Pub Date : 2021-11-28DOI: 10.1080/03605302.2022.2091453
A. Hyder, A. Segatti, Y. Sire, Changyou Wang
Abstract We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivière. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rather unusual heat flow for half-harmonic maps. We construct then weak solutions and prove their partial regularity in space and time via a Ginzburg-Landau approximation. The present paper complements the study initiated by Struwe and Chen-Lin.
{"title":"Partial regularity of the heat flow of half-harmonic maps and applications to harmonic maps with free boundary","authors":"A. Hyder, A. Segatti, Y. Sire, Changyou Wang","doi":"10.1080/03605302.2022.2091453","DOIUrl":"https://doi.org/10.1080/03605302.2022.2091453","url":null,"abstract":"Abstract We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivière. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rather unusual heat flow for half-harmonic maps. We construct then weak solutions and prove their partial regularity in space and time via a Ginzburg-Landau approximation. The present paper complements the study initiated by Struwe and Chen-Lin.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"1845 - 1882"},"PeriodicalIF":1.9,"publicationDate":"2021-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47651817","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-24DOI: 10.1080/03605302.2023.2175217
S. Steinerberger
Abstract We study the growth of Laplacian eigenfunctions on compact manifolds (M, g). Hörmander proved sharp polynomial bounds on 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 as the typical behavior. We propose a mechanism, centered around an analog of the spectral projector, for explaining the slow growth in the generic case: for to be large, it is necessary that either (1) several of the first n eigenfunctions were large in x 0 or (2) that is strongly correlated with a suitable linear combination of the first n eigenfunctions on most of the manifold or (3) both. An interesting byproduct is quantum entanglement for Laplacian eigenfunctions: the existence of two distinct points such that the sequences and do not behave like independent random variables. The existence of such points is not to be expected for generic manifolds but common for the classical manifolds and subtly intertwined with eigenfunction concentration.
{"title":"Quantum entanglement and the growth of Laplacian eigenfunctions","authors":"S. Steinerberger","doi":"10.1080/03605302.2023.2175217","DOIUrl":"https://doi.org/10.1080/03605302.2023.2175217","url":null,"abstract":"Abstract We study the growth of Laplacian eigenfunctions on compact manifolds (M, g). Hörmander proved sharp polynomial bounds on 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 as the typical behavior. We propose a mechanism, centered around an analog of the spectral projector, for explaining the slow growth in the generic case: for to be large, it is necessary that either (1) several of the first n eigenfunctions were large in x 0 or (2) that is strongly correlated with a suitable linear combination of the first n eigenfunctions on most of the manifold or (3) both. An interesting byproduct is quantum entanglement for Laplacian eigenfunctions: the existence of two distinct points such that the sequences and do not behave like independent random variables. The existence of such points is not to be expected for generic manifolds but common for the classical manifolds and subtly intertwined with eigenfunction concentration.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"511 - 541"},"PeriodicalIF":1.9,"publicationDate":"2021-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43836263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-20DOI: 10.1080/03605302.2023.2169937
Sky Cao, S. 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 arguments. In a companion work, we use the main results of this paper to propose a way toward the construction of 3D Yang–Mills measures.
{"title":"The Yang-Mills heat flow with random distributional initial data","authors":"Sky Cao, S. Chatterjee","doi":"10.1080/03605302.2023.2169937","DOIUrl":"https://doi.org/10.1080/03605302.2023.2169937","url":null,"abstract":"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 arguments. In a companion work, we use the main results of this paper to propose a way toward the construction of 3D Yang–Mills measures.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"209 - 251"},"PeriodicalIF":1.9,"publicationDate":"2021-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44725470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-17DOI: 10.1080/03605302.2022.2137678
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 exponents of the cocycle given by the damping term play the role of the Birkhoff limits of the scalar setting.
{"title":"Spectral asymptotics for the vectorial damped wave equation","authors":"Guillaume Klein","doi":"10.1080/03605302.2022.2137678","DOIUrl":"https://doi.org/10.1080/03605302.2022.2137678","url":null,"abstract":"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 exponents of the cocycle given by the damping term play the role of the Birkhoff limits of the scalar setting.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"1 - 21"},"PeriodicalIF":1.9,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47909859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-10DOI: 10.1080/03605302.2022.2070853
André de Laire, Salvador L'opez-Mart'inez
Abstract We consider the nonlocal Gross–Pitaevskii equation that models a Bose gas with general nonlocal interactions between particles in one spatial dimension, with constant density far away. We address the problem of the existence of traveling waves with nonvanishing conditions at infinity, i.e. dark solitons. Under general conditions on the interactions, we prove existence of dark solitons for almost every subsonic speed. Moreover, we show existence in the whole subsonic regime for a family of potentials. The proofs rely on a Mountain Pass argument combined with the so-called “monotonicity trick,” as well as on a priori estimates for the Palais–Smale sequences. Finally, we establish properties of the solitons such as exponential decay at infinity and analyticity.
