Pub Date : 2022-07-28DOI: 10.1080/03605302.2023.2215527
Alberto Chiarini, Giovanni Conforti, Giacomo Greco, Luca Tamanini
Abstract We show convergence of the gradients of the Schrödinger potentials to the (uniquely determined) gradient of Kantorovich potentials in the small-time limit under general assumptions on the marginals, which allow for unbounded densities and supports. Furthermore, we provide novel quantitative stability estimates for the optimal values and optimal couplings for the Schrödinger problem (SP), that we express in terms of a negative order weighted homogeneous Sobolev norm. The latter encodes the linearized behavior of the 2-Wasserstein distance between the marginals. The proofs of both results highlight for the first time the relevance of gradient bounds for Schrödinger potentials, that we establish here in full generality, in the analysis of the short-time behavior of Schrödinger bridges. Finally, we discuss how our results translate into the framework of quadratic Entropic Optimal Transport, that is a version of SP more suitable for applications in machine learning and data science.
{"title":"Gradient estimates for the Schrödinger potentials: convergence to the Brenier map and quantitative stability","authors":"Alberto Chiarini, Giovanni Conforti, Giacomo Greco, Luca Tamanini","doi":"10.1080/03605302.2023.2215527","DOIUrl":"https://doi.org/10.1080/03605302.2023.2215527","url":null,"abstract":"Abstract We show convergence of the gradients of the Schrödinger potentials to the (uniquely determined) gradient of Kantorovich potentials in the small-time limit under general assumptions on the marginals, which allow for unbounded densities and supports. Furthermore, we provide novel quantitative stability estimates for the optimal values and optimal couplings for the Schrödinger problem (SP), that we express in terms of a negative order weighted homogeneous Sobolev norm. The latter encodes the linearized behavior of the 2-Wasserstein distance between the marginals. The proofs of both results highlight for the first time the relevance of gradient bounds for Schrödinger potentials, that we establish here in full generality, in the analysis of the short-time behavior of Schrödinger bridges. Finally, we discuss how our results translate into the framework of quadratic Entropic Optimal Transport, that is a version of SP more suitable for applications in machine learning and data science.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"895 - 943"},"PeriodicalIF":1.9,"publicationDate":"2022-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42942787","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 : 2022-07-04DOI: 10.1080/03605302.2023.2180754
Nicholas Lohr
Abstract The main result of this article gives scaling asymptotics of the Wigner distributions of isotropic harmonic oscillator orbital coherent states concentrating along Hamiltonian orbits γ in shrinking tubes around γ in phase space. In particular, these Wigner distributions exhibit a hybrid semi-classical scaling. That is, simultaneously, we have an Airy scaling when the tube has radius normal to the energy surface Σ E , and a Gaussian scaling when the tube has radius tangent to Σ E .
{"title":"Scaling asymptotics of Wigner distributions of harmonic oscillator orbital coherent states","authors":"Nicholas Lohr","doi":"10.1080/03605302.2023.2180754","DOIUrl":"https://doi.org/10.1080/03605302.2023.2180754","url":null,"abstract":"Abstract The main result of this article gives scaling asymptotics of the Wigner distributions of isotropic harmonic oscillator orbital coherent states concentrating along Hamiltonian orbits γ in shrinking tubes around γ in phase space. In particular, these Wigner distributions exhibit a hybrid semi-classical scaling. That is, simultaneously, we have an Airy scaling when the tube has radius normal to the energy surface Σ E , and a Gaussian scaling when the tube has radius tangent to Σ E .","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"415 - 439"},"PeriodicalIF":1.9,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47897552","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 : 2022-06-30DOI: 10.1080/03605302.2022.2139725
G. Tarantello
Abstract We analyze a blow-up sequence of solutions for Liouville-type equations involving Dirac measures with “collapsing” poles. We consider the case where blow-up occurs exactly at a point where the poles coalesce. After proving that a” quantization” property still holds for the” blow-up mass,” we obtain precise pointwise estimates when blow-up occurs with the least blow-up mass. Interestingly, such estimates express the exact analogue of those previously obtained for solutions of “regular” Liouville equations where the “collapsing” Dirac measures are neglected. Such information will be used in a forthcoming paper to describe the asymptotic behavior of minimizers of the Donaldson functional introduced by Goncalves and Uhlenbeck in 2007, yielding to mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds.
