Pub Date : 2020-01-24DOI: 10.1142/s0219891621500260
Seungly Oh, A. Stefanov
For generalized Korteweg–De Vries (KdV) models with polynomial nonlinearity, we establish a local smoothing property in [Formula: see text] for [Formula: see text]. Such smoothing effect persists globally, provided that the [Formula: see text] norm does not blow up in finite time. More specifically, we show that a translate of the nonlinear part of the solution gains [Formula: see text] derivatives for [Formula: see text]. Following a new simple method, which is of independent interest, we establish that, for [Formula: see text], [Formula: see text] norm of a solution grows at most by [Formula: see text] if [Formula: see text] norm is a priori controlled.
{"title":"Smoothing and growth bound of periodic generalized Korteweg–De Vries equation","authors":"Seungly Oh, A. Stefanov","doi":"10.1142/s0219891621500260","DOIUrl":"https://doi.org/10.1142/s0219891621500260","url":null,"abstract":"For generalized Korteweg–De Vries (KdV) models with polynomial nonlinearity, we establish a local smoothing property in [Formula: see text] for [Formula: see text]. Such smoothing effect persists globally, provided that the [Formula: see text] norm does not blow up in finite time. More specifically, we show that a translate of the nonlinear part of the solution gains [Formula: see text] derivatives for [Formula: see text]. Following a new simple method, which is of independent interest, we establish that, for [Formula: see text], [Formula: see text] norm of a solution grows at most by [Formula: see text] if [Formula: see text] norm is a priori controlled.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45606017","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-03DOI: 10.1142/s0219891621500181
Antoine Benoit
We are interested in geometric optics expansions for linear hyperbolic systems of equations defined in the strip [Formula: see text]. More precisely the aim of this paper is to describe the influence of the boundary conditions on the behavior of the solution. This question has already been addressed in [A. Benoit, Wkb expansions for hyperbolic boundary value problems in a strip: Selfinteraction meets strong well-posedness, J. Inst. Math. Jussieu 19(5) (2020) 1629–1675] for stable boundary conditions. Here we do not require that the boundary conditions lead to strongly well-posed problems but only to weakly well-posed problems (that is loss(es) of derivatives are possible). The question is thus to determine what can be the minimal loss of derivatives in the energy estimate of the solution. The main result of this paper is to show, thanks to geometric optics expansions, that if the strip problem admits a boundary in the so-called [Formula: see text]-class of [S. Benzoni-Gavage, F. Rousset, D. Serre and K. Zumbrun, Generic types and transitions in hyperbolic initial-boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A 132(5) (2002) 1073–1104] then the loss of derivatives shall be at least increasing with the time of resolution. More precisely this loss is bounded by below by a step function increasing with respect to time which depends on the minimal time needed to perform a full regeneration of the wave packet.
{"title":"WKB expansions for weakly well-posed hyperbolic boundary value problems in a strip: Time depending loss of derivatives","authors":"Antoine Benoit","doi":"10.1142/s0219891621500181","DOIUrl":"https://doi.org/10.1142/s0219891621500181","url":null,"abstract":"We are interested in geometric optics expansions for linear hyperbolic systems of equations defined in the strip [Formula: see text]. More precisely the aim of this paper is to describe the influence of the boundary conditions on the behavior of the solution. This question has already been addressed in [A. Benoit, Wkb expansions for hyperbolic boundary value problems in a strip: Selfinteraction meets strong well-posedness, J. Inst. Math. Jussieu 19(5) (2020) 1629–1675] for stable boundary conditions. Here we do not require that the boundary conditions lead to strongly well-posed problems but only to weakly well-posed problems (that is loss(es) of derivatives are possible). The question is thus to determine what can be the minimal loss of derivatives in the energy estimate of the solution. The main result of this paper is to show, thanks to geometric optics expansions, that if the strip problem admits a boundary in the so-called [Formula: see text]-class of [S. Benzoni-Gavage, F. Rousset, D. Serre and K. Zumbrun, Generic types and transitions in hyperbolic initial-boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A 132(5) (2002) 1073–1104] then the loss of derivatives shall be at least increasing with the time of resolution. More precisely this loss is bounded by below by a step function increasing with respect to time which depends on the minimal time needed to perform a full regeneration of the wave packet.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2019-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45993363","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-01DOI: 10.1142/s021989161950022x
Yanni Zeng
We study the time asymptotic decay of solutions for a general system of hyperbolic–parabolic balance laws in one space dimension. The system has a physical viscosity matrix and a lower-order term f...
