Pub Date : 2021-02-26DOI: 10.1142/s0219891622500205
M. D. Francesco, Graziano Stivaletta
We consider the follow-the-leader particle approximation scheme for a [Formula: see text] scalar conservation law with non-negative compactly supported [Formula: see text] initial datum and with a [Formula: see text] concave flux, which is known to provide convergence towards the entropy solution [Formula: see text] to the corresponding Cauchy problem. We provide two novel contributions to this theory. First, we prove that the one-sided Lipschitz condition satisfied by the approximate density [Formula: see text] is a “discrete version of an entropy condition”; more precisely, under fairly general assumptions on [Formula: see text] (which imply concavity of [Formula: see text]) we prove that the continuum version [Formula: see text] of said condition allows to select a unique weak solution, despite [Formula: see text] is apparently weaker than the classical Oleinik–Hoff one-sided Lipschitz condition [Formula: see text]. Said result relies on an improved version of Hoff’s uniqueness. A byproduct of it is that the entropy condition is encoded in the particle scheme prior to the many-particle limit, which was never proven before. Second, we prove that in case [Formula: see text] the one-sided Lipschitz condition can be improved to a discrete version of the classical (and “sharp”) Oleinik–Hoff condition. In order to make the paper self-contained, we provide proofs (in some cases “alternative” ones) of all steps of the convergence of the particle scheme.
{"title":"The one-sided lipschitz condition in the follow-the-leader approximation of scalar conservation laws","authors":"M. D. Francesco, Graziano Stivaletta","doi":"10.1142/s0219891622500205","DOIUrl":"https://doi.org/10.1142/s0219891622500205","url":null,"abstract":"We consider the follow-the-leader particle approximation scheme for a [Formula: see text] scalar conservation law with non-negative compactly supported [Formula: see text] initial datum and with a [Formula: see text] concave flux, which is known to provide convergence towards the entropy solution [Formula: see text] to the corresponding Cauchy problem. We provide two novel contributions to this theory. First, we prove that the one-sided Lipschitz condition satisfied by the approximate density [Formula: see text] is a “discrete version of an entropy condition”; more precisely, under fairly general assumptions on [Formula: see text] (which imply concavity of [Formula: see text]) we prove that the continuum version [Formula: see text] of said condition allows to select a unique weak solution, despite [Formula: see text] is apparently weaker than the classical Oleinik–Hoff one-sided Lipschitz condition [Formula: see text]. Said result relies on an improved version of Hoff’s uniqueness. A byproduct of it is that the entropy condition is encoded in the particle scheme prior to the many-particle limit, which was never proven before. Second, we prove that in case [Formula: see text] the one-sided Lipschitz condition can be improved to a discrete version of the classical (and “sharp”) Oleinik–Hoff condition. In order to make the paper self-contained, we provide proofs (in some cases “alternative” ones) of all steps of the convergence of the particle scheme.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45692533","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 : 2021-02-17DOI: 10.1142/s0219891622500035
Y. Trakhinin
By using an equivalent form of the uniform Lopatinski condition for 1-shocks, we prove that the stability condition found by the energy method in [A. Morando, Y. Trakhinin and P. Trebeschi, Structural stability of shock waves in 2D compressible elastodynamics, Math. Ann. 378 (2020) 1471–1504] for the rectilinear shock waves in two-dimensional flows of compressible isentropic inviscid elastic materials is not only sufficient but also necessary for uniform stability (implying structural nonlinear stability of corresponding curved shock waves). The key point of our spectral analysis is a delicate study of the transition between uniform and weak stability. Moreover, we prove that the rectilinear shock waves are never violently unstable, i.e. they are always either uniformly or weakly stable.
