Pub Date : 2020-08-05DOI: 10.1142/s0219891621500156
Tomasz Cieślak, Jakub Siemianowski
We study here a Hamilton–Jacobi equation with a quadratic and degenerate Hamiltonian, which comes from the dynamics of a multipeakon in the Camassa–Holm equation. It is given by a quadratic form with a singular positive semi-definite matrix. We increase the regularity of the value function considered in earlier works, which is known to be the viscosity solution. We prove that for a two-peakon Hamiltonian such solutions are actually [Formula: see text]-Hölder continuous in space and time-Lipschitz continuous. The time-Lipschitz regularity is proven in any dimension [Formula: see text]. Such a regularity is already known in the one-dimensional case and, moreover it is the best possible, as shown earlier.
{"title":"Existence of viscosity solutions with the optimal regularity of a two-peakon Hamilton–Jacobi equation","authors":"Tomasz Cieślak, Jakub Siemianowski","doi":"10.1142/s0219891621500156","DOIUrl":"https://doi.org/10.1142/s0219891621500156","url":null,"abstract":"We study here a Hamilton–Jacobi equation with a quadratic and degenerate Hamiltonian, which comes from the dynamics of a multipeakon in the Camassa–Holm equation. It is given by a quadratic form with a singular positive semi-definite matrix. We increase the regularity of the value function considered in earlier works, which is known to be the viscosity solution. We prove that for a two-peakon Hamiltonian such solutions are actually [Formula: see text]-Hölder continuous in space and time-Lipschitz continuous. The time-Lipschitz regularity is proven in any dimension [Formula: see text]. Such a regularity is already known in the one-dimensional case and, moreover it is the best possible, as shown earlier.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45652161","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 : 2020-07-30DOI: 10.1142/S0219891621500272
Kihyun Kim, R. Schippa
New low regularity well-posedness results for the generalized Benjamin–Ono equations with quartic or higher nonlinearity and periodic boundary conditions are shown. We use the short-time Fourier transform restriction method and modified energies to overcome the derivative loss. Previously, Molinet–Ribaud established local well-posedness in [Formula: see text] via gauge transforms. We show local existence and a priori estimates in [Formula: see text], [Formula: see text], and local well-posedness in [Formula: see text], [Formula: see text] without using gauge transforms. In case of quartic nonlinearity we prove global existence of solutions conditional upon small initial data.
{"title":"Low regularity well-posedness for generalized Benjamin–Ono equations on the circle","authors":"Kihyun Kim, R. Schippa","doi":"10.1142/S0219891621500272","DOIUrl":"https://doi.org/10.1142/S0219891621500272","url":null,"abstract":"New low regularity well-posedness results for the generalized Benjamin–Ono equations with quartic or higher nonlinearity and periodic boundary conditions are shown. We use the short-time Fourier transform restriction method and modified energies to overcome the derivative loss. Previously, Molinet–Ribaud established local well-posedness in [Formula: see text] via gauge transforms. We show local existence and a priori estimates in [Formula: see text], [Formula: see text], and local well-posedness in [Formula: see text], [Formula: see text] without using gauge transforms. In case of quartic nonlinearity we prove global existence of solutions conditional upon small initial data.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47534446","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 : 2020-06-01DOI: 10.1142/s0219891620500071
C. Bauzet, V. Castel, J. Charrier
We are interested in multi-dimensional nonlinear scalar conservation laws forced by a multiplicative stochastic noise with a general time and space dependent flux-function. We address simultaneously theoretical and numerical issues in a general framework and consider a larger class of flux functions in comparison to the one in the literature. We establish existence and uniqueness of a stochastic entropy solution together with the convergence of a finite volume scheme. The novelty of this paper is the use of a numerical approximation (instead of a viscous one) in order to get, both, the existence and the uniqueness of solutions. The quantitative bounds in our uniqueness result constitute a preliminary step toward the establishment of strong error estimates. We also provide an [Formula: see text] stability result for the stochastic entropy solution.
{"title":"Existence and uniqueness result for an hyperbolic scalar conservation law with a stochastic force using a finite volume approximation","authors":"C. Bauzet, V. Castel, J. Charrier","doi":"10.1142/s0219891620500071","DOIUrl":"https://doi.org/10.1142/s0219891620500071","url":null,"abstract":"We are interested in multi-dimensional nonlinear scalar conservation laws forced by a multiplicative stochastic noise with a general time and space dependent flux-function. We address simultaneously theoretical and numerical issues in a general framework and consider a larger class of flux functions in comparison to the one in the literature. We establish existence and uniqueness of a stochastic entropy solution together with the convergence of a finite volume scheme. The novelty of this paper is the use of a numerical approximation (instead of a viscous one) in order to get, both, the existence and the uniqueness of solutions. The quantitative bounds in our uniqueness result constitute a preliminary step toward the establishment of strong error estimates. We also provide an [Formula: see text] stability result for the stochastic entropy solution.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/s0219891620500071","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47009377","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 : 2020-06-01DOI: 10.1142/s0219891620500101
Halit Sevki Aslan, M. Reissig
We study the global (in time) existence of small data solutions to some Cauchy problems for semilinear damped wave models with strong time-dependent oscillations. The goal is to understand the influence of strong oscillations in the coefficients on solutions to some semilinear models with an “effective-like” damping term, where the data are supposed to belong to different classes of regularity.