{"title":"On the blow-up analysis at collapsing poles for solutions of singular Liouville-type equations","authors":"G. Tarantello","doi":"10.1080/03605302.2022.2139725","DOIUrl":"https://doi.org/10.1080/03605302.2022.2139725","url":null,"abstract":"Abstract We analyze a blow-up sequence of solutions for Liouville-type equations involving Dirac measures with “collapsing” poles. We consider the case where blow-up occurs exactly at a point where the poles coalesce. After proving that a” quantization” property still holds for the” blow-up mass,” we obtain precise pointwise estimates when blow-up occurs with the least blow-up mass. Interestingly, such estimates express the exact analogue of those previously obtained for solutions of “regular” Liouville equations where the “collapsing” Dirac measures are neglected. Such information will be used in a forthcoming paper to describe the asymptotic behavior of minimizers of the Donaldson functional introduced by Goncalves and Uhlenbeck in 2007, yielding to mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"150 - 181"},"PeriodicalIF":1.9,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42857832","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 : 2022-06-07DOI: 10.1080/03605302.2023.2243491
Andrew Rout, Vedran Sohinger
Abstract In this paper, we give a microscopic derivation of Gibbs measures for the focusing cubic nonlinear Schrödinger equation on the one-dimensional torus from many-body quantum Gibbs states. Since we are not making any positivity assumptions on the interaction, it is necessary to introduce a truncation of the mass in the classical setting and of the rescaled particle number in the quantum setting. Our methods are based on a perturbative expansion of the interaction, similarly as in [1]. Due to the presence of the truncation, the obtained series have infinite radius of convergence. We treat the case of bounded, L 1 and delta function interaction potentials, without any sign assumptions. Within this framework, we also study time-dependent correlation functions. This is the first such known result in the focusing regime.
{"title":"A microscopic derivation of Gibbs measures for the 1D focusing cubic nonlinear Schrödinger equation","authors":"Andrew Rout, Vedran Sohinger","doi":"10.1080/03605302.2023.2243491","DOIUrl":"https://doi.org/10.1080/03605302.2023.2243491","url":null,"abstract":"Abstract In this paper, we give a microscopic derivation of Gibbs measures for the focusing cubic nonlinear Schrödinger equation on the one-dimensional torus from many-body quantum Gibbs states. Since we are not making any positivity assumptions on the interaction, it is necessary to introduce a truncation of the mass in the classical setting and of the rescaled particle number in the quantum setting. Our methods are based on a perturbative expansion of the interaction, similarly as in [1]. Due to the presence of the truncation, the obtained series have infinite radius of convergence. We treat the case of bounded, L 1 and delta function interaction potentials, without any sign assumptions. Within this framework, we also study time-dependent correlation functions. This is the first such known result in the focusing regime.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"1008 - 1055"},"PeriodicalIF":1.9,"publicationDate":"2022-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45121507","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 : 2022-05-11DOI: 10.1080/03605302.2022.2139723
King-Yeung Lam, Y. Lou, B. Perthame
Abstract The evolution of dispersal is a classical question in evolutionary biology, and it has been studied in a wide range of mathematical models. A selection-mutation model, in which the population is structured by space and a phenotypic trait, with the trait acting directly on the dispersal (diffusion) rate, was formulated by Perthame and Souganidis [Math. Model. Nat. Phenom. 11:154–166, 2016] to study the evolution of random dispersal toward the evolutionarily stable strategy. For the rare mutation limit, it was shown that the equilibrium population concentrates on a single trait associated to the smallest dispersal rate. In this paper, we consider the corresponding evolution equation and characterize the asymptotic behaviors of the time-dependent solutions in the rare mutation limit, under mild convexity assumptions on the underlying Hamiltonian function.