利用1-激波的一致Lopatinski条件的等价形式，证明了[A]中能量法得到的稳定性条件。Morando, Y. Trakhinin和P. Trebeschi，二维可压缩弹性动力学中激波的结构稳定性，数学。Ann. 378(2020) 1471-1504]对于可压缩等熵无粘弹性材料二维流动中直线激波的均匀稳定性不仅是充分的，而且是必要的(意味着相应弯曲激波的结构非线性稳定性)。我们光谱分析的重点是精细地研究均匀稳定和弱稳定之间的过渡。此外，我们还证明了直线激波从不剧烈不稳定，即它们总是均匀稳定或弱稳定的。
{"title":"On weak stability of shock waves in 2D compressible elastodynamics","authors":"Y. Trakhinin","doi":"10.1142/s0219891622500035","DOIUrl":"https://doi.org/10.1142/s0219891622500035","url":null,"abstract":"By using an equivalent form of the uniform Lopatinski condition for 1-shocks, we prove that the stability condition found by the energy method in [A. Morando, Y. Trakhinin and P. Trebeschi, Structural stability of shock waves in 2D compressible elastodynamics, Math. Ann. 378 (2020) 1471–1504] for the rectilinear shock waves in two-dimensional flows of compressible isentropic inviscid elastic materials is not only sufficient but also necessary for uniform stability (implying structural nonlinear stability of corresponding curved shock waves). The key point of our spectral analysis is a delicate study of the transition between uniform and weak stability. Moreover, we prove that the rectilinear shock waves are never violently unstable, i.e. they are always either uniformly or weakly stable.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48606984","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 : 2021-02-15DOI: 10.1142/s0219891622500047
Mashael Alammari, Stanley Snelson
We study general semilinear scalar-field equations on the real line with variable coefficients in the linear terms. These coefficients are uniformly small, but slowly decaying, perturbations of a constant-coefficient operator. We are motivated by the question of how these perturbations of the equation may change the stability properties of kink solutions (one-dimensional topological solitons). We prove existence of a stationary kink solution in our setting, and perform a detailed spectral analysis of the corresponding linearized operator, based on perturbing the linearized operator around the constant-coefficient kink. We derive a formula that allows us to check whether a discrete eigenvalue emerges from the essential spectrum under this perturbation. Known examples suggest that this extra eigenvalue may have an important influence on the long-time dynamics in a neighborhood of the kink. We also establish orbital stability of solitary-wave solutions in the variable-coefficient regime, despite the possible presence of negative eigenvalues in the linearization.
{"title":"Linear and orbital stability analysis for solitary-wave solutions of variable-coefficient scalar-field equations","authors":"Mashael Alammari, Stanley Snelson","doi":"10.1142/s0219891622500047","DOIUrl":"https://doi.org/10.1142/s0219891622500047","url":null,"abstract":"We study general semilinear scalar-field equations on the real line with variable coefficients in the linear terms. These coefficients are uniformly small, but slowly decaying, perturbations of a constant-coefficient operator. We are motivated by the question of how these perturbations of the equation may change the stability properties of kink solutions (one-dimensional topological solitons). We prove existence of a stationary kink solution in our setting, and perform a detailed spectral analysis of the corresponding linearized operator, based on perturbing the linearized operator around the constant-coefficient kink. We derive a formula that allows us to check whether a discrete eigenvalue emerges from the essential spectrum under this perturbation. Known examples suggest that this extra eigenvalue may have an important influence on the long-time dynamics in a neighborhood of the kink. We also establish orbital stability of solitary-wave solutions in the variable-coefficient regime, despite the possible presence of negative eigenvalues in the linearization.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46498937","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 : 2021-02-04DOI: 10.1142/s0219891621500223
T. Dao, H. Takeda
In this paper, we consider the Cauchy problem for semi-linear wave equations with structural damping term [Formula: see text], where [Formula: see text] is a constant. As is now well known, the linear principal part brings both the diffusion phenomenon and the regularity loss of solutions. This implies that, for the nonlinear problems, the choice of solution spaces plays an important role to obtain the global solutions with the sharp decay properties in time. Our main purpose in this paper is to prove the global (in time) existence of solutions for the small data and their decay properties for the supercritical nonlinearities.
