Pub Date : 2022-06-01DOI: 10.1142/s0219891622500060
G. Coclite, Lorenzo di Ruvo
The fifth-order Kudryashov–Sinelshchikov–Olver equation is a nonlinear partial differential equation, which describes the interactions between short waves and long waves. Here, we prove the global existence of solutions for the Cauchy problem associated with this equation.
{"title":"Global H4 solution for the fifth-order Kudryashov–Sinelshchikov–Olver equation","authors":"G. Coclite, Lorenzo di Ruvo","doi":"10.1142/s0219891622500060","DOIUrl":"https://doi.org/10.1142/s0219891622500060","url":null,"abstract":"The fifth-order Kudryashov–Sinelshchikov–Olver equation is a nonlinear partial differential equation, which describes the interactions between short waves and long waves. Here, we prove the global existence of solutions for the Cauchy problem associated with this equation.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63943879","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 : 2022-04-04DOI: 10.1142/s0219891623500108
Miriam Buck, S. Modena
We construct a large class of incompressible vector fields with Sobolev regularity, in dimension [Formula: see text], for which the chain rule problem has a negative answer. In particular, for any renormalization map [Formula: see text] (satisfying suitable assumptions) and any (distributional) renormalization defect [Formula: see text] of the form [Formula: see text], where [Formula: see text] is an [Formula: see text] vector field, we can construct an incompressible Sobolev vector field [Formula: see text] and a density [Formula: see text] for which [Formula: see text] but [Formula: see text], provided [Formula: see text].
{"title":"On the failure of the chain rule for the divergence of Sobolev vector fields","authors":"Miriam Buck, S. Modena","doi":"10.1142/s0219891623500108","DOIUrl":"https://doi.org/10.1142/s0219891623500108","url":null,"abstract":"We construct a large class of incompressible vector fields with Sobolev regularity, in dimension [Formula: see text], for which the chain rule problem has a negative answer. In particular, for any renormalization map [Formula: see text] (satisfying suitable assumptions) and any (distributional) renormalization defect [Formula: see text] of the form [Formula: see text], where [Formula: see text] is an [Formula: see text] vector field, we can construct an incompressible Sobolev vector field [Formula: see text] and a density [Formula: see text] for which [Formula: see text] but [Formula: see text], provided [Formula: see text].","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47359237","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 : 2021-11-29DOI: 10.1142/S0219891623500121
F. Bakharev, A. Enin, Yu. Petrova, N. Rastegaev
The solutions for a Riemann problem arising in chemical flooding models are studied using vanishing viscosity as an admissibility criterion. We show that when the flow function depends non-monotonically on the concentration of chemicals, non-classical undercompressive shocks appear. The monotonic dependence of the shock velocity on the ratio of dissipative coefficients is proven. For that purpose we provide the classification of the nullcline configurations for the traveling wave dynamical systems and analyze the saddle–saddle connections.
{"title":"Impact of dissipation ratio on vanishing viscosity solutions of the Riemann problem for chemical flooding model","authors":"F. Bakharev, A. Enin, Yu. Petrova, N. Rastegaev","doi":"10.1142/S0219891623500121","DOIUrl":"https://doi.org/10.1142/S0219891623500121","url":null,"abstract":"The solutions for a Riemann problem arising in chemical flooding models are studied using vanishing viscosity as an admissibility criterion. We show that when the flow function depends non-monotonically on the concentration of chemicals, non-classical undercompressive shocks appear. The monotonic dependence of the shock velocity on the ratio of dissipative coefficients is proven. For that purpose we provide the classification of the nullcline configurations for the traveling wave dynamical systems and analyze the saddle–saddle connections.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44209122","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 : 2021-11-03DOI: 10.1142/s021989162350008x
R. Ikehata
We consider the Cauchy problems in [Formula: see text] for the wave equation with a weighted [Formula: see text]-initial data. We derive sharp infinite time blowup estimates of the [Formula: see text]-norm of solutions in the case of [Formula: see text] and [Formula: see text]. Then, we apply it to the local energy decay estimates for [Formula: see text], which is not studied so completely when the [Formula: see text]th moment of the initial velocity does not vanish. The idea to derive them is strongly inspired from a technique used in [R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differ. Equ. 257 (2014) 2159–2177; R. Ikehata and M. Onodera, Remarks on large time behavior of the [Formula: see text]-norm of solutions to strongly damped wave equations, Differ. Integral Equ. 30 (2017) 505–520].
