Pub Date : 2023-01-11DOI: 10.1080/03605302.2022.2164303
Marius Müller
Abstract This erratum points out an error in “The Poisson equation involving surface measures” (Vol. 47 of Communications in Partial Differential Equations, (2022)) and provides a counterexample and discussion of the erroneous theorem.
{"title":"Erratum: the Poisson equation involving surface measures","authors":"Marius Müller","doi":"10.1080/03605302.2022.2164303","DOIUrl":"https://doi.org/10.1080/03605302.2022.2164303","url":null,"abstract":"Abstract This erratum points out an error in “The Poisson equation involving surface measures” (Vol. 47 of Communications in Partial Differential Equations, (2022)) and provides a counterexample and discussion of the erroneous theorem.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"350 - 353"},"PeriodicalIF":1.9,"publicationDate":"2023-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42933567","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 : 2022-11-29DOI: 10.1080/03605302.2023.2238953
Y. Achdou, Claude Le Bris
Abstract We study homogenization for a class of stationary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, supplemented at the origin with an effective Dirichlet condition that keeps track of the perturbation. Various comments and extensions are discussed.
{"title":"Homogenization of some periodic Hamilton-Jacobi equations with defects","authors":"Y. Achdou, Claude Le Bris","doi":"10.1080/03605302.2023.2238953","DOIUrl":"https://doi.org/10.1080/03605302.2023.2238953","url":null,"abstract":"Abstract We study homogenization for a class of stationary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, supplemented at the origin with an effective Dirichlet condition that keeps track of the perturbation. Various comments and extensions are discussed.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"944 - 986"},"PeriodicalIF":1.9,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43912390","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 : 2022-10-31DOI: 10.1080/03605302.2023.2212479
P. Druet, Katharina Hopf, A. Jüngel
Abstract We investigate degenerate cross-diffusion equations, with a rank-deficient diffusion-matrix, modelling multispecies population dynamics driven by partial pressure gradients. These equations have recently been found to arise in a mean-field limit of interacting stochastic particle systems. To date, their analysis in multiple space dimensions has been confined to the purely convective case with equal mobility coefficients. In this article, we introduce a normal form for an entropic class of such equations which reveals their structure of a symmetric hyperbolic–parabolic system. Due to the state-dependence of the range and kernel of the singular diffusive matrix, our way of rewriting the equations is different from that classically used for symmetric second-order systems with a nullspace invariance property. By means of this change of variables, we solve the Cauchy problem for short times and positive initial data in for
{"title":"Hyperbolic–parabolic normal form and local classical solutions for cross-diffusion systems with incomplete diffusion","authors":"P. Druet, Katharina Hopf, A. Jüngel","doi":"10.1080/03605302.2023.2212479","DOIUrl":"https://doi.org/10.1080/03605302.2023.2212479","url":null,"abstract":"Abstract We investigate degenerate cross-diffusion equations, with a rank-deficient diffusion-matrix, modelling multispecies population dynamics driven by partial pressure gradients. These equations have recently been found to arise in a mean-field limit of interacting stochastic particle systems. To date, their analysis in multiple space dimensions has been confined to the purely convective case with equal mobility coefficients. In this article, we introduce a normal form for an entropic class of such equations which reveals their structure of a symmetric hyperbolic–parabolic system. Due to the state-dependence of the range and kernel of the singular diffusive matrix, our way of rewriting the equations is different from that classically used for symmetric second-order systems with a nullspace invariance property. By means of this change of variables, we solve the Cauchy problem for short times and positive initial data in for","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"863 - 894"},"PeriodicalIF":1.9,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45966347","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 : 2022-10-25DOI: 10.1080/03605302.2022.2129383
S. Schochet, Xin Xu
Abstract Uniform existence of solutions to initial-value problems and convergence of appropriately filtered solutions are proven for a special class of three-scale singular limit equations, without any restriction on the initial data. The uniform existence is proven using a novel system of energy estimates. The convergence result is based on a detailed analysis of the fastest-scale oscillations, which unlike in two-scale systems have no explicit solution formula.
