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Uniform Skoda integrability and Calabi–Yau degeneration Anal. PDE (IF 1.8) Pub Date : 2024-08-21 Yang Li
We study polarised algebraic degenerations of Calabi–Yau manifolds. We prove a uniform Skoda-type estimate and a uniform L∞-estimate for the Calabi–Yau Kähler potentials.
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Unique continuation for the heat operator with potentials in weak spaces Anal. PDE (IF 1.8) Pub Date : 2024-08-21 Eunhee Jeong, Sanghyuk Lee, Jaehyeon Ryu
We prove the strong unique continuation property for the differential inequality |(∂t + Δ)u(x,t)|≤ V (x,t)|u(x,t)|, with V contained in weak spaces. In particular, we establish the strong unique continuation property for V ∈ Lt∞Lx[t]d∕2,∞, which has been left open since the works of Escauriaza (2000) and Escauriaza and Vega (2001). Our results are consequences of the Carleman estimates for the heat
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Nonnegative Ricci curvature and minimal graphs with linear growth Anal. PDE (IF 1.8) Pub Date : 2024-08-21 Giulio Colombo, Eddygledson S. Gama, Luciano Mari, Marco Rigoli
We study minimal graphs with linear growth on complete manifolds Mm with Ric ≥ 0. Under the further assumption that the (m−2)-th Ricci curvature in radial direction is bounded below by Cr(x)−2, we prove that any such graph, if nonconstant, forces tangent cones at infinity of M to split off a line. Note that M is not required to have Euclidean volume growth. We also show that M may not split off any
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Nonlinear periodic waves on the Einstein cylinder Anal. PDE (IF 1.8) Pub Date : 2024-08-21 Athanasios Chatzikaleas, Jacques Smulevici
Motivated by the study of small amplitude nonlinear waves in the anti-de Sitter spacetime and in particular the conjectured existence of periodic in time solutions to the Einstein equations, we construct families of arbitrary small time-periodic solutions to the conformal cubic wave equation and the spherically symmetric Yang–Mills equations on the Einstein cylinder ℝ × 𝕊3. For the conformal cubic
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Host–Kra factors for ⊕ p∈Pℤ∕pℤ actions and finite-dimensional nilpotent systems Anal. PDE (IF 1.8) Pub Date : 2024-08-21 Or Shalom
Let 𝒫 be a countable multiset of primes and let G = ⊕ p∈Pℤ∕pℤ. We study the universal characteristic factors associated with the Gowers–Host–Kra seminorms for the group G. We show that the universal characteristic factor of order < k + 1 is a factor of an inverse limit of finite-dimensional k-step nilpotent homogeneous spaces. The latter is a counterpart of a k-step nilsystem where the homogeneous
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A fast point charge interacting with the screened Vlasov–Poisson system Anal. PDE (IF 1.8) Pub Date : 2024-08-21 Richard M. Höfer, Raphael Winter
We consider the long-time behavior of a fast, charged particle interacting with an initially spatially homogeneous background plasma. The background is modeled by the screened Vlasov–Poisson equations, whereas the interaction potential of the point charge is assumed to be smooth. We rigorously prove the validity of the stopping power theory in physics, which predicts a decrease of the velocity V (t)
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Haagerup’s phase transition at polydisc slicing Anal. PDE (IF 1.8) Pub Date : 2024-08-21 Giorgos Chasapis, Salil Singh, Tomasz Tkocz
We establish a sharp comparison inequality between the negative moments and the second moment of the magnitude of sums of independent random vectors uniform on three-dimensional Euclidean spheres. This provides a probabilistic extension of the Oleszkiewicz–Pełczyński polydisc slicing result. The Haagerup-type phase transition occurs exactly when the p-norm recovers volume, in contrast to the real case
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A substitute for Kazhdan’s property (T) for universal nonlattices Anal. PDE (IF 1.8) Pub Date : 2024-08-21 Narutaka Ozawa
The well-known theorem of Shalom–Vaserstein and Ershov–Jaikin-Zapirain states that the group EL n(ℛ), generated by elementary matrices over a finitely generated commutative ring ℛ, has Kazhdan’s property (T) as soon as n ≥ 3. This is no longer true if the ring ℛ is replaced by a commutative rng (a ring but without the identity) due to nilpotent quotients EL n(ℛ∕ℛk). We prove that even in such a
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Trigonometric chaos and Xp inequalities, I : Balanced Fourier truncations over discrete groups Anal. PDE (IF 1.8) Pub Date : 2024-08-21 Antonio Ismael Cano-Mármol, José M. Conde-Alonso, Javier Parcet
We investigate Lp-estimates for balanced averages of Fourier truncations in group algebras, in terms of “differential operators” acting on them. Our results extend a fundamental inequality of Naor for the hypercube (with profound consequences in metric geometry) to discrete groups. Different inequalities are established in terms of “directional derivatives” which are constructed via affine representations
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Beurling–Carleson sets, inner functions and a semilinear equation Anal. PDE (IF 1.8) Pub Date : 2024-08-21 Oleg Ivrii, Artur Nicolau
Beurling–Carleson sets have appeared in a number of areas of complex analysis such as boundary zero sets of analytic functions, inner functions with derivative in the Nevanlinna class, cyclicity in weighted Bergman spaces, Fuchsian groups of Widom-type and the corona problem in quotient Banach algebras. After surveying these developments, we give a general definition of Beurling–Carleson sets and discuss
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Projective embedding of stably degenerating sequences of hyperbolic Riemann surfaces Anal. PDE (IF 1.8) Pub Date : 2024-07-19 Jingzhou Sun
Given a sequence of genus g ≥ 2 curves converging to a punctured Riemann surface with complete metric of constant Gaussian curvature − 1, we prove that the Kodaira embedding using an orthonormal basis of the Bergman space of sections of a pluricanonical bundle also converges to the embedding of the limit space together with extra complex projective lines.