{"title":"Existence and decay of traveling waves for the nonlocal Gross–Pitaevskii equation","authors":"André de Laire, Salvador L'opez-Mart'inez","doi":"10.1080/03605302.2022.2070853","DOIUrl":"https://doi.org/10.1080/03605302.2022.2070853","url":null,"abstract":"Abstract We consider the nonlocal Gross–Pitaevskii equation that models a Bose gas with general nonlocal interactions between particles in one spatial dimension, with constant density far away. We address the problem of the existence of traveling waves with nonvanishing conditions at infinity, i.e. dark solitons. Under general conditions on the interactions, we prove existence of dark solitons for almost every subsonic speed. Moreover, we show existence in the whole subsonic regime for a family of potentials. The proofs rely on a Mountain Pass argument combined with the so-called “monotonicity trick,” as well as on a priori estimates for the Palais–Smale sequences. Finally, we establish properties of the solitons such as exponential decay at infinity and analyticity.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"1732 - 1794"},"PeriodicalIF":1.9,"publicationDate":"2021-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48840789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.1080/03605302.2023.2190525
S. Cygan, A. Marciniak-Czochra, G. 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 systems (2021)], regular (i.e. sufficiently smooth) stationary solutions are shown to exist, however, all of them are unstable. The goal of this work is to construct discontinuous stationary solutions to general reaction-diffusion-ODE systems and to find sufficient conditions for their stability.
{"title":"Stable discontinuous stationary solutions to reaction-diffusion-ODE systems","authors":"S. Cygan, A. Marciniak-Czochra, G. Karch, Kanako Suzuki","doi":"10.1080/03605302.2023.2190525","DOIUrl":"https://doi.org/10.1080/03605302.2023.2190525","url":null,"abstract":"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 systems (2021)], regular (i.e. sufficiently smooth) stationary solutions are shown to exist, however, all of them are unstable. The goal of this work is to construct discontinuous stationary solutions to general reaction-diffusion-ODE systems and to find sufficient conditions for their stability.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"478 - 510"},"PeriodicalIF":1.9,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44355220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-26DOI: 10.1080/03605302.2023.2191326
B. Cassano, Valentina Franceschi, D. Krejčiřík, D. 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 inequalities for the Heisenberg sub-Laplacian. In particular, we establish a sub-Riemannian analogue of Laptev and Weidl sub-criticality result for magnetic Laplacians in the plane. Instrumental for our argument is the validity of a Hardy-type inequality for the Folland–Stein operator, that we prove in this article and has an interest on its own.
{"title":"Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group","authors":"B. Cassano, Valentina Franceschi, D. Krejčiřík, D. Prandi","doi":"10.1080/03605302.2023.2191326","DOIUrl":"https://doi.org/10.1080/03605302.2023.2191326","url":null,"abstract":"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 inequalities for the Heisenberg sub-Laplacian. In particular, we establish a sub-Riemannian analogue of Laptev and Weidl sub-criticality result for magnetic Laplacians in the plane. Instrumental for our argument is the validity of a Hardy-type inequality for the Folland–Stein operator, that we prove in this article and has an interest on its own.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"711 - 752"},"PeriodicalIF":1.9,"publicationDate":"2021-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49010097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-13DOI: 10.1080/03605302.2022.2051188
M. Cassier, C. Hazard, P. Joly
Abstract This paper is concerned with the time-dependent Maxwell’s equations for a plane interface between a negative material described by the Drude model and the vacuum, which fill, respectively, two complementary half-spaces. In a first paper, we have constructed a generalized Fourier transform which diagonalizes the Hamiltonian that represents the propagation of transverse electric waves. In this second paper, we use this transform to prove the limiting absorption and limiting amplitude principles, which concern, respectively, the behavior of the resolvent near the continuous spectrum and the long time response of the medium to a time-harmonic source of prescribed frequency. This paper also underlines the existence of an interface resonance which occurs when there exists a particular frequency characterized by a ratio of permittivities and permeabilities equal to −1 across the interface. At this frequency, the response of the system to a harmonic forcing term blows up linearly in time. Such a resonance is unusual for wave problem in unbounded domains and corresponds to a non-zero embedded eigenvalue of infinite multiplicity of the underlying operator. This is the time counterpart of the ill-posdness of the corresponding harmonic problem.
