n of ℝn, n ≥ 2, at any point where the map is conformal. For n = 2 this generalizes the classical Schwarz–Pick lemma, and for n ≥ 3 it gives the optimal Schwarz–Pick lemma for conformal minimal discs 𝔻 → 𝔹n. This implies that conformal harmonic maps M → 𝔹n from any hyperbolic
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An improved regularity criterion and absence of splash-like singularities for g-SQG patches Anal. PDE (IF 2.2) Pub Date : 2024-04-24 Junekey Jeon, Andrej Zlatoš
We prove that splash-like singularities cannot occur for sufficiently regular patch solutions to the generalized surface quasi-geostrophic equation on the plane or half-plane with parameter α ≤ 1 4. This includes potential touches of more than two patch boundary segments in the same location, an eventuality that has not been excluded previously and presents nontrivial complications (in fact, if we
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Spectral gap for obstacle scattering in dimension 2 Anal. PDE (IF 2.2) Pub Date : 2024-04-24 Lucas Vacossin
We study the problem of scattering by several strictly convex obstacles, with smooth boundary and satisfying a noneclipse condition. We show, in dimension 2 only, the existence of a spectral gap for the meromorphic continuation of the Laplace operator outside the obstacles. The proof of this result relies on a reduction to an open hyperbolic quantum map, achieved by Nonnenmacher et al. (Ann. of Math
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On a spatially inhomogeneous nonlinear Fokker–Planck equation : Cauchy problem and diffusion asymptotics Anal. PDE (IF 2.2) Pub Date : 2024-03-06 Francesca Anceschi, Yuzhe Zhu
We investigate the Cauchy problem and the diffusion asymptotics for a spatially inhomogeneous kinetic model associated to a nonlinear Fokker–Planck operator. We derive the global well-posedness result with instantaneous smoothness effect, when the initial data lies below a Maxwellian. The proof relies on the hypoelliptic analog of classical parabolic theory, as well as a positivity-spreading result
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Strichartz inequalities with white noise potential on compact surfaces Anal. PDE (IF 2.2) Pub Date : 2024-03-06 Antoine Mouzard, Immanuel Zachhuber
We prove Strichartz inequalities for the Schrödinger equation and the wave equation with multiplicative noise on a two-dimensional manifold. This relies on the Anderson Hamiltonian described using high-order paracontrolled calculus. As an application, it gives a low-regularity solution theory for the associated nonlinear equations.
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Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential Anal. PDE (IF 2.2) Pub Date : 2024-03-06 Sylvester Eriksson-Bique, Elefterios Soultanis
We represent minimal upper gradients of Newtonian functions, in the range 1 ≤ p < ∞, by maximal directional derivatives along “generic” curves passing through a given point, using plan-modulus duality and disintegration techniques. As an application we introduce the notion of p-weak charts and prove that every Newtonian function admits a differential with respect to such charts, yielding a linear approximation
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Smooth extensions for inertial manifolds of semilinear parabolic equations Anal. PDE (IF 2.2) Pub Date : 2024-03-06 Anna Kostianko, Sergey Zelik
The paper is devoted to a comprehensive study of smoothness of inertial manifolds (IMs) for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than C1,𝜀-regularity for such manifolds (for some positive, but small 𝜀). Nevertheless, as shown in the paper, under natural assumptions, the obstacles to the existence of a Cn-smooth inertial manifold (where n ∈
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Semiclassical eigenvalue estimates under magnetic steps Anal. PDE (IF 2.2) Pub Date : 2024-03-06 Wafaa Assaad, Bernard Helffer, Ayman Kachmar
We establish accurate eigenvalue asymptotics and, as a by-product, sharp estimates of the splitting between two consecutive eigenvalues for the Dirichlet magnetic Laplacian with a nonuniform magnetic field having a jump discontinuity along a smooth curve. The asymptotics hold in the semiclassical limit, which also corresponds to a large magnetic field limit and is valid under a geometric assumption
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Necessary density conditions for sampling and interpolation in spectral subspaces of elliptic differential operators Anal. PDE (IF 2.2) Pub Date : 2024-03-06 Karlheinz Gröchenig, Andreas Klotz
We prove necessary density conditions for sampling in spectral subspaces of a second-order uniformly elliptic differential operator on ℝd with slowly oscillating symbol. For constant-coefficient operators, these are precisely Landau’s necessary density conditions for bandlimited functions, but for more general elliptic differential operators it has been unknown whether such a critical density even