{"title":"L2-blowup estimates of the wave equation and its application to local energy decay","authors":"R. Ikehata","doi":"10.1142/s021989162350008x","DOIUrl":"https://doi.org/10.1142/s021989162350008x","url":null,"abstract":"We consider the Cauchy problems in [Formula: see text] for the wave equation with a weighted [Formula: see text]-initial data. We derive sharp infinite time blowup estimates of the [Formula: see text]-norm of solutions in the case of [Formula: see text] and [Formula: see text]. Then, we apply it to the local energy decay estimates for [Formula: see text], which is not studied so completely when the [Formula: see text]th moment of the initial velocity does not vanish. The idea to derive them is strongly inspired from a technique used in [R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differ. Equ. 257 (2014) 2159–2177; R. Ikehata and M. Onodera, Remarks on large time behavior of the [Formula: see text]-norm of solutions to strongly damped wave equations, Differ. Integral Equ. 30 (2017) 505–520].","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45176265","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 : 2021-10-07DOI: 10.1142/S0219891622500138
Sergio Spagnolo Giovanni Taglialatela
In this paper, we consider the Cauchy problem for higher-order weakly hyperbolic equations assuming that the principal symbol depends only on one space variable and the characteristic roots [Formula: see text] verify an inequality like [Formula: see text] We prove that the Cauchy problem is well-posed in [Formula: see text] if the operators with frozen coefficients are uniformly hyperbolic in the sense of Gårding.
{"title":"The Cauchy problem for properly hyperbolic equations in one space variable","authors":"Sergio Spagnolo Giovanni Taglialatela","doi":"10.1142/S0219891622500138","DOIUrl":"https://doi.org/10.1142/S0219891622500138","url":null,"abstract":"In this paper, we consider the Cauchy problem for higher-order weakly hyperbolic equations assuming that the principal symbol depends only on one space variable and the characteristic roots [Formula: see text] verify an inequality like [Formula: see text] We prove that the Cauchy problem is well-posed in [Formula: see text] if the operators with frozen coefficients are uniformly hyperbolic in the sense of Gårding.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43484404","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 : 2021-09-01DOI: 10.1142/s021989162150020x
T. Pichard, N. Aguillon, B. Després, E. Godlewski, M. Ndjinga
Motivated by the modeling of boiling two-phase flows, we study systems of balance laws with a source term defined as a discontinuous function of the unknown. Due to this discontinuous source term, the classical theory of partial differential equations (PDEs) is not sufficient here. Restricting to a simpler system with linear fluxes, a notion of generalized solution is developed. An important point in the construction of a solution is that the curve along which the source jumps, which we call the boiling curve, must never be tangent to the characteristics. This leads to exhibit sufficient conditions which ensure the existence and uniqueness of a solution in two different situations: first when the initial data is smooth and such that the boiling curve is either overcharacteristic or subcharacteristic; then with discontinuous initial data in the case of Riemann problems. A numerical illustration is given in this last case.
{"title":"Existence and uniqueness of generalized solutions to hyperbolic systems with linear fluxes and stiff sources","authors":"T. Pichard, N. Aguillon, B. Després, E. Godlewski, M. Ndjinga","doi":"10.1142/s021989162150020x","DOIUrl":"https://doi.org/10.1142/s021989162150020x","url":null,"abstract":"Motivated by the modeling of boiling two-phase flows, we study systems of balance laws with a source term defined as a discontinuous function of the unknown. Due to this discontinuous source term, the classical theory of partial differential equations (PDEs) is not sufficient here. Restricting to a simpler system with linear fluxes, a notion of generalized solution is developed. An important point in the construction of a solution is that the curve along which the source jumps, which we call the boiling curve, must never be tangent to the characteristics. This leads to exhibit sufficient conditions which ensure the existence and uniqueness of a solution in two different situations: first when the initial data is smooth and such that the boiling curve is either overcharacteristic or subcharacteristic; then with discontinuous initial data in the case of Riemann problems. A numerical illustration is given in this last case.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45394069","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 : 2021-09-01DOI: 10.1142/s0219891621500211
Huali Zhang
We prove the local existence, uniqueness and stability of local solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial data of velocity, density, specific vorticity [Formula: see text] and the spatial derivative of specific vorticity [Formula: see text].