{"title":"Global existence results for semi-linear structurally damped wave equations with nonlinear convection","authors":"T. Dao, H. Takeda","doi":"10.1142/s0219891621500223","DOIUrl":"https://doi.org/10.1142/s0219891621500223","url":null,"abstract":"In this paper, we consider the Cauchy problem for semi-linear wave equations with structural damping term [Formula: see text], where [Formula: see text] is a constant. As is now well known, the linear principal part brings both the diffusion phenomenon and the regularity loss of solutions. This implies that, for the nonlinear problems, the choice of solution spaces plays an important role to obtain the global solutions with the sharp decay properties in time. Our main purpose in this paper is to prove the global (in time) existence of solutions for the small data and their decay properties for the supercritical nonlinearities.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44825787","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 : 2020-12-28DOI: 10.1142/S0219891622500175
H. Barzegar, David Fajman
It is shown that Milne models (a subclass of Friedmann–Lematre–Robertson–Walker (FLRW) spacetimes with negative spatial curvature) are nonlinearly stable in the set of solutions to the Einstein–Vlasov–Maxwell system, describing universes with ensembles of collisionless self-gravitating, charged particles. The system contains various slowly decaying borderline terms in the mutually coupled equations describing the propagation of particles and Maxwell fields. The effects of those terms are controlled using a suitable hierarchy based on the energy density of the matter fields.
{"title":"Stable cosmologies with collisionless charged matter","authors":"H. Barzegar, David Fajman","doi":"10.1142/S0219891622500175","DOIUrl":"https://doi.org/10.1142/S0219891622500175","url":null,"abstract":"It is shown that Milne models (a subclass of Friedmann–Lematre–Robertson–Walker (FLRW) spacetimes with negative spatial curvature) are nonlinearly stable in the set of solutions to the Einstein–Vlasov–Maxwell system, describing universes with ensembles of collisionless self-gravitating, charged particles. The system contains various slowly decaying borderline terms in the mutually coupled equations describing the propagation of particles and Maxwell fields. The effects of those terms are controlled using a suitable hierarchy based on the energy density of the matter fields.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"123 9","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41284485","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 : 2020-12-18DOI: 10.1142/s0219891623500133
K. Koumatos, Corrado Lattanzio, S. Spirito, A. Tzavaras
We consider nonlinear viscoelastic materials of Kelvin–Voigt-type with stored energies satisfying an Andrews–Ball condition, allowing for nonconvexity in a compact set. Existence of weak solutions with deformation gradients in [Formula: see text] is established for energies of any superquadratic growth. In two space dimensions, weak solutions notably turn out to be unique in this class. Conservation of energy for weak solutions in two and three dimensions, as well as global regularity for smooth initial data in two dimensions are established under additional mild restrictions on the growth of the stored energy.
{"title":"Existence and uniqueness for a viscoelastic Kelvin–Voigt model with nonconvex stored energy","authors":"K. Koumatos, Corrado Lattanzio, S. Spirito, A. Tzavaras","doi":"10.1142/s0219891623500133","DOIUrl":"https://doi.org/10.1142/s0219891623500133","url":null,"abstract":"We consider nonlinear viscoelastic materials of Kelvin–Voigt-type with stored energies satisfying an Andrews–Ball condition, allowing for nonconvexity in a compact set. Existence of weak solutions with deformation gradients in [Formula: see text] is established for energies of any superquadratic growth. In two space dimensions, weak solutions notably turn out to be unique in this class. Conservation of energy for weak solutions in two and three dimensions, as well as global regularity for smooth initial data in two dimensions are established under additional mild restrictions on the growth of the stored energy.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44139194","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/S0219891620500277
H. Jenssen, J. Ridder
Let [Formula: see text] be a convex function satisfying [Formula: see text], [Formula: see text] for [Formula: see text], and [Formula: see text]. Consider the unique entropy admissible (i.e. Kružkov) solution [Formula: see text] of the scalar, 1-d Cauchy problem [Formula: see text], [Formula: see text]. For compactly supported data [Formula: see text] with bounded [Formula: see text]-variation, we realize the solution [Formula: see text] as a limit of front-tracking approximations and show that the [Formula: see text]-variation of (the right continuous version of) [Formula: see text] is non-increasing in time. We also establish the natural time-continuity estimate [Formula: see text] for [Formula: see text], where [Formula: see text] depends on [Formula: see text]. Finally, according to a theorem of Goffman–Moran–Waterman, any regulated function of compact support has bounded [Formula: see text]-variation for some [Formula: see text]. As a corollary we thus have: if [Formula: see text] is a regulated function, so is [Formula: see text] for all [Formula: see text].