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On blowup for the supercritical quadratic wave equation Anal. PDE (IF 2.2) Pub Date : 2024-03-06 Elek Csobo, Irfan Glogić, Birgit Schörkhuber
We study singularity formation for the quadratic wave equation in the energy supercritical case, i.e., for d ≥ 7. We find in closed form a new, nontrivial, radial, self-similar blow-up solution u∗ which exists for all d ≥ 7. For d = 9, we study the stability of u∗ without any symmetry assumptions on the initial data and show that there is a family of perturbations which lead to blowup via u∗ . In similarity
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Arnold’s variational principle and its application to the stability of planar vortices Anal. PDE (IF 2.2) Pub Date : 2024-03-06 Thierry Gallay, Vladimír Šverák
We consider variational principles related to V. I. Arnold’s stability criteria for steady-state solutions of the two-dimensional incompressible Euler equation. Our goal is to investigate under which conditions the quadratic forms defined by the second variation of the associated functionals can be used in the stability analysis, both for the Euler evolution and for the Navier–Stokes equation at low
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Explicit formula of radiation fields of free waves with applications on channel of energy Anal. PDE (IF 2.2) Pub Date : 2024-03-06 Liang Li, Ruipeng Shen, Lijuan Wei
We give a few explicit formulas regarding the radiation fields of linear free waves. We then apply these formulas on the channel-of-energy theory. We characterize all the radial weakly nonradiative solutions in all dimensions and give a few new exterior energy estimates.
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On L∞ estimates for Monge–Ampère and Hessian equations on nef classes Anal. PDE (IF 2.2) Pub Date : 2024-03-06 Bin Guo, Duong H. Phong, Freid Tong, Chuwen Wang
The PDE approach developed earlier by the first three authors for L∞ estimates for fully nonlinear equations on Kähler manifolds is shown to apply as well to Monge–Ampère and Hessian equations on nef classes. In particular, one obtains a new proof of the estimates of Boucksom, Eyssidieux, Guedj and Zeriahi (2010) and Fu, Guo and Song (2020) for the Monge–Ampère equation, together with their generalization
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The prescribed curvature problem for entire hypersurfaces in Minkowski space Anal. PDE (IF 2.2) Pub Date : 2024-02-05 Changyu Ren, Zhizhang Wang, Ling Xiao
We prove three results in this paper: First, we prove, for a wide class of functions φ ∈ C2(𝕊n−1) and ψ(X,ν) ∈ C2(ℝn+1× ℍn), there exists a unique, entire, strictly convex, spacelike hypersurface ℳu satisfying σk(κ[ℳu]) = ψ(X,ν) and u(x) →|x| + φ(x∕|x|) as |x|→∞. Second, when k = n−1,n−2, we show the existence and uniqueness of an entire, k-convex, spacelike hypersurface ℳu satisfying σk(κ[ℳu]) =
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Anisotropic micropolar fluids subject to a uniform microtorque: the stable case Anal. PDE (IF 2.2) Pub Date : 2024-02-05 Antoine Remond-Tiedrez, Ian Tice
We study a three-dimensional, incompressible, viscous, micropolar fluid with anisotropic microstructure on a periodic domain. Subject to a uniform microtorque, this system admits a unique nontrivial equilibrium. We prove that when the microstructure is inertially oblate (i.e., pancake-like) this equilibrium is nonlinearly asymptotically stable. Our proof employs a nonlinear energy method built from
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Strong ill-posedness for SQG in critical Sobolev spaces Anal. PDE (IF 2.2) Pub Date : 2024-02-05 In-Jee Jeong, Junha Kim
We prove that the inviscid surface quasigeostrophic (SQG) equations are strongly ill-posed in critical Sobolev spaces: there exists an initial data H2(𝕋2) without any solutions in Lt∞H2 . Moreover, we prove strong critical norm inflation for C∞-smooth data. Our proof is robust and extends to give similar ill-posedness results for the family of modified SQG equations which interpolate the SQG with
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Large-scale regularity for the stationary Navier–Stokes equations over non-Lipschitz boundaries Anal. PDE (IF 2.2) Pub Date : 2024-02-05 Mitsuo Higaki, Christophe Prange, Jinping Zhuge
We address the large-scale regularity theory for the stationary Navier–Stokes equations in highly oscillating bumpy John domains. These domains are very rough, possibly with fractals or cusps, at the microscopic scale, but are amenable to the mathematical analysis of the Navier–Stokes equations. We prove a large-scale Calderón–Zygmund estimate, a large-scale Lipschitz estimate, and large-scale higher-order
