Pub Date : 2023-09-01DOI: 10.1142/s0219891623500169
Yongcai Geng
In this paper, via the regularity of sonic speed, we are concerned with the well and ill-posedness problems of the relativistic Euler equations with free boundary. First, we deduce the physical vacuum condition of relativistic Euler equations, which means that the sonic speed [Formula: see text] behaves like a half power of distance to the vacuum boundary [Formula: see text], satisfying [Formula: see text], it belongs to H[Formula: see text]lder continuous. Then, for [Formula: see text], this case means that the sonic speed belongs to [Formula: see text] smooth across the vacuum boundary, it is proved from both Lagrangian and Eulerian coordinates points of view. Finally, for the cases [Formula: see text] and [Formula: see text], the boundary behaviors are verified ill-posed by the unbounded acceleration of the fluid near the vacuum boundary. In this paper, the uniform bounds of velocity [Formula: see text] with respect to [Formula: see text] and the upper bounds for the square of sonic speed [Formula: see text] are very important in the proof of no matter whether well or ill-posedness because this will enable us to avoid many difficulties in the mathematical structure of relativistic fluids especially near the vacuum boundary. It is our innovation that distinguishes from non-relativistic Euler equations [J. Jang and N. Masmoudi, Well and ill-posedness for compressible Euler equations with vacuum, J. Math. Phys. 53 (2012) 1–11].
本文通过声速的正则性，关注自由边界相对论欧拉方程的好求与错求问题。首先，我们推导相对论欧拉方程的物理真空条件，即声速[式：见正文]表现为到真空边界[式：见正文]距离的半幂，满足[式：见正文]，它属于H[式：见正文]lder连续。然后，对于[公式：见正文]，这种情况意味着声速在真空边界上属于[公式：见正文]光滑，这从拉格朗日坐标和欧拉坐标的角度都得到了证明。最后，对于[公式：见正文]和[公式：见正文]两种情况，由于流体在真空边界附近的加速度无约束，边界行为得到了验证。在本文中，相对于[公式：见正文]的速度均匀界[公式：见正文]和声速平方的上界[公式：见正文]在证明好摆性或错摆性中都非常重要，因为这将使我们避免相对论流体数学结构中的许多困难，尤其是在真空边界附近。这是我们区别于非相对论欧拉方程的创新之处 [J. Jang and N. Masmoud]。Jang and N. Masmoudi, Well and ill-posedness for compressible Euler equations with vacuum, J. Math. Phys.53 (2012) 1-11].
{"title":"Well and ill-posedness of free boundary problems to relativistic Euler equations","authors":"Yongcai Geng","doi":"10.1142/s0219891623500169","DOIUrl":"https://doi.org/10.1142/s0219891623500169","url":null,"abstract":"In this paper, via the regularity of sonic speed, we are concerned with the well and ill-posedness problems of the relativistic Euler equations with free boundary. First, we deduce the physical vacuum condition of relativistic Euler equations, which means that the sonic speed [Formula: see text] behaves like a half power of distance to the vacuum boundary [Formula: see text], satisfying [Formula: see text], it belongs to H[Formula: see text]lder continuous. Then, for [Formula: see text], this case means that the sonic speed belongs to [Formula: see text] smooth across the vacuum boundary, it is proved from both Lagrangian and Eulerian coordinates points of view. Finally, for the cases [Formula: see text] and [Formula: see text], the boundary behaviors are verified ill-posed by the unbounded acceleration of the fluid near the vacuum boundary. In this paper, the uniform bounds of velocity [Formula: see text] with respect to [Formula: see text] and the upper bounds for the square of sonic speed [Formula: see text] are very important in the proof of no matter whether well or ill-posedness because this will enable us to avoid many difficulties in the mathematical structure of relativistic fluids especially near the vacuum boundary. It is our innovation that distinguishes from non-relativistic Euler equations [J. Jang and N. Masmoudi, Well and ill-posedness for compressible Euler equations with vacuum, J. Math. Phys. 53 (2012) 1–11].","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"27 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139343411","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 : 2023-09-01DOI: 10.1142/s0219891623500182
Peder Aursand, Anders Nordli
We derive a novel two-component generalization of the nonlinear variational wave equation as a model for the director field of a nematic liquid crystal with a variable order parameter. The equation admits classical solutions locally in time. We prove that a special semilinear case is globally well-posed. We show that a particular long time asymptotic expansion around a constant state in a moving frame satisfies the two-component Hunter–Saxton system.