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Uniqueness of excited states to −Δu + u−u3 = 0 in three dimensions Anal. PDE (IF 1.8) Pub Date : 2024-07-19 Alex Cohen, Zhenhao Li, Wilhelm Schlag
We prove the uniqueness of several excited states to the ODE ÿ(t) + (2∕t)ẏ(t) + f(y(t)) = 0, y(0) = b, and ẏ(0) = 0, for the model nonlinearity f(y) = y3 − y. The n-th excited state is a solution with exactly n zeros and which tends to 0 as t →∞. These represent all smooth radial nonzero solutions to the PDE Δu + f(u) = 0 in H1. We interpret the ODE as a damped oscillator governed by a double-well
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On the spectrum of nondegenerate magnetic Laplacians Anal. PDE (IF 1.8) Pub Date : 2024-07-19 Laurent Charles
We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is nondegenerate. Under a general condition, the Laplacian acting on high tensor powers of the bundle exhibits gaps and clusters of eigenvalues. We prove that for each cluster the number of eigenvalues that it contains is given by a Riemann–Roch number. We also give a pointwise description of the Schwartz kernel
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Variational methods for the kinetic Fokker–Planck equation Anal. PDE (IF 1.8) Pub Date : 2024-07-19 Dallas Albritton, Scott Armstrong, Jean-Christophe Mourrat, Matthew Novack
We develop a functional-analytic approach to the study of the Kramers and kinetic Fokker–Planck equations which parallels the classical H1 theory of uniformly elliptic equations. In particular, we identify a function space analogous to H1 and develop a well-posedness theory for weak solutions in this space. In the case of a conservative force, we identify the weak solution as the minimizer of a uniformly
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Improved endpoint bounds for the lacunary spherical maximal operator Anal. PDE (IF 1.8) Pub Date : 2024-07-19 Laura Cladek, Benjamin Krause
We prove new endpoint bounds for the lacunary spherical maximal operator and as a consequence obtain almost everywhere pointwise convergence of lacunary spherical means for functions locally in Llog log log L(log log log log L)1+𝜖 for any 𝜖 > 0.
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Global well-posedness for a system of quasilinear wave equations on a product space Anal. PDE (IF 1.8) Pub Date : 2024-07-19 Cécile Huneau, Annalaura Stingo
We consider a system of quasilinear wave equations on the product space ℝ1+3 × 𝕊1 , which we want to see as a toy model for the Einstein equations with additional compact dimensions. We show global existence of solutions for small and regular initial data with polynomial decay at infinity. The method combines energy estimates on hyperboloids inside the light cone and weighted energy estimates outside
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Existence of resonances for Schrödinger operators on hyperbolic space Anal. PDE (IF 1.8) Pub Date : 2024-07-19 David Borthwick, Yiran Wang
We prove existence results and lower bounds for the resonances of Schrödinger operators associated to smooth, compactly support potentials on hyperbolic space. The results are derived from a combination of heat and wave trace expansions and asymptotics of the scattering phase.
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Characterization of rectifiability via Lusin-type approximation Anal. PDE (IF 1.8) Pub Date : 2024-07-19 Andrea Marchese, Andrea Merlo
We prove that a Radon measure μ on ℝn can be written as μ = ∑ i=0nμi, where each of the μi is an i-dimensional rectifiable measure if and only if, for every Lipschitz function f : ℝn → ℝ and every 𝜀 > 0, there exists a function g of class C1 such that μ({x ∈ ℝn : g(x)≠f(x)}) < 𝜀.