{"title":"Toward uniform existence and convergence theorems for three-scale systems of hyperbolic PDEs with general initial data","authors":"S. Schochet, Xin Xu","doi":"10.1080/03605302.2022.2129383","DOIUrl":"https://doi.org/10.1080/03605302.2022.2129383","url":null,"abstract":"Abstract Uniform existence of solutions to initial-value problems and convergence of appropriately filtered solutions are proven for a special class of three-scale singular limit equations, without any restriction on the initial data. The uniform existence is proven using a novel system of energy estimates. The convergence result is based on a detailed analysis of the fastest-scale oscillations, which unlike in two-scale systems have no explicit solution formula.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"2401 - 2443"},"PeriodicalIF":1.9,"publicationDate":"2022-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45054278","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 : 2022-10-18DOI: 10.1080/03605302.2022.2122836
A. Stevens, M. Winkler
Abstract The degenerate Keller-Segel type system is considered in balls with R > 0 and m > 1. Our main results reveal that throughout the entire degeneracy range the interplay between degenerate diffusion and cross-diffusive attraction herein can enforce persistent localization of solutions inside a compact subset of Ω, no matter whether solutions remain bounded or blow up. More precisely, it is shown that for arbitrary and one can find such that if and is nonnegative and radially symmetric with and then a corresponding zero-flux type initial-boundary value problem admits a radial weak solution (u, v), extensible up to a maximal time and satisfying if which has the additional property that In particular, this conclusion is seen to be valid whenever u 0 is radially nonincreasing with
{"title":"Taxis-driven persistent localization in a degenerate Keller-Segel system","authors":"A. Stevens, M. Winkler","doi":"10.1080/03605302.2022.2122836","DOIUrl":"https://doi.org/10.1080/03605302.2022.2122836","url":null,"abstract":"Abstract The degenerate Keller-Segel type system is considered in balls with R > 0 and m > 1. Our main results reveal that throughout the entire degeneracy range the interplay between degenerate diffusion and cross-diffusive attraction herein can enforce persistent localization of solutions inside a compact subset of Ω, no matter whether solutions remain bounded or blow up. More precisely, it is shown that for arbitrary and one can find such that if and is nonnegative and radially symmetric with and then a corresponding zero-flux type initial-boundary value problem admits a radial weak solution (u, v), extensible up to a maximal time and satisfying if which has the additional property that In particular, this conclusion is seen to be valid whenever u 0 is radially nonincreasing with","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"2341 - 2362"},"PeriodicalIF":1.9,"publicationDate":"2022-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59338677","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 : 2022-09-16DOI: 10.1080/03605302.2023.2241546
Jason Murphy
Abstract We follow up on work of Strauss, Weder, and Watanabe concerning scattering and inverse scattering for nonlinear Schrödinger equations with nonlinearities of the form .
{"title":"Recovery of a spatially-dependent coefficient from the NLS scattering map","authors":"Jason Murphy","doi":"10.1080/03605302.2023.2241546","DOIUrl":"https://doi.org/10.1080/03605302.2023.2241546","url":null,"abstract":"Abstract We follow up on work of Strauss, Weder, and Watanabe concerning scattering and inverse scattering for nonlinear Schrödinger equations with nonlinearities of the form .","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"991 - 1007"},"PeriodicalIF":1.9,"publicationDate":"2022-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46557667","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 : 2022-09-08DOI: 10.1080/03605302.2022.2116717
C. Cheng, Tai-Ping Liu, Shih-Hsien Yu
Abstract The purpose of the present paper is to study the structure of Green’s function for heat equation in several spatial dimensions and with rough heat conductivity coefficient. We take the heat conductivity coefficient to be of bounded variation in the x direction and study the dispersion in the (y, z) direction. The goal is to understand the coupling of dissipation across rough heat conductivity and the multi-dimensional dispersion in the Green’s function A series of exponential functions of path integral with coefficients over a field of complex analytic functions around imaginary axis are formulated in the Laplace and Fourier transforms variables. The Green’s function in the transformed variables is written as the sum of these integrals over random paths. The integral over a random path is rearranged through the reflection property over a variation of heat conductivity coefficient and become a simple form in terms of path phase and amplitude. The complex analytic and combinatorics method is then used to yield a precise pointwise structure of the Green’s function in the physical domain
{"title":"Green’s function of heat equation for heterogeneous media in 3-D","authors":"C. Cheng, Tai-Ping Liu, Shih-Hsien Yu","doi":"10.1080/03605302.2022.2116717","DOIUrl":"https://doi.org/10.1080/03605302.2022.2116717","url":null,"abstract":"Abstract The purpose of the present paper is to study the structure of Green’s function for heat equation in several spatial dimensions and with rough heat conductivity coefficient. We take the heat conductivity coefficient to be of bounded variation in the x direction and study the dispersion in the (y, z) direction. The goal is to understand the coupling of dissipation across rough heat conductivity and the multi-dimensional dispersion in the Green’s function A series of exponential functions of path integral with coefficients over a field of complex analytic functions around imaginary axis are formulated in the Laplace and Fourier transforms variables. The Green’s function in the transformed variables is written as the sum of these integrals over random paths. The integral over a random path is rearranged through the reflection property over a variation of heat conductivity coefficient and become a simple form in terms of path phase and amplitude. The complex analytic and combinatorics method is then used to yield a precise pointwise structure of the Green’s function in the physical domain","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"2180 - 2216"},"PeriodicalIF":1.9,"publicationDate":"2022-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48161808","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 : 2022-09-07DOI: 10.1080/03605302.2023.2175218
Alexandre Rege
Abstract We present two results regarding the three-dimensional Vlasov–Poisson system in the full space with an external magnetic field. First, we investigate the propagation of velocity moments for solutions to the system when the magnetic field is uniform and time-dependent. We combine the classical moment approach with an induction procedure depending on the cyclotron period This allows us to obtain, like in the unmagnetized case, the propagation of velocity moments of order k > 2 in the full space case and of order k > 3 in the periodic case. Second, this time taking a general magnetic field that depends on both time and position, we manage to extend a result by Miot [A uniqueness criterion for unbounded solutions to the Vlasov–Poisson system, 2016] regarding uniqueness for Vlasov–Poisson to the magnetized framework.
