Pub Date : 2020-12-01DOI: 10.1142/S0219891620500204
Alexander Keimer, M. Singh, Tanya Veeravalli
We study the initial value problem and the initial boundary value problem for nonlocal conservation laws. The nonlocal term is realized via a spatial integration of the solution between specified boundaries and affects the flux function of a given “local” conservation law in a multiplicative way. For a strictly convex flux function and strictly positive nonlocal impact we prove existence and uniqueness of weak entropy solutions relying on a fixed-point argument for the nonlocal term and an explicit Lax–Hopf-type solution formula for the corresponding Hamilton–Jacobi (HJ) equation. Using the developed theory for HJ equations, we obtain a semi-explicit Lax–Hopf-type formula for the solution of the corresponding nonlocal HJ equation and a semi-explicit Lax–Oleinik-type formula for the nonlocal conservation law.
{"title":"Existence and uniqueness results for a class of nonlocal conservation laws by means of a Lax–Hopf-type solution formula","authors":"Alexander Keimer, M. Singh, Tanya Veeravalli","doi":"10.1142/S0219891620500204","DOIUrl":"https://doi.org/10.1142/S0219891620500204","url":null,"abstract":"We study the initial value problem and the initial boundary value problem for nonlocal conservation laws. The nonlocal term is realized via a spatial integration of the solution between specified boundaries and affects the flux function of a given “local” conservation law in a multiplicative way. For a strictly convex flux function and strictly positive nonlocal impact we prove existence and uniqueness of weak entropy solutions relying on a fixed-point argument for the nonlocal term and an explicit Lax–Hopf-type solution formula for the corresponding Hamilton–Jacobi (HJ) equation. Using the developed theory for HJ equations, we obtain a semi-explicit Lax–Hopf-type formula for the solution of the corresponding nonlocal HJ equation and a semi-explicit Lax–Oleinik-type formula for the nonlocal conservation law.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"17 1","pages":"677-705"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44761555","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 : 2020-12-01DOI: 10.1142/S0219891620500216
M. Ohnawa, Masahiro Suzuki
We prove the unique existence of time-periodic solutions to general hyperbolic equations with periodic external forces autonomous or nonautonomous over a domain bounded by two parallel planes, prov...
证明了在两个平行平面上具有自治或非自治周期外力的一般双曲型方程的时间周期解的唯一存在性。
{"title":"Time-periodic solutions of symmetric hyperbolic systems","authors":"M. Ohnawa, Masahiro Suzuki","doi":"10.1142/S0219891620500216","DOIUrl":"https://doi.org/10.1142/S0219891620500216","url":null,"abstract":"We prove the unique existence of time-periodic solutions to general hyperbolic equations with periodic external forces autonomous or nonautonomous over a domain bounded by two parallel planes, prov...","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"17 1","pages":"707-726"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42956443","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 : 2020-11-04DOI: 10.1142/s0219891620500149
Hao-Guang Li, Yachun Li
We consider the Cauchy problem for the three-dimensional, compressible radiation hydrodynamic equations. We establish the existence and uniqueness of local strong solutions for large initial data s...
考虑三维可压缩辐射流体动力学方程的柯西问题。我们建立了大初始数据的局部强解的存在唯一性。
{"title":"Local existence and Serrin-type blow-up criterion for strong solutions to the radiation hydrodynamic equations","authors":"Hao-Guang Li, Yachun Li","doi":"10.1142/s0219891620500149","DOIUrl":"https://doi.org/10.1142/s0219891620500149","url":null,"abstract":"We consider the Cauchy problem for the three-dimensional, compressible radiation hydrodynamic equations. We establish the existence and uniqueness of local strong solutions for large initial data s...","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"17 1","pages":"501-557"},"PeriodicalIF":0.7,"publicationDate":"2020-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46646033","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 : 2020-10-17DOI: 10.1142/s0219891622500023
E. Panov
Under a precise genuine nonlinearity assumption we establish the decay of entropy solutions of a multidimensional scalar conservation law with merely continuous flux and with initial data being a sum of periodic function and a function vanishing at infinity (in the sense of measure).