{"title":"Influence of strong time-dependent oscillations in semilinear damped wave models","authors":"Halit Sevki Aslan, M. Reissig","doi":"10.1142/s0219891620500101","DOIUrl":"https://doi.org/10.1142/s0219891620500101","url":null,"abstract":"We study the global (in time) existence of small data solutions to some Cauchy problems for semilinear damped wave models with strong time-dependent oscillations. The goal is to understand the influence of strong oscillations in the coefficients on solutions to some semilinear models with an “effective-like” damping term, where the data are supposed to belong to different classes of regularity.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"17 1","pages":"395-442"},"PeriodicalIF":0.7,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/s0219891620500101","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46291061","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 : 2020-05-26DOI: 10.1142/s0219891621500193
Pascal Millet
The main topic of this paper is the Goursat problem at the horizon for the Klein–Gordon equation on the De Sitter–Kerr metric when the angular momentum (per unit of mass) of the black hole is small. Indeed, we solve the Goursat problem for fixed angular momentum [Formula: see text] of the field (with the restriction that [Formula: see text] is not zero in the case of a massless field).
{"title":"The Goursat problem at the horizons for the Klein–Gordon equation on the de Sitter–Kerr metric","authors":"Pascal Millet","doi":"10.1142/s0219891621500193","DOIUrl":"https://doi.org/10.1142/s0219891621500193","url":null,"abstract":"The main topic of this paper is the Goursat problem at the horizon for the Klein–Gordon equation on the De Sitter–Kerr metric when the angular momentum (per unit of mass) of the black hole is small. Indeed, we solve the Goursat problem for fixed angular momentum [Formula: see text] of the field (with the restriction that [Formula: see text] is not zero in the case of a massless field).","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45127770","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 : 2020-05-12DOI: 10.1142/s021989162050006x
M. S. Goudiaby, G. Kreiss
We consider a subcritical flow in a sudden expansion canal. The flow is given by 1D Saint-Venant equations on each side of the expansion together with mass conservation and Borda–Carnot conditions ...
{"title":"Existence result for the coupling of shallow water and Borda–Carnot equations with Riemann data","authors":"M. S. Goudiaby, G. Kreiss","doi":"10.1142/s021989162050006x","DOIUrl":"https://doi.org/10.1142/s021989162050006x","url":null,"abstract":"We consider a subcritical flow in a sudden expansion canal. The flow is given by 1D Saint-Venant equations on each side of the expansion together with mass conservation and Borda–Carnot conditions ...","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"17 1","pages":"185-212"},"PeriodicalIF":0.7,"publicationDate":"2020-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/s021989162050006x","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47757085","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 : 2020-03-01DOI: 10.1142/s0219891620500058
Mark E. Williams
We prove energy estimates for exact solutions to a class of linear, weakly stable, first-order hyperbolic boundary problems with “large”, oscillatory, zeroth-order coefficients, that is, coefficients whose amplitude is large, [Formula: see text], compared to the wavelength of the oscillations, [Formula: see text]. The methods that have been used previously to prove useful energy estimates for weakly stable problems with oscillatory coefficients (e.g. simultaneous diagonalization of first-order and zeroth-order parts) all appear to fail in the presence of such large coefficients. We show that our estimates provide a way to “justify geometric optics”, that is, a way to decide whether or not approximate solutions, constructed for example by geometric optics, are close to the exact solutions on a time interval independent of [Formula: see text]. Systems of this general type arise in some classical problems of “strongly nonlinear geometric optics” coming from fluid mechanics. Special assumptions that we make here do not yet allow us to treat the latter problems, but we believe the present analysis will provide some guidance on how to attack more general cases.
{"title":"Weakly stable hyperbolic boundary problems with large oscillatory coefficients: Simple cascades","authors":"Mark E. Williams","doi":"10.1142/s0219891620500058","DOIUrl":"https://doi.org/10.1142/s0219891620500058","url":null,"abstract":"We prove energy estimates for exact solutions to a class of linear, weakly stable, first-order hyperbolic boundary problems with “large”, oscillatory, zeroth-order coefficients, that is, coefficients whose amplitude is large, [Formula: see text], compared to the wavelength of the oscillations, [Formula: see text]. The methods that have been used previously to prove useful energy estimates for weakly stable problems with oscillatory coefficients (e.g. simultaneous diagonalization of first-order and zeroth-order parts) all appear to fail in the presence of such large coefficients. We show that our estimates provide a way to “justify geometric optics”, that is, a way to decide whether or not approximate solutions, constructed for example by geometric optics, are close to the exact solutions on a time interval independent of [Formula: see text]. Systems of this general type arise in some classical problems of “strongly nonlinear geometric optics” coming from fluid mechanics. Special assumptions that we make here do not yet allow us to treat the latter problems, but we believe the present analysis will provide some guidance on how to attack more general cases.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"17 1","pages":"141-183"},"PeriodicalIF":0.7,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/s0219891620500058","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43145610","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 : 2020-03-01DOI: 10.1142/s0219891620500046
L. Ferreira, J. E. Pérez-López
We show global-in-time well-posedness and self-similarity for the semilinear wave equation with nonlinearity [Formula: see text] in a time-weighted framework based on the larger family of homogeneous Besov spaces [Formula: see text] for [Formula: see text]. As a consequence, in some cases of the power [Formula: see text], we cover a initial-data class larger than in some previous results. Our approach relies on dispersive-type estimates and a suitable [Formula: see text]-product estimate in Besov spaces.