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On the endpoint regularity in Onsager’s conjecture Anal. PDE (IF 1.8) Pub Date : 2024-07-19 Philip Isett
Onsager’s conjecture states that the conservation of energy may fail for three-dimensional incompressible Euler flows with Hölder regularity below 1 3. This conjecture was recently solved by the author, yet the endpoint case remains an interesting open question with further connections to turbulence theory. In this work, we construct energy nonconserving solutions to the three-dimensional incompressible
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Extreme temporal intermittency in the linear Sobolev transport: Almost smooth nonunique solutions Anal. PDE (IF 1.8) Pub Date : 2024-07-19 Alexey Cheskidov, Xiaoyutao Luo
We revisit the notion of temporal intermittency to obtain sharp nonuniqueness results for linear transport equations. We construct divergence-free vector fields with sharp Sobolev regularity Lt1W1,p for all p < ∞ in space dimensions d ≥ 2 whose transport equations admit nonunique weak solutions belonging to LtpCk for all p < ∞ and k ∈ ℕ. In particular, our result shows that the time-integrability assumption
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Lp-polarity, Mahler volumes, and the isotropic constant Anal. PDE (IF 1.8) Pub Date : 2024-07-19 Bo Berndtsson, Vlassis Mastrantonis, Yanir A. Rubinstein
This article introduces Lp versions of the support function of a convex body K and associates to these canonical Lp-polar bodies K∘,p and Mahler volumes ℳp(K). Classical polarity is then seen as L∞-polarity. This one-parameter generalization of polarity leads to a generalization of the Mahler conjectures, with a subtle advantage over the original conjecture: conjectural uniqueness of extremizers for
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Strong cosmic censorship in the presence of matter: the decisive effect of horizon oscillations on the black hole interior geometry Anal. PDE (IF 1.8) Pub Date : 2024-06-20 Christoph Kehle, Maxime Van de Moortel
Motivated by the strong cosmic censorship conjecture in the presence of matter, we study the Einstein equations coupled with a charged/massive scalar field with spherically symmetric characteristic data relaxing to a Reissner–Nordström event horizon. Contrary to the vacuum case, the relaxation rate is conjectured to be slow (nonintegrable), opening the possibility that the matter fields and the metric
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A semiclassical Birkhoff normal form for constant-rank magnetic fields Anal. PDE (IF 1.8) Pub Date : 2024-06-20 Léo Morin
This paper deals with classical and semiclassical nonvanishing magnetic fields on a Riemannian manifold of arbitrary dimension. We assume that the magnetic field B = dA has constant rank and admits a discrete well. On the classical part, we exhibit a harmonic oscillator for the Hamiltonian H = |p − A(q)|2 near the zero-energy surface: the cyclotron motion. On the semiclassical part, we describe the
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Blow-up of solutions of critical elliptic equations in three dimensions Anal. PDE (IF 1.8) Pub Date : 2024-06-20 Rupert L. Frank, Tobias König, Hynek Kovařík
We describe the asymptotic behavior of positive solutions u𝜀 of the equation −Δu + au = 3u5−𝜀 in Ω ⊂ ℝ3 with a homogeneous Dirichlet boundary condition. The function a is assumed to be critical in the sense of Hebey and Vaugon, and the functions u𝜀 are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up
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A determination of the blowup solutions to the focusing, quintic NLS with mass equal to the mass of the soliton Anal. PDE (IF 1.8) Pub Date : 2024-06-20 Benjamin Dodson
We prove the only blowup solutions to the focusing, quintic nonlinear Schrödinger equation with mass equal to the mass of the soliton are rescaled solitons or the pseudoconformal transformation of those solitons.
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Uniform stability in the Euclidean isoperimetric problem for the Allen–Cahn energy Anal. PDE (IF 1.8) Pub Date : 2024-06-20 Francesco Maggi, Daniel Restrepo
We consider the isoperimetric problem defined on the whole ℝn by the Allen–Cahn energy functional. For nondegenerate double-well potentials, we prove sharp quantitative stability inequalities of quadratic type which are uniform in the length scale of the phase transitions. We also derive a rigidity theorem for critical points analogous to the classical Alexandrov theorem for constant mean curvature
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Connectivity conditions and boundary Poincaré inequalities Anal. PDE (IF 1.8) Pub Date : 2024-06-20 Olli Tapiola, Xavier Tolsa
Inspired by recent work of Mourgoglou and the second author, and earlier work of Hofmann, Mitrea and Taylor, we consider connections between the local John condition, the Harnack chain condition and weak boundary Poincaré inequalities in open sets Ω ⊂ ℝn+1 , with codimension-1 Ahlfors–David regular boundaries. First, we prove that if Ω satisfies both the local John condition and the exterior corkscrew
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The singular strata of a free-boundary problem for harmonic measure Anal. PDE (IF 1.8) Pub Date : 2024-05-17 Sean McCurdy
We obtain quantitative estimates on the fine structure of the singular set of the mutual boundary ∂Ω± for pairs of complementary domains Ω+,Ω−⊂ ℝn which arise in a class of two-sided free boundary problems for harmonic measure. These estimates give new insight into the structure of the mutual boundary ∂Ω±.