{"title":"A Hamilton-Jacobi approach to evolution of dispersal","authors":"King-Yeung Lam, Y. Lou, B. Perthame","doi":"10.1080/03605302.2022.2139723","DOIUrl":"https://doi.org/10.1080/03605302.2022.2139723","url":null,"abstract":"Abstract The evolution of dispersal is a classical question in evolutionary biology, and it has been studied in a wide range of mathematical models. A selection-mutation model, in which the population is structured by space and a phenotypic trait, with the trait acting directly on the dispersal (diffusion) rate, was formulated by Perthame and Souganidis [Math. Model. Nat. Phenom. 11:154–166, 2016] to study the evolution of random dispersal toward the evolutionarily stable strategy. For the rare mutation limit, it was shown that the equilibrium population concentrates on a single trait associated to the smallest dispersal rate. In this paper, we consider the corresponding evolution equation and characterize the asymptotic behaviors of the time-dependent solutions in the rare mutation limit, under mild convexity assumptions on the underlying Hamiltonian function.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"86 - 118"},"PeriodicalIF":1.9,"publicationDate":"2022-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43656126","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 : 2022-05-10DOI: 10.1080/03605302.2023.2183409
I. Capuzzo Dolcetta, A. Davini
Abstract We study the asymptotic behavior of the viscosity solutions of the Hamilton-Jacobi (HJ) equation as the positive discount factor λ tends to 0, where is the perturbation of a Hamiltonian –periodic in the space variable and convex and coercive in the momentum, by a compactly supported potential The constant c(G) appearing above is defined as the infimum of values for which the HJ equation in admits bounded viscosity subsolutions. We prove that the functions locally uniformly converge, for to a specific solution of the critical equation We identify in terms of projected Mather measures for G and of the limit to the unperturbed periodic problem. Our work also includes a qualitative analysis of the critical equation with a weak KAM theoretic flavor.
{"title":"On the vanishing discount approximation for compactly supported perturbations of periodic Hamiltonians: the 1d case","authors":"I. Capuzzo Dolcetta, A. Davini","doi":"10.1080/03605302.2023.2183409","DOIUrl":"https://doi.org/10.1080/03605302.2023.2183409","url":null,"abstract":"Abstract We study the asymptotic behavior of the viscosity solutions of the Hamilton-Jacobi (HJ) equation as the positive discount factor λ tends to 0, where is the perturbation of a Hamiltonian –periodic in the space variable and convex and coercive in the momentum, by a compactly supported potential The constant c(G) appearing above is defined as the infimum of values for which the HJ equation in admits bounded viscosity subsolutions. We prove that the functions locally uniformly converge, for to a specific solution of the critical equation We identify in terms of projected Mather measures for G and of the limit to the unperturbed periodic problem. Our work also includes a qualitative analysis of the critical equation with a weak KAM theoretic flavor.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"576 - 622"},"PeriodicalIF":1.9,"publicationDate":"2022-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48236363","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 : 2022-04-12DOI: 10.1080/03605302.2023.2191265
N. David
Abstract We consider a (degenerate) cross-diffusion model of tumour growth structured by phenotypic trait. We prove the existence of weak solutions and the incompressible limit as the pressure becomes stiff extending methods recently introduced in the context of two-species cross-diffusion systems. In the stiff-pressure limit, the compressible model generates a free boundary problem of Hele-Shaw type. Moreover, we prove a new L 4-bound on the pressure gradient.
{"title":"Phenotypic heterogeneity in a model of tumour growth: existence of solutions and incompressible limit","authors":"N. David","doi":"10.1080/03605302.2023.2191265","DOIUrl":"https://doi.org/10.1080/03605302.2023.2191265","url":null,"abstract":"Abstract We consider a (degenerate) cross-diffusion model of tumour growth structured by phenotypic trait. We prove the existence of weak solutions and the incompressible limit as the pressure becomes stiff extending methods recently introduced in the context of two-species cross-diffusion systems. In the stiff-pressure limit, the compressible model generates a free boundary problem of Hele-Shaw type. Moreover, we prove a new L 4-bound on the pressure gradient.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"678 - 710"},"PeriodicalIF":1.9,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43923487","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 : 2022-04-06DOI: 10.1080/03605302.2022.2059676
Edgard A. Pimentel, Makson S. Santos, E. Teixeira
Abstract We prove higher-order fractional Sobolev regularity for fully nonlinear, uniformly elliptic equations in the presence of unbounded source terms. More precisely, we show the existence of a universal number depending only on ellipticity constants and dimension, such that if u is a viscosity solution of then, with appropriate estimates. Our strategy suggests a sort of fractional feature of fully nonlinear diffusion processes, as what we actually show is that for a universal constant We believe our techniques are flexible and can be adapted to various models and contexts.