{"title":"A two-component nonlinear variational wave system","authors":"Peder Aursand, Anders Nordli","doi":"10.1142/s0219891623500182","DOIUrl":"https://doi.org/10.1142/s0219891623500182","url":null,"abstract":"We derive a novel two-component generalization of the nonlinear variational wave equation as a model for the director field of a nematic liquid crystal with a variable order parameter. The equation admits classical solutions locally in time. We prove that a special semilinear case is globally well-posed. We show that a particular long time asymptotic expansion around a constant state in a moving frame satisfies the two-component Hunter–Saxton system.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135894829","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 : 2023-09-01DOI: 10.1142/s0219891623500170
William Golding, Sam Krupa, Alexis Vasseur
In this paper, we study the [Formula: see text]-contraction property of small extremal shocks for 1-d systems of hyperbolic conservation laws endowed with a single convex entropy, when subjected to large perturbations. We show that the weight coefficient [Formula: see text] can be chosen with amplitude proportional to the size of the shock. The main result of this paper is a key building block in the companion paper [G. Chen, S. G. Krupa and A. F. Vasseur, Uniqueness and weak-BV stability for [Formula: see text] conservation laws, Arch. Ration. Mech. Anal. 246(1) (2022) 299–332] in which uniqueness and BV-weak stability results for [Formula: see text] systems of hyperbolic conservation laws are proved.
{"title":"Sharp a-contraction estimates for small extremal shocks","authors":"William Golding, Sam Krupa, Alexis Vasseur","doi":"10.1142/s0219891623500170","DOIUrl":"https://doi.org/10.1142/s0219891623500170","url":null,"abstract":"In this paper, we study the [Formula: see text]-contraction property of small extremal shocks for 1-d systems of hyperbolic conservation laws endowed with a single convex entropy, when subjected to large perturbations. We show that the weight coefficient [Formula: see text] can be chosen with amplitude proportional to the size of the shock. The main result of this paper is a key building block in the companion paper [G. Chen, S. G. Krupa and A. F. Vasseur, Uniqueness and weak-BV stability for [Formula: see text] conservation laws, Arch. Ration. Mech. Anal. 246(1) (2022) 299–332] in which uniqueness and BV-weak stability results for [Formula: see text] systems of hyperbolic conservation laws are proved.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135894813","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 : 2023-09-01DOI: 10.1142/s0219891623500200
R. Borsche, M. Garavello, B. Gunarso
This paper deals with the well-posedness on a network of a Temple system of nonlinear hyperbolic balance laws. Temple systems are characterized by the fact that shock and rarefaction curves coincide. This study is motivated by a model for traffic, recently proposed, inspired by kinetic considerations. The proof of the well-posedness is based on the wave-front tracking procedure, on the pseudo-polygonal technique and on the operator splitting method.
{"title":"Temple system on networks","authors":"R. Borsche, M. Garavello, B. Gunarso","doi":"10.1142/s0219891623500200","DOIUrl":"https://doi.org/10.1142/s0219891623500200","url":null,"abstract":"This paper deals with the well-posedness on a network of a Temple system of nonlinear hyperbolic balance laws. Temple systems are characterized by the fact that shock and rarefaction curves coincide. This study is motivated by a model for traffic, recently proposed, inspired by kinetic considerations. The proof of the well-posedness is based on the wave-front tracking procedure, on the pseudo-polygonal technique and on the operator splitting method.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"40 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139344934","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 : 2023-06-01DOI: 10.1142/s0219891623500091
Suprio Bhar, Imran H. Biswas, Saibal Khan, G. Vallet
This paper is concerned with sample paths and path-based properties of the entropy solution to conservation laws with stochastic forcing. We derive a series of uniform maximal-type estimates for the viscous perturbation and establish the existence of stochastic entropy solution that has Hölder continuous sample paths. This information is then carefully choreographed with Kružkov’s technique to obtain stronger continuous dependence estimates, based on the nonlinearities, for the sample paths of the solutions. Finally, convergence of sample paths is established for vanishing viscosity approximation along with an explicit rate of convergence.