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On a family of fully nonlinear integrodifferential operators : from fractional Laplacian to nonlocal Monge–Ampère Anal. PDE (IF 2.2) Pub Date : 2024-02-05 Luis A. Caffarelli, María Soria-Carro
We introduce a new family of intermediate operators between the fractional Laplacian and the nonlocal Monge–Ampère introduced by Caffarelli and Silvestre that are given by infimums of integrodifferential operators. Using rearrangement techniques, we obtain representation formulas and give a connection to optimal transport. Finally, we consider a global Poisson problem prescribing data at infinity,
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Propagation of singularities for gravity-capillary water waves Anal. PDE (IF 2.2) Pub Date : 2024-02-05 Hui Zhu
We obtain two results of propagation for the gravity-capillary water wave system. The first result shows the propagation of oscillations and the spatial decay at infinity; the second result shows a microlocal smoothing effect under the nontrapping condition of the initial free surface. These results extend the works of Craig, Kappeler and Strauss (1995), Wunsch (1999) and Nakamura (2005) to quasilinear
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Shift equivalences through the lens of Cuntz–Krieger algebras Anal. PDE (IF 2.2) Pub Date : 2024-02-05 Toke Meier Carlsen, Adam Dor-On, Søren Eilers
Motivated by Williams’ problem of measuring novel differences between shift equivalence (SE) and strong shift equivalence (SSE), we introduce three equivalence relations that provide new ways to obstruct SSE while merely assuming SE. Our shift equivalence relations arise from studying graph C*-algebras, where a variety of intermediary equivalence relations naturally arise. As a consequence we realize
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Higher rank quantum-classical correspondence Anal. PDE (IF 2.2) Pub Date : 2023-12-11 Joachim Hilgert, Tobias Weich, Lasse L. Wolf
For a compact Riemannian locally symmetric space Γ∖G∕K of arbitrary rank we determine the location of certain Ruelle–Taylor resonances for the Weyl chamber action. We provide a Weyl-lower bound on an appropriate counting function for the Ruelle–Taylor resonances and establish a spectral gap which is uniform in Γ if G∕K is irreducible of higher rank. This is achieved by proving a quantum-classical correspondence
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Growth of high Lp norms for eigenfunctions : an application of geodesic beams Anal. PDE (IF 2.2) Pub Date : 2023-12-11 Yaiza Canzani, Jeffrey Galkowski
This work concerns Lp norms of high energy Laplace eigenfunctions: (−Δg − λ2)ϕλ = 0, ∥ϕλ∥L2 = 1. Sogge (1988) gave optimal estimates on the growth of ∥ϕλ∥Lp for a general compact Riemannian manifold. Here we give general dynamical conditions guaranteeing quantitative improvements in Lp estimates for p > pc, where pc is the critical exponent. We also apply results of an earlier paper (Canzani and Galkowski
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Perturbed interpolation formulae and applications Anal. PDE (IF 2.2) Pub Date : 2023-12-11 João P. G. Ramos, Mateus Sousa
We employ functional analysis techniques in order to deduce some versions of classical and recent interpolation results in Fourier analysis with perturbed nodes. As an application of our techniques, we obtain generalizations of Kadec’s 1 4-theorem for interpolation formulae in the Paley–Wiener space both in the real and complex cases, as well as versions of the recent interpolation result of Radchenko
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Nonexistence of the box dimension for dynamically invariant sets Anal. PDE (IF 2.2) Pub Date : 2023-12-11 Natalia Jurga
One of the key challenges in the dimension theory of smooth dynamical systems lies in establishing whether or not the Hausdorff, lower and upper box dimensions coincide for invariant sets. For sets invariant under conformal dynamics, these three dimensions always coincide. On the other hand, considerable attention has been given to examples of sets invariant under nonconformal dynamics whose Hausdorff
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Decoupling inequalities for short generalized Dirichlet sequences Anal. PDE (IF 2.2) Pub Date : 2023-12-11 Yuqiu Fu, Larry Guth, Dominique Maldague
We study decoupling theory for functions on ℝ with Fourier transform supported in a neighborhood of short Dirichlet sequences {log n}n=N+1N+N1∕2 , as well as sequences with similar convexity properties. We utilize the wave packet structure of functions with frequency support near an arithmetic progression.