{"title":"Fractional Sobolev regularity for fully nonlinear elliptic equations","authors":"Edgard A. Pimentel, Makson S. Santos, E. Teixeira","doi":"10.1080/03605302.2022.2059676","DOIUrl":"https://doi.org/10.1080/03605302.2022.2059676","url":null,"abstract":"Abstract We prove higher-order fractional Sobolev regularity for fully nonlinear, uniformly elliptic equations in the presence of unbounded source terms. More precisely, we show the existence of a universal number depending only on ellipticity constants and dimension, such that if u is a viscosity solution of then, with appropriate estimates. Our strategy suggests a sort of fractional feature of fully nonlinear diffusion processes, as what we actually show is that for a universal constant We believe our techniques are flexible and can be adapted to various models and contexts.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"1539 - 1558"},"PeriodicalIF":1.9,"publicationDate":"2022-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43226099","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 : 2022-04-05DOI: 10.1080/03605302.2022.2084628
A. Bressan, S. Galtung, Katrin Grunert, K. Nguyen
Abstract This paper provides an asymptotic description of a solution to the Burgers-Hilbert equation in a neighborhood of a point where two shocks interact. The solution is obtained as the sum of a function with H 2 regularity away from the shocks plus a corrector term having an asymptotic behavior like close to each shock. A key step in the analysis is the construction of piecewise smooth solutions with a single shock for a general class of initial data.
{"title":"Shock interactions for the Burgers-Hilbert equation","authors":"A. Bressan, S. Galtung, Katrin Grunert, K. Nguyen","doi":"10.1080/03605302.2022.2084628","DOIUrl":"https://doi.org/10.1080/03605302.2022.2084628","url":null,"abstract":"Abstract This paper provides an asymptotic description of a solution to the Burgers-Hilbert equation in a neighborhood of a point where two shocks interact. The solution is obtained as the sum of a function with H 2 regularity away from the shocks plus a corrector term having an asymptotic behavior like close to each shock. A key step in the analysis is the construction of piecewise smooth solutions with a single shock for a general class of initial data.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"1795 - 1844"},"PeriodicalIF":1.9,"publicationDate":"2022-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48607242","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 : 2022-04-04DOI: 10.1080/03605302.2023.2169939
Madelyne M. Brown
Abstract Let (M, g) be a smooth, compact, Riemannian manifold and a sequence of L 2-normalized Laplace eigenfunctions on M. For a smooth submanifold we consider the growth of the restricted eigenfunctions by testing them against a sequence of functions on H whose wavefront set avoids That is, we study what we call the generalized Fourier coefficients: We give an explicit bound on these coefficients depending on how the defect measures for the two collections of functions and ψh relate. This allows us to get a little– o improvement whenever the collection of recurrent directions over the wavefront set of ψh is small. To obtain our estimates, we utilize geodesic beam techniques.
{"title":"On the growth of generalized Fourier coefficients of restricted eigenfunctions","authors":"Madelyne M. Brown","doi":"10.1080/03605302.2023.2169939","DOIUrl":"https://doi.org/10.1080/03605302.2023.2169939","url":null,"abstract":"Abstract Let (M, g) be a smooth, compact, Riemannian manifold and a sequence of L 2-normalized Laplace eigenfunctions on M. For a smooth submanifold we consider the growth of the restricted eigenfunctions by testing them against a sequence of functions on H whose wavefront set avoids That is, we study what we call the generalized Fourier coefficients: We give an explicit bound on these coefficients depending on how the defect measures for the two collections of functions and ψh relate. This allows us to get a little– o improvement whenever the collection of recurrent directions over the wavefront set of ψh is small. To obtain our estimates, we utilize geodesic beam techniques.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"252 - 285"},"PeriodicalIF":1.9,"publicationDate":"2022-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45088392","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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