{"title":"Kolmogorov continuity and stability of sample paths of entropy solutions of stochastic conservation laws","authors":"Suprio Bhar, Imran H. Biswas, Saibal Khan, G. Vallet","doi":"10.1142/s0219891623500091","DOIUrl":"https://doi.org/10.1142/s0219891623500091","url":null,"abstract":"This paper is concerned with sample paths and path-based properties of the entropy solution to conservation laws with stochastic forcing. We derive a series of uniform maximal-type estimates for the viscous perturbation and establish the existence of stochastic entropy solution that has Hölder continuous sample paths. This information is then carefully choreographed with Kružkov’s technique to obtain stronger continuous dependence estimates, based on the nonlinearities, for the sample paths of the solutions. Finally, convergence of sample paths is established for vanishing viscosity approximation along with an explicit rate of convergence.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49082178","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 : 2023-06-01DOI: 10.1142/s0219891623500145
N. Popivanov, T. Popov, I. Witt
In the 1950s, Protter proposed multi-dimensional analogues of the classical Guderley–Morawetz problem for mixed-type hyperbolic-elliptic equations on the plane that models transonic flows in fluid dynamics. The multi-dimensional variants turn out to be different from the two-dimensional case and the situation there is still not clear. Here, we study Protter problems in the hyperbolic part of the domain. Unlike the planar analogues, the four-dimensional variant is not well-posed for classical solutions. The problem is not Fredholm — there is an infinite number of necessary conditions for classical solvability. Alternatively, the notion of a generalized solution that may have singularities was introduced. It is known that for smooth right-hand sides, the uniquely determined generalized solution may have a power-type growth at one boundary point. The singularity is isolated at the vertex of the boundary characteristic light cone and does not propagate along the cone. Here, we construct a new singular solution with an exponential growth at the point where the singularity appears.
{"title":"Solutions with exponential singularity for (3 + 1)-D Protter problems","authors":"N. Popivanov, T. Popov, I. Witt","doi":"10.1142/s0219891623500145","DOIUrl":"https://doi.org/10.1142/s0219891623500145","url":null,"abstract":"In the 1950s, Protter proposed multi-dimensional analogues of the classical Guderley–Morawetz problem for mixed-type hyperbolic-elliptic equations on the plane that models transonic flows in fluid dynamics. The multi-dimensional variants turn out to be different from the two-dimensional case and the situation there is still not clear. Here, we study Protter problems in the hyperbolic part of the domain. Unlike the planar analogues, the four-dimensional variant is not well-posed for classical solutions. The problem is not Fredholm — there is an infinite number of necessary conditions for classical solvability. Alternatively, the notion of a generalized solution that may have singularities was introduced. It is known that for smooth right-hand sides, the uniquely determined generalized solution may have a power-type growth at one boundary point. The singularity is isolated at the vertex of the boundary characteristic light cone and does not propagate along the cone. Here, we construct a new singular solution with an exponential growth at the point where the singularity appears.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49657726","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 : 2023-06-01DOI: 10.1142/s021989162350011x
Armand Coudray
We study the peeling for the wave equation on the Vaidya spacetime following the approach developed by Mason and Nicolas in Mason–Nicolas 2009. The idea is to encode the regularity at null infinity of the rescaled field, characterized by Sobolev-type norms, in terms of corresponding function spaces of initial data. All function spaces are obtained from energy fluxes associated with an observer constructed from the Morawetz vector field on Minkowski spacetime. We combine conformal techniques and energy estimates to obtain the optimal classes of initial data ensuring a given regularity of the rescaled field. The classes of data are equivalent to those obtained on Minkowski and Schwarzschild spacetimes in that they impose the same decay at infinity and regularity.
{"title":"Peeling-off behavior of wave equation in the Vaidya spacetime","authors":"Armand Coudray","doi":"10.1142/s021989162350011x","DOIUrl":"https://doi.org/10.1142/s021989162350011x","url":null,"abstract":"We study the peeling for the wave equation on the Vaidya spacetime following the approach developed by Mason and Nicolas in Mason–Nicolas 2009. The idea is to encode the regularity at null infinity of the rescaled field, characterized by Sobolev-type norms, in terms of corresponding function spaces of initial data. All function spaces are obtained from energy fluxes associated with an observer constructed from the Morawetz vector field on Minkowski spacetime. We combine conformal techniques and energy estimates to obtain the optimal classes of initial data ensuring a given regularity of the rescaled field. The classes of data are equivalent to those obtained on Minkowski and Schwarzschild spacetimes in that they impose the same decay at infinity and regularity.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46538302","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 : 2023-06-01DOI: 10.1142/s0219891623500157
Chueh-Hsin Chang, Tai-Ping Liu
We construct the viscous profile of the Navier–Stokes equations for compressible media under certain sufficient local hypotheses of the constitutive relation. Our result applies to shocks of arbitrary strength and generalizes the classical work of Gilbarg for the convex constitutive relation of Bethe–Weyl.