{"title":"Spectral theory for Maxwell’s equations at the interface of a metamaterial. Part II: Limiting absorption, limiting amplitude principles and interface resonance","authors":"M. Cassier, C. Hazard, P. Joly","doi":"10.1080/03605302.2022.2051188","DOIUrl":"https://doi.org/10.1080/03605302.2022.2051188","url":null,"abstract":"Abstract This paper is concerned with the time-dependent Maxwell’s equations for a plane interface between a negative material described by the Drude model and the vacuum, which fill, respectively, two complementary half-spaces. In a first paper, we have constructed a generalized Fourier transform which diagonalizes the Hamiltonian that represents the propagation of transverse electric waves. In this second paper, we use this transform to prove the limiting absorption and limiting amplitude principles, which concern, respectively, the behavior of the resolvent near the continuous spectrum and the long time response of the medium to a time-harmonic source of prescribed frequency. This paper also underlines the existence of an interface resonance which occurs when there exists a particular frequency characterized by a ratio of permittivities and permeabilities equal to −1 across the interface. At this frequency, the response of the system to a harmonic forcing term blows up linearly in time. Such a resonance is unusual for wave problem in unbounded domains and corresponds to a non-zero embedded eigenvalue of infinite multiplicity of the underlying operator. This is the time counterpart of the ill-posdness of the corresponding harmonic problem.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"1217 - 1295"},"PeriodicalIF":1.9,"publicationDate":"2021-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41487394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-10DOI: 10.1080/03605302.2021.1983596
S. Kawashima, Y. Shibata, Jiang Xu
Abstract In this paper, we are concerned with generally symmetric hyperbolic-parabolic systems with Korteweg-type dispersion. Referring to those classical efforts by Kawashima et al., we formulate new structural conditions for the Korteweg-type dispersion and develop the dissipative mechanism of “regularity-gain type.” As an application, it is checked that several concrete model systems (e.g., the compressible Navier-Stokes(-Fourier)-Korteweg system) satisfy the general structural conditions. In addition, the optimality of our general theory on the dissipative structure is also verified by calculating the asymptotic expansions of eigenvalues.
{"title":"Dissipative structure for symmetric hyperbolic-parabolic systems with Korteweg-type dispersion","authors":"S. Kawashima, Y. Shibata, Jiang Xu","doi":"10.1080/03605302.2021.1983596","DOIUrl":"https://doi.org/10.1080/03605302.2021.1983596","url":null,"abstract":"Abstract In this paper, we are concerned with generally symmetric hyperbolic-parabolic systems with Korteweg-type dispersion. Referring to those classical efforts by Kawashima et al., we formulate new structural conditions for the Korteweg-type dispersion and develop the dissipative mechanism of “regularity-gain type.” As an application, it is checked that several concrete model systems (e.g., the compressible Navier-Stokes(-Fourier)-Korteweg system) satisfy the general structural conditions. In addition, the optimality of our general theory on the dissipative structure is also verified by calculating the asymptotic expansions of eigenvalues.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"378 - 400"},"PeriodicalIF":1.9,"publicationDate":"2021-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44204708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-08DOI: 10.1080/03605302.2022.2056705
Gael Diebou Yomgne, H. Koch
Abstract Weakly harmonic maps from a domain (the upper half-space or a bounded domain, ) into a smooth closed manifold are studied. Prescribing small Dirichlet data in either of the classes or we establish solvability of the resulting boundary value problems by means of a nonvariational method. As a by-product, solutions are shown to be locally smooth, Moreover, we show that boundary data can be chosen large in the underlying topologies if Ω is smooth and bounded by perturbing strictly stable smooth harmonic maps.
{"title":"Dirichlet problem for weakly harmonic maps with rough data","authors":"Gael Diebou Yomgne, H. Koch","doi":"10.1080/03605302.2022.2056705","DOIUrl":"https://doi.org/10.1080/03605302.2022.2056705","url":null,"abstract":"Abstract Weakly harmonic maps from a domain (the upper half-space or a bounded domain, ) into a smooth closed manifold are studied. Prescribing small Dirichlet data in either of the classes or we establish solvability of the resulting boundary value problems by means of a nonvariational method. As a by-product, solutions are shown to be locally smooth, Moreover, we show that boundary data can be chosen large in the underlying topologies if Ω is smooth and bounded by perturbing strictly stable smooth harmonic maps.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"1504 - 1535"},"PeriodicalIF":1.9,"publicationDate":"2021-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49070242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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