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Global well-posedness of Vlasov–Poisson-type systems in bounded domains Anal. PDE (IF 2.2) Pub Date : 2023-12-11 Ludovic Cesbron, Mikaela Iacobelli
In this paper we prove global existence of classical solutions to the Vlasov–Poisson and ionic Vlasov–Poisson models in bounded domains. On the boundary, we consider the specular reflection boundary condition for the Vlasov equation and either homogeneous Dirichlet or Neumann conditions for the Poisson equations.
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Overdetermined boundary problems with nonconstant Dirichlet and Neumann data Anal. PDE (IF 2.2) Pub Date : 2023-11-11 Miguel Domínguez-Vázquez, Alberto Enciso, Daniel Peralta-Salas
We consider the overdetermined boundary problem for a general second-order semilinear elliptic equation on bounded domains of ℝn , where one prescribes both the Dirichlet and Neumann data of the solution. We are interested in the case where the data are not necessarily constant and where the coefficients of the equation can depend on the position, so that the overdetermined problem does not generally
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Monge–Ampère gravitation as a Γ-limit of good rate functions Anal. PDE (IF 2.2) Pub Date : 2023-11-11 Luigi Ambrosio, Aymeric Baradat, Yann Brenier
Monge–Ampère gravitation is a modification of the classical Newtonian gravitation where the linear Poisson equation is replaced by the nonlinear Monge–Ampère equation. This paper is concerned with the rigorous derivation of Monge–Ampère gravitation for a finite number of particles from the stochastic model of a Brownian point cloud, following the formal ideas of a recent work by Brenier (Bull. Inst
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IDA and Hankel operators on Fock spaces Anal. PDE (IF 2.2) Pub Date : 2023-11-11 Zhangjian Hu, Jani A. Virtanen
We introduce a new space IDA of locally integrable functions whose integral distance to holomorphic functions is finite, and use it to completely characterize boundedness and compactness of Hankel operators on weighted Fock spaces. As an application, for bounded symbols, we show that the Hankel operator Hf is compact if and only if Hf¯ is compact, which complements the classical compactness result
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Global stability of spacetimes with supersymmetric compactifications Anal. PDE (IF 2.2) Pub Date : 2023-11-11 Lars Andersson, Pieter Blue, Zoe Wyatt, Shing-Tung Yau
This paper proves the stability, with respect to the evolution determined by the vacuum Einstein equations, of the Cartesian product of higher-dimensional Minkowski space with a compact, Ricci-flat Riemannian manifold that admits a spin structure and a nonzero parallel spinor. Such a product includes the example of Calabi–Yau and other special holonomy compactifications, which play a central role in
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Stability of traveling waves for the Burgers–Hilbert equation Anal. PDE (IF 2.2) Pub Date : 2023-11-11 Ángel Castro, Diego Córdoba, Fan Zheng
We consider smooth solutions of the Burgers–Hilbert equation that are a small perturbation δ from a global periodic traveling wave with small amplitude 𝜖. We use a modified energy method to prove the existence time of smooth solutions on a time scale of 1∕(𝜖δ), with 0 < δ ≪ 𝜖 ≪ 1, and on a time scale of 𝜖∕δ2, with 0 < δ ≪ 𝜖2 ≪ 1. Moreover, we show that the traveling wave exists for an amplitude
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Defining the spectral position of a Neumann domain Anal. PDE (IF 2.2) Pub Date : 2023-11-11 Ram Band, Graham Cox, Sebastian K. Egger
A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition into Neumann domains, a.k.a. a Morse–Smale complex. This partition is generated by gradient flow lines of the eigenfunction, which bound the so-called Neumann domains. We prove that the Neumann Laplacian defined on a Neumann domain is self-adjoint and has a purely discrete spectrum. In addition, we prove
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A uniqueness result for the two-vortex traveling wave in the nonlinear Schrödinger equation Anal. PDE (IF 2.2) Pub Date : 2023-11-11 David Chiron, Eliot Pacherie
For the nonlinear Schrödinger equation in dimension 2, the existence of a global minimizer of the energy at fixed momentum has been established by Bethuel, Gravejat and Saut (2009) (see also work of Chiron and Mariş (2017)). This minimizer is a traveling wave for the nonlinear Schrödinger equation. For large momenta, the propagation speed is small and the minimizer behaves like two well-separated vortices
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Classification of convex ancient free-boundary curve-shortening flows in the disc Anal. PDE (IF 2.2) Pub Date : 2023-11-11 Theodora Bourni, Mat Langford
Using a combination of direct geometric methods and an analysis of the linearization of the flow about the horizontal bisector, we prove that there exists a unique (modulo rotations about the origin) convex ancient curve-shortening flow in the disc with free boundary on the circle. This appears to be the first result of its kind in the free-boundary setting.