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On complete embedded translating solitons of the mean curvature flow that are of finite genus Anal. PDE (IF 1.8) Pub Date : 2024-05-17 Graham Smith
We desingularise the union of three Grim paraboloids along Costa–Hoffman–Meeks surfaces in order to obtain complete embedded translating solitons of the mean curvature flow with three ends and arbitrary finite genus.
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Hausdorff measure bounds for nodal sets of Steklov eigenfunctions Anal. PDE (IF 1.8) Pub Date : 2024-05-17 Stefano Decio
We study nodal sets of Steklov eigenfunctions in a bounded domain with 𝒞2 boundary. Our first result is a lower bound for the Hausdorff measure of the nodal set: we show that, for uλ a Steklov eigenfunction with eigenvalue λ≠0, we have ℋd−1({uλ = 0}) ≥ cΩ, where cΩ is independent of λ. We also prove an almost sharp upper bound, namely, ℋd−1({uλ = 0}) ≤ CΩλlog (λ + e).
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On full asymptotics of real analytic torsions for compact locally symmetric orbifolds Anal. PDE (IF 1.8) Pub Date : 2024-05-17 Bingxiao Liu
We consider a certain sequence of flat vector bundles on a compact locally symmetric orbifold, and we evaluate explicitly the associated asymptotic Ray–Singer real analytic torsion. The basic idea is to computing the heat trace via Selberg’s trace formula, so that a key point in this paper is to evaluate the orbital integrals associated with nontrivial elliptic elements. For that purpose, we deduce
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The Landau equation as a gradient Flow Anal. PDE (IF 1.8) Pub Date : 2024-05-17 José A. Carrillo, Matias G. Delgadino, Laurent Desvillettes, Jeremy S.-H. Wu
We propose a gradient flow perspective to the spatially homogeneous Landau equation for soft potentials. We construct a tailored metric on the space of probability measures based on the entropy dissipation of the Landau equation. Under this metric, the Landau equation can be characterized as the gradient flow of the Boltzmann entropy. In particular, we characterize the dynamics of the PDE through a
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Degenerating hyperbolic surfaces and spectral gaps for large genus Anal. PDE (IF 1.8) Pub Date : 2024-05-17 Yunhui Wu, Haohao Zhang, Xuwen Zhu
We study the differences of two consecutive eigenvalues λi − λi−1, i up to 2g − 2, for the Laplacian on hyperbolic surfaces of genus g, and show that the supremum of such spectral gaps over the moduli space has infimum limit at least 1 4 as the genus goes to infinity. A min-max principle for eigenvalues on degenerating hyperbolic surfaces is also established.
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Plateau flow or the heat flow for half-harmonic maps Anal. PDE (IF 1.8) Pub Date : 2024-05-17 Michael Struwe
Using the interpretation of the half-Laplacian on S1 as the Dirichlet-to-Neumann operator for the Laplace equation on the ball B, we devise a classical approach to the heat flow for half-harmonic maps from S1 to a closed target manifold N ⊂ ℝn, recently studied by Wettstein, and for arbitrary finite-energy data we obtain a result fully analogous to the author’s 1985 results for the harmonic map heat
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Noncommutative maximal operators with rough kernels Anal. PDE (IF 1.8) Pub Date : 2024-05-17 Xudong Lai
This paper is devoted to the study of noncommutative maximal operators with rough kernels. More precisely, we prove the weak-type (1,1) boundedness for noncommutative maximal operators with rough kernels. The proof of the weak-type (1,1) estimate is based on the noncommutative Calderón–Zygmund decomposition. To deal with the rough kernel, we use the microlocal decomposition in the proofs of both the
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Structure of sets with nearly maximal Favard length Anal. PDE (IF 1.8) Pub Date : 2024-05-17 Alan Chang, Damian Dąbrowski, Tuomas Orponen, Michele Villa
Let E ⊂ B(1) ⊂ ℝ2 be an ℋ1 measurable set with ℋ1(E) < ∞, and let L ⊂ ℝ2 be a line segment with ℋ1(L) = ℋ1(E). It is not hard to see that Fav (E) ≤ Fav (L). We prove that in the case of near equality, that is, Fav (E) ≥ Fav (L) − δ, the set E can be covered by an 𝜖-Lipschitz graph, up to a set of length 𝜖. The dependence between 𝜖 and δ is polynomial: in fact, the conclusions hold with 𝜖
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Schauder estimates for equations with cone metrics, II Anal. PDE (IF 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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 1.8) 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