{"title":"Global well-posedness and self-similarity for semilinear wave equations in a time-weighted framework of Besov type","authors":"L. Ferreira, J. E. Pérez-López","doi":"10.1142/s0219891620500046","DOIUrl":"https://doi.org/10.1142/s0219891620500046","url":null,"abstract":"We show global-in-time well-posedness and self-similarity for the semilinear wave equation with nonlinearity [Formula: see text] in a time-weighted framework based on the larger family of homogeneous Besov spaces [Formula: see text] for [Formula: see text]. As a consequence, in some cases of the power [Formula: see text], we cover a initial-data class larger than in some previous results. Our approach relies on dispersive-type estimates and a suitable [Formula: see text]-product estimate in Besov spaces.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"17 1","pages":"123-139"},"PeriodicalIF":0.7,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/s0219891620500046","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43504630","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 : 2020-03-01DOI: 10.1142/s0219891620500034
F. Colombini, T. Nishitani
We consider the Cauchy problem for second-order differential operators with two independent variables [Formula: see text]. Assuming that [Formula: see text] is a nonnegative [Formula: see text] function and [Formula: see text] is a nonnegative Gevrey function of order [Formula: see text], we prove that the Cauchy problem for [Formula: see text] is well-posed in the Gevrey class of any order [Formula: see text] with [Formula: see text].
{"title":"On the Cauchy problem for Dt2 − Dx(b(t)a(x))Dx","authors":"F. Colombini, T. Nishitani","doi":"10.1142/s0219891620500034","DOIUrl":"https://doi.org/10.1142/s0219891620500034","url":null,"abstract":"We consider the Cauchy problem for second-order differential operators with two independent variables [Formula: see text]. Assuming that [Formula: see text] is a nonnegative [Formula: see text] function and [Formula: see text] is a nonnegative Gevrey function of order [Formula: see text], we prove that the Cauchy problem for [Formula: see text] is well-posed in the Gevrey class of any order [Formula: see text] with [Formula: see text].","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"17 1","pages":"75-122"},"PeriodicalIF":0.7,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/s0219891620500034","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48627697","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 : 2020-02-13DOI: 10.1142/s0219891622500011
Elena Giorgi
We derive the equations governing the linear stability of Kerr–Newman spacetime to coupled electromagnetic-gravitational perturbations. The equations generalize the celebrated Teukolsky equation for curvature perturbations of Kerr, and the Regge–Wheeler equation for metric perturbations of Reissner–Nordström. Because of the “apparent indissolubility of the coupling between the spin-1 and spin-2 fields”, as put by Chandrasekhar, the stability of Kerr–Newman spacetime cannot be obtained through standard decomposition in modes. Due to the impossibility to decouple the modes of the gravitational and electromagnetic fields, the equations governing the linear stability of Kerr–Newman have not been previously derived. Using a tensorial approach that was applied to Kerr, we produce a set of generalized Regge–Wheeler equations for perturbations of Kerr–Newman, which are suitable for the study of linearized stability by physical space methods. The physical space analysis overcomes the issue of coupling of spin-1 and spin-2 fields and represents the first step towards an analytical proof of the stability of the Kerr–Newman black hole.
{"title":"Electromagnetic-gravitational perturbations of Kerr–Newman spacetime: The Teukolsky and Regge–Wheeler equations","authors":"Elena Giorgi","doi":"10.1142/s0219891622500011","DOIUrl":"https://doi.org/10.1142/s0219891622500011","url":null,"abstract":"We derive the equations governing the linear stability of Kerr–Newman spacetime to coupled electromagnetic-gravitational perturbations. The equations generalize the celebrated Teukolsky equation for curvature perturbations of Kerr, and the Regge–Wheeler equation for metric perturbations of Reissner–Nordström. Because of the “apparent indissolubility of the coupling between the spin-1 and spin-2 fields”, as put by Chandrasekhar, the stability of Kerr–Newman spacetime cannot be obtained through standard decomposition in modes. Due to the impossibility to decouple the modes of the gravitational and electromagnetic fields, the equations governing the linear stability of Kerr–Newman have not been previously derived. Using a tensorial approach that was applied to Kerr, we produce a set of generalized Regge–Wheeler equations for perturbations of Kerr–Newman, which are suitable for the study of linearized stability by physical space methods. The physical space analysis overcomes the issue of coupling of spin-1 and spin-2 fields and represents the first step towards an analytical proof of the stability of the Kerr–Newman black hole.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47975541","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}
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