研究一维双曲-抛物平衡律一般系统解的时间渐近衰减问题。系统具有物理粘度矩阵和低阶项f…
{"title":"Lp time asymptotic decay for general hyperbolic–parabolic balance laws with applications","authors":"Yanni Zeng","doi":"10.1142/s021989161950022x","DOIUrl":"https://doi.org/10.1142/s021989161950022x","url":null,"abstract":"We study the time asymptotic decay of solutions for a general system of hyperbolic–parabolic balance laws in one space dimension. The system has a physical viscosity matrix and a lower-order term f...","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"16 1","pages":"663-700"},"PeriodicalIF":0.7,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/s021989161950022x","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45161967","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-01DOI: 10.1142/s0219891619500231
Xiaopeng Zhao
We study the density-dependent incompressible Cahn–Hilliard–Navier–Stokes system, which describes a two-phase flow of two incompressible fluids with different densities. We establish the local exis...
{"title":"Strong solutions to the density-dependent incompressible Cahn–Hilliard–Navier–Stokes system","authors":"Xiaopeng Zhao","doi":"10.1142/s0219891619500231","DOIUrl":"https://doi.org/10.1142/s0219891619500231","url":null,"abstract":"We study the density-dependent incompressible Cahn–Hilliard–Navier–Stokes system, which describes a two-phase flow of two incompressible fluids with different densities. We establish the local exis...","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"16 1","pages":"701-742"},"PeriodicalIF":0.7,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/s0219891619500231","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47365128","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-01DOI: 10.1142/s0219891619500218
Zhen Wang, Xing-Ping Wu
We establish a well-posedness theory and a blow-up criterion for the Chaplygin gas equations in ℝN for any dimension N ≥ 1. First, given ω = 1 ρ, ℋ↪𝒞0,1, we prove the well-posedness property for so...
对于任意维数N≥1的chplygin气体方程，我们建立了适定性理论和爆破判据。首先，在给定ω = 1 ρ， h ' ' '𝒞0,1的条件下，证明了h = 1 ρ， h ' ' '𝒞0,1的适定性。
{"title":"Well-posedness and blow-up criterion for the Chaplygin gas equations in ℝN","authors":"Zhen Wang, Xing-Ping Wu","doi":"10.1142/s0219891619500218","DOIUrl":"https://doi.org/10.1142/s0219891619500218","url":null,"abstract":"We establish a well-posedness theory and a blow-up criterion for the Chaplygin gas equations in ℝN for any dimension N ≥ 1. First, given ω = 1 ρ, ℋ↪𝒞0,1, we prove the well-posedness property for so...","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"16 1","pages":"639-661"},"PeriodicalIF":0.7,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/s0219891619500218","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42755872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-10-21DOI: 10.1142/s0219891619500164
Yachun Li, Zhaoyang Shang
We consider the planar compressible magnetohydrodynamics (MHD) system for a viscous and heat-conducting ideal polytropic gas, when the viscosity, magnetic diffusion and heat conductivity depend on the specific volume [Formula: see text] and the temperature [Formula: see text]. For technical reasons, the viscosity coefficients, magnetic diffusion and heat conductivity are assumed to be proportional to [Formula: see text] where [Formula: see text] is a non-degenerate and smooth function satisfying some natural conditions. We prove the existence and uniqueness of the global-in-time classical solution to the initial-boundary value problem when general large initial data are prescribed and the exponent [Formula: see text] is sufficiently small. A similar result is also established for planar Hall-MHD equations.
{"title":"Global large solutions to planar magnetohydrodynamics equations with temperature-dependent coefficients","authors":"Yachun Li, Zhaoyang Shang","doi":"10.1142/s0219891619500164","DOIUrl":"https://doi.org/10.1142/s0219891619500164","url":null,"abstract":"We consider the planar compressible magnetohydrodynamics (MHD) system for a viscous and heat-conducting ideal polytropic gas, when the viscosity, magnetic diffusion and heat conductivity depend on the specific volume [Formula: see text] and the temperature [Formula: see text]. For technical reasons, the viscosity coefficients, magnetic diffusion and heat conductivity are assumed to be proportional to [Formula: see text] where [Formula: see text] is a non-degenerate and smooth function satisfying some natural conditions. We prove the existence and uniqueness of the global-in-time classical solution to the initial-boundary value problem when general large initial data are prescribed and the exponent [Formula: see text] is sufficiently small. A similar result is also established for planar Hall-MHD equations.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2019-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/s0219891619500164","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47857152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-10-09DOI: 10.1142/S0219891620500241
H. Pecher
The local well-posedness problem is considered for the Dirac–Klein–Gordon system in two space dimensions for data in Fourier–Lebesgue spaces [Formula: see text], where [Formula: see text] and [Formula: see text] and [Formula: see text] denote dual exponents. We lower the regularity assumptions on the data with respect to scaling improving the results of d’Ancona et al. in the classical case [Formula: see text]. Crucial is the fact that the nonlinearities fulfill a null condition as detected by these authors.