{"title":"On decay of entropy solutions to multidimensional conservation laws in the case of perturbed periodic initial data","authors":"E. Panov","doi":"10.1142/s0219891622500023","DOIUrl":"https://doi.org/10.1142/s0219891622500023","url":null,"abstract":"Under a precise genuine nonlinearity assumption we establish the decay of entropy solutions of a multidimensional scalar conservation law with merely continuous flux and with initial data being a sum of periodic function and a function vanishing at infinity (in the sense of measure).","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43548135","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 : 2020-09-01DOI: 10.1142/s0219891620500174
Jianjun Chen, G. Lai, Wancheng Sheng
The expansion of a wedge of magnetic fluid into vacuum is studied in this paper. The magnetic fluid away from the sharp corner of a wedge expands into the vacuum as two plane-symmetric rarefaction waves, and the problem can be reduced to the interaction of these two rarefaction waves. In order to determine the flow in the interaction zone, we formulate a Goursat problem for the two-dimensional, self-similar Euler equations of magnetohydrodynamic. This system is of mixed type, and the type at each point is determined by the local fluid velocity and the local magneto-acoustic speed. We establish that the system is uniformly hyperbolic in the interaction zone when the half-angle of the wedge is less than some angle [Formula: see text], while the existence of a global classical solution to the Goursat problem is proven by a method of characteristic decomposition.
{"title":"On the rarefaction waves of the two-dimensional compressible Euler equations for magnetohydrodynamics","authors":"Jianjun Chen, G. Lai, Wancheng Sheng","doi":"10.1142/s0219891620500174","DOIUrl":"https://doi.org/10.1142/s0219891620500174","url":null,"abstract":"The expansion of a wedge of magnetic fluid into vacuum is studied in this paper. The magnetic fluid away from the sharp corner of a wedge expands into the vacuum as two plane-symmetric rarefaction waves, and the problem can be reduced to the interaction of these two rarefaction waves. In order to determine the flow in the interaction zone, we formulate a Goursat problem for the two-dimensional, self-similar Euler equations of magnetohydrodynamic. This system is of mixed type, and the type at each point is determined by the local fluid velocity and the local magneto-acoustic speed. We establish that the system is uniformly hyperbolic in the interaction zone when the half-angle of the wedge is less than some angle [Formula: see text], while the existence of a global classical solution to the Goursat problem is proven by a method of characteristic decomposition.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"17 1","pages":"591-612"},"PeriodicalIF":0.7,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/s0219891620500174","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43456182","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 : 2020-09-01DOI: 10.1142/s0219891620500186
Changhua Wei, Yuzhu Wang
We study here the Cauchy problem associated with the isentropic and compressible Euler equations for Chaplygin gases. Based on the new formulation of the compressible Euler equations in J. Luk and J. Speck [The hidden null structure of the compressible Euler equations and a prelude to applications, J. Hyperbolic Differ. Equ. 17 (2020) 1–60] we show that the wave system satisfied by the modified density and the velocity for Chaplygin gases satisfies the weak null condition. We then prove the global existence of smooth solutions to the irrotational and isentropic Chaplygin gases without introducing a potential function, when the initial data are small perturbations to a constant state.
{"title":"Global smooth solutions to 3D irrotational Euler equations for Chaplygin gases","authors":"Changhua Wei, Yuzhu Wang","doi":"10.1142/s0219891620500186","DOIUrl":"https://doi.org/10.1142/s0219891620500186","url":null,"abstract":"We study here the Cauchy problem associated with the isentropic and compressible Euler equations for Chaplygin gases. Based on the new formulation of the compressible Euler equations in J. Luk and J. Speck [The hidden null structure of the compressible Euler equations and a prelude to applications, J. Hyperbolic Differ. Equ. 17 (2020) 1–60] we show that the wave system satisfied by the modified density and the velocity for Chaplygin gases satisfies the weak null condition. We then prove the global existence of smooth solutions to the irrotational and isentropic Chaplygin gases without introducing a potential function, when the initial data are small perturbations to a constant state.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"17 1","pages":"613-637"},"PeriodicalIF":0.7,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/s0219891620500186","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46378551","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 : 2020-09-01DOI: 10.1142/s0219891620500137
L. Aloui, A. Arama
We consider a system of two coupled wave equations in an exterior domain, where only one equation is directly damped. We prove that the solutions are [Formula: see text]-approximated by special functions, classified into three patterns depending on the values of the damping and the coupling terms, as well as on the speeds of the waves. In particular, when the damping term is sufficiently large, the waves are asymptotically equal to solutions of parabolic-type equations as [Formula: see text]. This result generalizes the standard diffusion phenomenon for directly damped hyperbolic systems.