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Ground state properties in the quasiclassical regime Anal. PDE (IF 2.2) Pub Date : 2023-10-16 Michele Correggi, Marco Falconi, Marco Olivieri
We study the ground state energy and ground states of systems coupling nonrelativistic quantum particles and force-carrying Bose fields, such as radiation, in the quasiclassical approximation. The latter is very useful whenever the force-carrying field has a very large number of excitations and thus behaves in a semiclassical way, while the nonrelativistic particles, on the other hand, retain their
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A characterization of the Razak–Jacelon algebra Anal. PDE (IF 2.2) Pub Date : 2023-10-16 Norio Nawata
Combining Elliott, Gong, Lin and Niu’s result and Castillejos and Evington’s result, we see that if A is a simple separable nuclear monotracial C ∗-algebra, then A ⊗𝒲 is isomorphic to 𝒲, where 𝒲 is the Razak–Jacelon algebra. In this paper, we give another proof of this. In particular, we show that if 𝒟 is a simple separable nuclear monotracial M2∞-stable C ∗-algebra which is KK-equivalent to
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Inverse problems for nonlinear magnetic Schrödinger equations on conformally transversally anisotropic manifolds Anal. PDE (IF 2.2) Pub Date : 2023-10-16 Katya Krupchyk, Gunther Uhlmann
We study the inverse boundary problem for a nonlinear magnetic Schrödinger operator on a conformally transversally anisotropic Riemannian manifold of dimension n ≥ 3. Under suitable assumptions on the nonlinearity, we show that the knowledge of the Dirichlet-to-Neumann map on the boundary of the manifold determines the nonlinear magnetic and electric potentials uniquely. No assumptions on the transversal
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Discrete velocity Boltzmann equations in the plane: stationary solutions Anal. PDE (IF 2.2) Pub Date : 2023-10-16 Leif Arkeryd, Anne Nouri
We prove the existence of stationary mild solutions for normal discrete velocity Boltzmann equations in the plane with no pair of colinear interacting velocities and given ingoing boundary values. We remove an important restriction from a previous paper that all velocities point into the same half-space. A key property is L1 compactness of integrated collision frequency for a sequence of approximations
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Bosons in a double well: two-mode approximation and fluctuations Anal. PDE (IF 2.2) Pub Date : 2023-10-16 Alessandro Olgiati, Nicolas Rougerie, Dominique Spehner
We study the ground state for many interacting bosons in a double-well potential, in a joint limit where the particle number and the distance between the potential wells both go to infinity. Two single-particle orbitals (one for each well) are macroscopically occupied, and we are concerned with deriving the corresponding effective Bose–Hubbard Hamiltonian. We prove an energy expansion, including the
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A general notion of uniform ellipticity and the regularity of the stress field for elliptic equations in divergence form Anal. PDE (IF 2.2) Pub Date : 2023-10-16 Umberto Guarnotta, Sunra Mosconi
For solutions of Div (DF(Du)) = f we show that the quasiconformality of z↦DF(z) is the key property leading to the Sobolev regularity of the stress field DF(Du), in relation with the summability of f. This class of nonlinearities encodes in a general way the notion of uniform ellipticity and encompasses all known instances where the stress field is known to be Sobolev regular. We provide examples
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Simplices in thin subsets of Euclidean spaces Anal. PDE (IF 2.2) Pub Date : 2023-09-21 Alex Iosevich, Ákos Magyar
Let Δ be a nondegenerate simplex on k vertices. We prove that there exists a threshold sk < k such that any set A ⊆ ℝk of Hausdorff dimension dim A ≥ sk necessarily contains a similar copy of the simplex Δ.