{"title":"Propagation of velocity moments and uniqueness for the magnetized Vlasov–Poisson system","authors":"Alexandre Rege","doi":"10.1080/03605302.2023.2175218","DOIUrl":"https://doi.org/10.1080/03605302.2023.2175218","url":null,"abstract":"Abstract We present two results regarding the three-dimensional Vlasov–Poisson system in the full space with an external magnetic field. First, we investigate the propagation of velocity moments for solutions to the system when the magnetic field is uniform and time-dependent. We combine the classical moment approach with an induction procedure depending on the cyclotron period This allows us to obtain, like in the unmagnetized case, the propagation of velocity moments of order k > 2 in the full space case and of order k > 3 in the periodic case. Second, this time taking a general magnetic field that depends on both time and position, we manage to extend a result by Miot [A uniqueness criterion for unbounded solutions to the Vlasov–Poisson system, 2016] regarding uniqueness for Vlasov–Poisson to the magnetized framework.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"386 - 414"},"PeriodicalIF":1.9,"publicationDate":"2022-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45841889","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 : 2022-08-17DOI: 10.1080/03605302.2023.2212477
Alex H. Ardila, Jason Murphy
Abstract We study the cubic-quintic NLS in three space dimensions. It is known that scattering holds for solutions with mass-energy in a region corresponding to positive virial, the boundary of which is delineated both by ground state solitons and by certain rescalings thereof. We classify the possible behaviors of solutions on the part of the boundary attained solely by solitons. In particular, we show that non-soliton solutions either scatter in both time directions or coincide (modulo symmetries) with a special solution, which scatters in one time direction and converges exponentially to the soliton in the other.
{"title":"Threshold solutions for the 3d cubic-quintic NLS","authors":"Alex H. Ardila, Jason Murphy","doi":"10.1080/03605302.2023.2212477","DOIUrl":"https://doi.org/10.1080/03605302.2023.2212477","url":null,"abstract":"Abstract We study the cubic-quintic NLS in three space dimensions. It is known that scattering holds for solutions with mass-energy in a region corresponding to positive virial, the boundary of which is delineated both by ground state solitons and by certain rescalings thereof. We classify the possible behaviors of solutions on the part of the boundary attained solely by solitons. In particular, we show that non-soliton solutions either scatter in both time directions or coincide (modulo symmetries) with a special solution, which scatters in one time direction and converges exponentially to the soliton in the other.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"819 - 862"},"PeriodicalIF":1.9,"publicationDate":"2022-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42435389","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 : 2022-08-16DOI: 10.1080/03605302.2023.2202733
E. Feireisl, Arnab Roy, A. Zarnescu
Abstract We consider the motion of a small rigid object immersed in a viscous compressible fluid in the 3-dimensional Eucleidean space. Assuming the object is a ball of a small radius ε we show that the behavior of the fluid is not influenced by the object in the asymptotic limit The result holds for the isentropic pressure law for any under mild assumptions concerning the rigid body density. In particular, the latter may be bounded as soon as The proof uses a new method of construction of the test functions in the weak formulation of the problem, and, in particular, a new form of the so-called Bogovskii operator.
{"title":"On the motion of a small rigid body in a viscous compressible fluid","authors":"E. Feireisl, Arnab Roy, A. Zarnescu","doi":"10.1080/03605302.2023.2202733","DOIUrl":"https://doi.org/10.1080/03605302.2023.2202733","url":null,"abstract":"Abstract We consider the motion of a small rigid object immersed in a viscous compressible fluid in the 3-dimensional Eucleidean space. Assuming the object is a ball of a small radius ε we show that the behavior of the fluid is not influenced by the object in the asymptotic limit The result holds for the isentropic pressure law for any under mild assumptions concerning the rigid body density. In particular, the latter may be bounded as soon as The proof uses a new method of construction of the test functions in the weak formulation of the problem, and, in particular, a new form of the so-called Bogovskii operator.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"794 - 818"},"PeriodicalIF":1.9,"publicationDate":"2022-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44618489","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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