{"title":"Local well-posedness of the two-dimensional Dirac–Klein–Gordon equations in Fourier–Lebesgue spaces","authors":"H. Pecher","doi":"10.1142/S0219891620500241","DOIUrl":"https://doi.org/10.1142/S0219891620500241","url":null,"abstract":"The local well-posedness problem is considered for the Dirac–Klein–Gordon system in two space dimensions for data in Fourier–Lebesgue spaces [Formula: see text], where [Formula: see text] and [Formula: see text] and [Formula: see text] denote dual exponents. We lower the regularity assumptions on the data with respect to scaling improving the results of d’Ancona et al. in the classical case [Formula: see text]. Crucial is the fact that the nonlinearities fulfill a null condition as detected by these authors.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2019-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46045807","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-08-21DOI: 10.1142/S0219891619500127
Achenef Tesfahun
We study the growth-in-time of higher order Sobolev norms of solutions to the Dirac–Klein–Gordon (DKG) equations in one space dimension. We show that these norms grow at most polynomially-in-time. The main ingredients in the proof are the upside-down [Formula: see text]-method which was introduced by Colliander, Keel, Staffilani, Takaoka and Tao, and bilinear null-form estimates.
研究一维Dirac-Klein-Gordon (DKG)方程解的高阶Sobolev范数随时间的增长。我们证明这些范数最多以多项式随时间增长。证明的主要成分是由Colliander, Keel, Staffilani, Takaoka和Tao引入的倒立[公式:见文本]方法，以及双线性零形式估计。
{"title":"Growth-in-time of higher Sobolev norms of solutions to the 1D Dirac–Klein–Gordon system","authors":"Achenef Tesfahun","doi":"10.1142/S0219891619500127","DOIUrl":"https://doi.org/10.1142/S0219891619500127","url":null,"abstract":"We study the growth-in-time of higher order Sobolev norms of solutions to the Dirac–Klein–Gordon (DKG) equations in one space dimension. We show that these norms grow at most polynomially-in-time. The main ingredients in the proof are the upside-down [Formula: see text]-method which was introduced by Colliander, Keel, Staffilani, Takaoka and Tao, and bilinear null-form estimates.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2019-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/S0219891619500127","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41906778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-08-21DOI: 10.1142/S0219891619500085
D. Kong, Qi Liu, Changming Song
We investigate a dissipative hyperbolic geometry flow in two space variables for which a new nonlinear wave equation is derived. Based on an energy method, the global existence of solutions to the dissipative hyperbolic geometry flow is established. Furthermore, the scalar curvature of the metric remains uniformly bounded. Moreover, under suitable assumptions, we establish the global existence of classical solutions to the Cauchy problem, and we show that the solution and its derivative decay to zero as the time tends to infinity. In addition, the scalar curvature of the solution metric converges to the one of the flat metric at an algebraic rate.
{"title":"Classical solutions to a dissipative hyperbolic geometry flow in two space variables","authors":"D. Kong, Qi Liu, Changming Song","doi":"10.1142/S0219891619500085","DOIUrl":"https://doi.org/10.1142/S0219891619500085","url":null,"abstract":"We investigate a dissipative hyperbolic geometry flow in two space variables for which a new nonlinear wave equation is derived. Based on an energy method, the global existence of solutions to the dissipative hyperbolic geometry flow is established. Furthermore, the scalar curvature of the metric remains uniformly bounded. Moreover, under suitable assumptions, we establish the global existence of classical solutions to the Cauchy problem, and we show that the solution and its derivative decay to zero as the time tends to infinity. In addition, the scalar curvature of the solution metric converges to the one of the flat metric at an algebraic rate.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2019-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/S0219891619500085","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45788218","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-08-21DOI: 10.1142/S0219891619500097
A. Barbagallo, V. Esposito
We establish several a priori estimates of local or global nature in Sobolev spaces with general exponent [Formula: see text] for a class of second-order hyperbolic operators with double characteristics in presence of a transition in a domain of the Euclidian space [Formula: see text].
{"title":"A priori estimates in Sobolev spaces for a class of hyperbolic operators in presence of transition","authors":"A. Barbagallo, V. Esposito","doi":"10.1142/S0219891619500097","DOIUrl":"https://doi.org/10.1142/S0219891619500097","url":null,"abstract":"We establish several a priori estimates of local or global nature in Sobolev spaces with general exponent [Formula: see text] for a class of second-order hyperbolic operators with double characteristics in presence of a transition in a domain of the Euclidian space [Formula: see text].","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2019-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/S0219891619500097","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48089808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：