{"title":"Diffusion phenomenon for indirectly damped hyperbolic systems coupled by velocities in exterior domains","authors":"L. Aloui, A. Arama","doi":"10.1142/s0219891620500137","DOIUrl":"https://doi.org/10.1142/s0219891620500137","url":null,"abstract":"We consider a system of two coupled wave equations in an exterior domain, where only one equation is directly damped. We prove that the solutions are [Formula: see text]-approximated by special functions, classified into three patterns depending on the values of the damping and the coupling terms, as well as on the speeds of the waves. In particular, when the damping term is sufficiently large, the waves are asymptotically equal to solutions of parabolic-type equations as [Formula: see text]. This result generalizes the standard diffusion phenomenon for directly damped hyperbolic systems.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"17 1","pages":"475-500"},"PeriodicalIF":0.7,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49203646","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 : 2020-09-01DOI: 10.1142/s0219891620500150
Bora Moon, Ji-Mi Yim
We study the large time behavior of solutions to the space-time monopole equations in [Formula: see text] dimensions. We establish that the solutions will tend to traveling wave solution when time tends to infinity.
{"title":"Large time behavior of solutions to space-time monopole equations in 1 + 1 dimensions","authors":"Bora Moon, Ji-Mi Yim","doi":"10.1142/s0219891620500150","DOIUrl":"https://doi.org/10.1142/s0219891620500150","url":null,"abstract":"We study the large time behavior of solutions to the space-time monopole equations in [Formula: see text] dimensions. We establish that the solutions will tend to traveling wave solution when time tends to infinity.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"17 1","pages":"559-568"},"PeriodicalIF":0.7,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42334434","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 : 2020-08-23DOI: 10.1142/S0219891622500163
Louis Dongbing Cha, Arick Shao
A key feature of [Formula: see text]-dimensional nonlinear wave equations is that they admit left or right traveling waves, under appropriate algebraic conditions on the nonlinearities. In this paper, we prove global stability of such traveling wave solutions for [Formula: see text]-dimensional systems of nonlinear wave equations, given a certain asymptotic null condition and sufficient decay for the traveling wave. We first consider semilinear systems as a simpler model problem; we then proceed to treat more general quasilinear systems.
{"title":"Global stability of traveling waves for (1 + 1)-dimensional systems of quasilinear wave equations","authors":"Louis Dongbing Cha, Arick Shao","doi":"10.1142/S0219891622500163","DOIUrl":"https://doi.org/10.1142/S0219891622500163","url":null,"abstract":"A key feature of [Formula: see text]-dimensional nonlinear wave equations is that they admit left or right traveling waves, under appropriate algebraic conditions on the nonlinearities. In this paper, we prove global stability of such traveling wave solutions for [Formula: see text]-dimensional systems of nonlinear wave equations, given a certain asymptotic null condition and sufficient decay for the traveling wave. We first consider semilinear systems as a simpler model problem; we then proceed to treat more general quasilinear systems.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47123877","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 : 2020-08-07DOI: 10.1142/s0219891620500095
Luigi Forcella, Lysianne Hari
We consider the pure-power defocusing nonlinear Klein–Gordon equation, in the H1-subcritical case, posed on the product space ℝd × 𝕋, where 𝕋 is the one-dimensional flat torus. In this framework, w...
{"title":"Large data scattering for NLKG on waveguide ℝd × 𝕋","authors":"Luigi Forcella, Lysianne Hari","doi":"10.1142/s0219891620500095","DOIUrl":"https://doi.org/10.1142/s0219891620500095","url":null,"abstract":"We consider the pure-power defocusing nonlinear Klein–Gordon equation, in the H1-subcritical case, posed on the product space ℝd × 𝕋, where 𝕋 is the one-dimensional flat torus. In this framework, w...","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"17 1","pages":"355-394"},"PeriodicalIF":0.7,"publicationDate":"2020-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44676072","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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