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Resonances for Schrödinger operators on infinite cylinders and other products Anal. PDE (IF 2.2) Pub Date : 2023-09-21 T. J. Christiansen
We study the resonances of Schrödinger operators on the infinite product X = ℝd × 𝕊1, where d is odd, 𝕊1 is the unit circle, and the potential V lies in Lc∞(X). This paper shows that at high energy, resonances of the Schrödinger operator −Δ + V on X = ℝd × 𝕊1 which are near the continuous spectrum are approximated by the resonances of −Δ + V 0 on X, where the potential V 0 is given by averaging V
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A structure theorem for elliptic and parabolic operators with applications to homogenization of operators of Kolmogorov type Anal. PDE (IF 2.2) Pub Date : 2023-09-21 Malte Litsgård, Kaj Nyström
We consider the operators ∇ X ⋅ (A(X)∇ X),∇ X ⋅ (A(X)∇ X) − ∂t,∇ X ⋅ (A(X)∇ X) + X ⋅∇ Y − ∂t, where X ∈Ω, (X,t) ∈Ω × ℝ and (X,Y,t) ∈Ω × ℝm × ℝ, respectively, and where Ω ⊂ ℝm is an (unbounded) Lipschitz domain with defining function ψ : ℝm−1 → ℝ being Lipschitz with constant bounded by M. Assume that the elliptic measure associated to the first of these operators is mutually absolutely continuous
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Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flows Anal. PDE (IF 2.2) Pub Date : 2023-09-21 Li-Juan Cheng, Anton Thalmaier
Let M be a differentiable manifold endowed with a family of complete Riemannian metrics g(t) evolving under a geometric flow over the time interval [0,T[. We give a probabilistic representation for the derivative of the corresponding conjugate semigroup on M which is generated by a Schrödinger-type operator. With the help of this derivative formula, we derive fundamental Harnack-type inequalities in
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The regularity of the boundary of vortex patches for some nonlinear transport equations Anal. PDE (IF 2.2) Pub Date : 2023-09-21 Juan Carlos Cantero, Joan Mateu, Joan Orobitg, Joan Verdera
We prove the persistence of boundary smoothness of vortex patches for a nonlinear transport equation in ℝn with velocity field given by convolution of the density with an odd kernel, homogeneous of degree − (n − 1) and of class C2(ℝn ∖{0}, ℝn). This allows the velocity field to have nontrivial divergence. The quasigeostrophic equation in ℝ3 and the Cauchy transport equation in the plane are examples
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Directional square functions Anal. PDE (IF 2.2) Pub Date : 2023-09-21 Natalia Accomazzo, Francesco Di Plinio, Paul Hagelstein, Ioannis Parissis, Luz Roncal
Quantitative formulations of Fefferman’s counterexample for the ball multiplier are naturally linked to square function estimates for conical and directional multipliers. We develop a novel framework for these square function estimates, based on a directional embedding theorem for Carleson sequences and multiparameter time-frequency analysis techniques. As applications we prove sharp or quantified
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Partial regularity for Navier–Stokes and liquid crystals inequalities without maximum principle Anal. PDE (IF 2.2) Pub Date : 2023-09-21 Gabriel S. Koch
In 1985, V. Scheffer discussed partial regularity results for what he called solutions to the Navier–Stokes inequality. These maps essentially satisfy the incompressibility condition as well as the local and global energy inequalities and the pressure equation which may be derived formally from the Navier–Stokes system of equations, but they are not required to satisfy the Navier–Stokes system itself
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The Cauchy problem for the infinitesimal model in the regime of small variance Anal. PDE (IF 2.2) Pub Date : 2023-08-23 Florian Patout
We study the asymptotic behavior of solutions of the Cauchy problem associated to a quantitative genetics model with a sexual mode of reproduction. It combines trait-dependent mortality and a nonlinear integral reproduction operator, the infinitesimal model. A parameter describes the standard deviation between the offspring and the mean parental traits. We show that under mild assumptions upon the
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The “good” Boussinesq equation : long-time asymptotics Anal. PDE (IF 2.2) Pub Date : 2023-08-23 Christophe Charlier, Jonatan Lenells, Deng-Shan Wang
We consider the initial-value problem for the “good” Boussinesq equation on the line. Using inverse scattering techniques, the solution can be expressed in terms of the solution of a 3 × 3-matrix Riemann–Hilbert problem. We establish formulas for the long-time asymptotics of the solution by performing a Deift–Zhou steepest descent analysis of a regularized version of this Riemann–Hilbert problem. Our