{"title":"Shock profiles of Navier–Stokes equations for compressible medium","authors":"Chueh-Hsin Chang, Tai-Ping Liu","doi":"10.1142/s0219891623500157","DOIUrl":"https://doi.org/10.1142/s0219891623500157","url":null,"abstract":"We construct the viscous profile of the Navier–Stokes equations for compressible media under certain sufficient local hypotheses of the constitutive relation. Our result applies to shocks of arbitrary strength and generalizes the classical work of Gilbarg for the convex constitutive relation of Bethe–Weyl.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44815793","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 : 2023-03-01DOI: 10.1142/s0219891623500078
Yanni Zeng
We consider a [Formula: see text] system of hyperbolic balance laws that is the converted form under inverse Hopf–Cole transformation of a Keller–Segel type chemotaxis model. We study Cauchy problem when Cauchy data connect two different end-states as [Formula: see text]. The background wave is a diffusive contact wave of the reduced system. We establish global existence of solution and study the time asymptotic behavior. In the special case where the cellular population initially approaches its stable equilibrium value as [Formula: see text], we obtain nonlinear stability of the diffusive contact wave under smallness assumption. In the general case where the population initially does not approach to its stable equilibrium value at least at one of the far fields, we use a correction function in the time asymptotic ansatz, and show that the population approaches logistically to its stable equilibrium value. Our result shows two significant differences when comparing to Euler equations with damping. The first one is the existence of a secondary wave in the time asymptotic ansatz. This implies that our solutions converge to the diffusive contact wave slower than those of Euler equations with damping. The second one is that the correction function logistically grows rather than exponentially decays.
{"title":"Convergence to a diffusive contact wave for solutions to a system of hyperbolic balance laws","authors":"Yanni Zeng","doi":"10.1142/s0219891623500078","DOIUrl":"https://doi.org/10.1142/s0219891623500078","url":null,"abstract":"We consider a [Formula: see text] system of hyperbolic balance laws that is the converted form under inverse Hopf–Cole transformation of a Keller–Segel type chemotaxis model. We study Cauchy problem when Cauchy data connect two different end-states as [Formula: see text]. The background wave is a diffusive contact wave of the reduced system. We establish global existence of solution and study the time asymptotic behavior. In the special case where the cellular population initially approaches its stable equilibrium value as [Formula: see text], we obtain nonlinear stability of the diffusive contact wave under smallness assumption. In the general case where the population initially does not approach to its stable equilibrium value at least at one of the far fields, we use a correction function in the time asymptotic ansatz, and show that the population approaches logistically to its stable equilibrium value. Our result shows two significant differences when comparing to Euler equations with damping. The first one is the existence of a secondary wave in the time asymptotic ansatz. This implies that our solutions converge to the diffusive contact wave slower than those of Euler equations with damping. The second one is that the correction function logistically grows rather than exponentially decays.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46622178","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 : 2023-03-01DOI: 10.1142/s0219891623500017
J. Hua, Lirong Xia
We investigate the uniqueness of entropy solution to 2D Riemann problem of compressible isentropic Euler system with maximum density constraint. The constraint is imposed with a singular pressure. Given initial data for which the standard self-similar solution consists of one shock or one shock and one rarefaction wave, it turns out that there exist infinitely many admissible weak solutions. This extends the result of Markfelder and Klingenberg in [S. Markfelder and C. Klingenberg, The Riemann problem for the multidimensional isentropic system of gas dynamics is ill-posed if it contains a shock, Arch. Ration. Mech. Anal. 227(3) (2018) 967–994] for classical Euler system to the case with maximum density constraint. Also some estimates on the density of these solutions are given to describe the behavior of solutions near congestion.
{"title":"The non-uniqueness of admissible solutions to 2D Riemann problem of compressible isentropic Euler system with maximum density constraint","authors":"J. Hua, Lirong Xia","doi":"10.1142/s0219891623500017","DOIUrl":"https://doi.org/10.1142/s0219891623500017","url":null,"abstract":"We investigate the uniqueness of entropy solution to 2D Riemann problem of compressible isentropic Euler system with maximum density constraint. The constraint is imposed with a singular pressure. Given initial data for which the standard self-similar solution consists of one shock or one shock and one rarefaction wave, it turns out that there exist infinitely many admissible weak solutions. This extends the result of Markfelder and Klingenberg in [S. Markfelder and C. Klingenberg, The Riemann problem for the multidimensional isentropic system of gas dynamics is ill-posed if it contains a shock, Arch. Ration. Mech. Anal. 227(3) (2018) 967–994] for classical Euler system to the case with maximum density constraint. Also some estimates on the density of these solutions are given to describe the behavior of solutions near congestion.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45680105","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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