{"title":"Local existence with low regularity for the 2D compressible Euler equations","authors":"Huali Zhang","doi":"10.1142/s0219891621500211","DOIUrl":"https://doi.org/10.1142/s0219891621500211","url":null,"abstract":"We prove the local existence, uniqueness and stability of local solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial data of velocity, density, specific vorticity [Formula: see text] and the spatial derivative of specific vorticity [Formula: see text].","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48214899","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 : 2021-09-01DOI: 10.1142/s0219891621500235
Tai-Ping Liu
Shock waves of arbitrary strength in the Euler equations for compressible media are studied. The admissibility condition for a shock wave is shown to be equivalent to its formation according to the entropy production criterion. The Riemann problem with large data has a unique admissible solutions. These quantitative results are based on the exact global expressions for the basic physical variables as the states move along the Hugoniot and wave curves.
{"title":"Shock waves in Euler equations for compressible medium","authors":"Tai-Ping Liu","doi":"10.1142/s0219891621500235","DOIUrl":"https://doi.org/10.1142/s0219891621500235","url":null,"abstract":"Shock waves of arbitrary strength in the Euler equations for compressible media are studied. The admissibility condition for a shock wave is shown to be equivalent to its formation according to the entropy production criterion. The Riemann problem with large data has a unique admissible solutions. These quantitative results are based on the exact global expressions for the basic physical variables as the states move along the Hugoniot and wave curves.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47732177","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 : 2021-07-22DOI: 10.1142/s0219891623500042
Tomasz Dkebiec, Jack W. D. Skipper, E. Wiedemann
We prove via convex integration a result that allows to pass from a so-called subsolution of the isentropic Euler equations (in space dimension at least 2) to exact weak solutions. The method is closely related to the incompressible scheme established by De Lellis–Székelyhidi, in particular, we only perturb momenta and not densities. Surprisingly, though, this turns out not to be a restriction, as can be seen from our simple characterization of the [Formula: see text]-convex hull of the constitutive set. An important application of our scheme has been exhibited in recent work by Gallenmüller–Wiedemann.
{"title":"A general convex integration scheme for the isentropic compressible Euler equations","authors":"Tomasz Dkebiec, Jack W. D. Skipper, E. Wiedemann","doi":"10.1142/s0219891623500042","DOIUrl":"https://doi.org/10.1142/s0219891623500042","url":null,"abstract":"We prove via convex integration a result that allows to pass from a so-called subsolution of the isentropic Euler equations (in space dimension at least 2) to exact weak solutions. The method is closely related to the incompressible scheme established by De Lellis–Székelyhidi, in particular, we only perturb momenta and not densities. Surprisingly, though, this turns out not to be a restriction, as can be seen from our simple characterization of the [Formula: see text]-convex hull of the constitutive set. An important application of our scheme has been exhibited in recent work by Gallenmüller–Wiedemann.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42959723","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 : 2021-07-19DOI: 10.1142/s0219891621500119
Qing Han, Lin Zhang
We study the Maxwell equation and the spin-2 field equation in Bondi–Sachs coordinates associated with an asymptotically flat Lorentzian metric. We consider the mixed boundary/initial value problem, where the initial data are imposed on a null hypersurface and a boundary value is prescribed on a timelike hypersurface. We establish Sobolev [Formula: see text] space-time estimates for solutions and their asymptotic expansions at the null infinity.
{"title":"On the null-timelike boundary for Maxwell and spin-2 fields in asymptotically flat spaces","authors":"Qing Han, Lin Zhang","doi":"10.1142/s0219891621500119","DOIUrl":"https://doi.org/10.1142/s0219891621500119","url":null,"abstract":"We study the Maxwell equation and the spin-2 field equation in Bondi–Sachs coordinates associated with an asymptotically flat Lorentzian metric. We consider the mixed boundary/initial value problem, where the initial data are imposed on a null hypersurface and a boundary value is prescribed on a timelike hypersurface. We establish Sobolev [Formula: see text] space-time estimates for solutions and their asymptotic expansions at the null infinity.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44296170","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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