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Quantitative Obata’s theorem Anal. PDE (IF 2.2) Pub Date : 2023-08-23 Fabio Cavalletti, Andrea Mondino, Daniele Semola
We prove a quantitative version of Obata’s theorem involving the shape of functions with null mean value when compared with the cosine of distance functions from single points. The deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise. These results are obtained in the general framework of (possibly nonsmooth) metric measure spaces with curvature-dimension
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Couniversality and controlled maps on product systems over right LCM semigroups Anal. PDE (IF 2.2) Pub Date : 2023-08-23 Evgenios T.A. Kakariadis, Elias G. Katsoulis, Marcelo Laca, Xin Li
We study the structure of C∗-algebras associated with compactly aligned product systems over group embeddable right LCM semigroups. Towards this end we employ controlled maps and a controlled elimination method that associates the original cores to those of the controlling pair, and we combine these with applications of the C∗-envelope theory for cosystems of nonselfadjoint operator algebras recently
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Nonuniform stability of damped contraction semigroups Anal. PDE (IF 2.2) Pub Date : 2023-08-12 Ralph Chill, Lassi Paunonen, David Seifert, Reinhard Stahn, Yuri Tomilov
We investigate the stability properties of strongly continuous semigroups generated by operators of the form A − BB∗ , where A is the generator of a contraction semigroup and B is a possibly unbounded operator. Such systems arise naturally in the study of hyperbolic partial differential equations with damping on the boundary or inside the spatial domain. As our main results we present general sufficient
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Long time solutions for quasilinear Hamiltonian perturbations of Schrödinger and Klein–Gordon equations on tori Anal. PDE (IF 2.2) Pub Date : 2023-08-12 Roberto Feola, Benoît Grébert, Felice Iandoli
We consider quasilinear, Hamiltonian perturbations of the cubic Schrödinger and of the cubic (derivative) Klein–Gordon equations on the d-dimensional torus. If 𝜖 ≪ 1 is the size of the initial datum, we prove that the lifespan of solutions is strictly larger than the local existence time 𝜖−2. More precisely, concerning the Schrödinger equation we show that the lifespan is at least of order O(𝜖−4)
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An extension problem, trace Hardy and Hardy’s inequalities for the Ornstein–Uhlenbeck operator Anal. PDE (IF 2.2) Pub Date : 2023-08-12 Pritam Ganguly, Ramesh Manna, Sundaram Thangavelu
We study an extension problem for the Ornstein–Uhlenbeck operator L = −Δ + 2x ⋅∇ + n, and we obtain various characterisations of the solution of the same. We use a particular solution of that extension problem to prove a trace Hardy inequality for L from which Hardy’s inequality for fractional powers of L is obtained. We also prove an isometry property of the solution operator associated to the extension
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On the well-posedness problem for the derivative nonlinear Schrödinger equation Anal. PDE (IF 2.2) Pub Date : 2023-08-12 Rowan Killip, Maria Ntekoume, Monica Vişan
We consider the derivative nonlinear Schrödinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and L2-critical with respect to scaling. We first discuss whether ensembles of orbits with L2-equicontinuous initial data remain equicontinuous under evolution. We prove that this is true under the restriction M(q) = ∫ |q|2 < 4π
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Exponential integrability in Gauss space Anal. PDE (IF 2.2) Pub Date : 2023-08-12 Paata Ivanisvili, Ryan Russell
Talagrand showed that finiteness of 𝔼 e|∇ f(X)|2∕2 implies finiteness of 𝔼 ef(X)−𝔼f(X), where X is the standard Gaussian vector in ℝn and f is a smooth function. However, in this paper we show that finiteness of 𝔼 e|∇ f|2∕2 (1 + |∇ f|)−1 implies finiteness of 𝔼 ef(X)−𝔼f(X), and we also obtain quantitative bounds log 𝔼 ef−𝔼f ≤ 10 𝔼 e|∇ f|2∕2 (1 + |∇ f|)−1. Moreover, the extra factor
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Strong semiclassical limits from Hartree and Hartree–Fock to Vlasov–Poisson equations Anal. PDE (IF 2.2) Pub Date : 2023-06-15 Laurent Lafleche, Chiara Saffirio
We consider the semiclassical limit from the Hartree to the Vlasov equation with general singular interaction potential including the Coulomb and gravitational interactions, and we prove explicit bounds in the strong topologies of Schatten norms. Moreover, in the case of fermions, we provide estimates on the size of the exchange term in the Hartree–Fock equation and also obtain a rate of convergence