{"title":"On Φ-variation for 1-d scalar conservation laws","authors":"H. Jenssen, J. Ridder","doi":"10.1142/S0219891620500277","DOIUrl":"https://doi.org/10.1142/S0219891620500277","url":null,"abstract":"Let [Formula: see text] be a convex function satisfying [Formula: see text], [Formula: see text] for [Formula: see text], and [Formula: see text]. Consider the unique entropy admissible (i.e. Kružkov) solution [Formula: see text] of the scalar, 1-d Cauchy problem [Formula: see text], [Formula: see text]. For compactly supported data [Formula: see text] with bounded [Formula: see text]-variation, we realize the solution [Formula: see text] as a limit of front-tracking approximations and show that the [Formula: see text]-variation of (the right continuous version of) [Formula: see text] is non-increasing in time. We also establish the natural time-continuity estimate [Formula: see text] for [Formula: see text], where [Formula: see text] depends on [Formula: see text]. Finally, according to a theorem of Goffman–Moran–Waterman, any regulated function of compact support has bounded [Formula: see text]-variation for some [Formula: see text]. As a corollary we thus have: if [Formula: see text] is a regulated function, so is [Formula: see text] for all [Formula: see text].","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49018863","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/S0219891620500228
A. Arora, S. Roudenko
We study the generalized Hartree equation, which is a nonlinear Schrodinger-type equation with a nonlocal potential iut + Δu + (|x|−b ∗|u|p)|u|p−2u = 0,x ∈ ℝN. We establish the local well-posedness...
{"title":"Well-posedness and blow-up properties for the generalized Hartree equation","authors":"A. Arora, S. Roudenko","doi":"10.1142/S0219891620500228","DOIUrl":"https://doi.org/10.1142/S0219891620500228","url":null,"abstract":"We study the generalized Hartree equation, which is a nonlinear Schrodinger-type equation with a nonlocal potential iut + Δu + (|x|−b ∗|u|p)|u|p−2u = 0,x ∈ ℝN. We establish the local well-posedness...","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"17 1","pages":"727-763"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48759943","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/S0219891620500253
J. Dias
Following Dias et al. [Vanishing viscosity with short wave-long wave interactions for multi-D scalar conservation laws, J. Differential Equations 251 (2007) 555–563], we study the linearized stability of a pair [Formula: see text], where [Formula: see text] is a shock profile for a family of quasilinear hyperbolic conservation laws in [Formula: see text] coupled with a semilinear Schrödinger equation.
{"title":"Linear stability of shock profiles for a quasilinear Benney system in ℝ2 × ℝ+","authors":"J. Dias","doi":"10.1142/S0219891620500253","DOIUrl":"https://doi.org/10.1142/S0219891620500253","url":null,"abstract":"Following Dias et al. [Vanishing viscosity with short wave-long wave interactions for multi-D scalar conservation laws, J. Differential Equations 251 (2007) 555–563], we study the linearized stability of a pair [Formula: see text], where [Formula: see text] is a shock profile for a family of quasilinear hyperbolic conservation laws in [Formula: see text] coupled with a semilinear Schrödinger equation.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"17 1","pages":"797-807"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41565747","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/S0219891620500265
Paulo Amorim, F. Berthelin, T. Goudon
We consider a non-local scalar conservation law in two space dimensions which arises as the formal hydrodynamic limit of a Fokker–Planck equation. This Fokker–Planck equation is, in turn, the kinetic description of an individual-based model describing the navigation of self-propelled particles in a pheromone landscape. The pheromone may be linked to the agent distribution itself, leading to a nonlinear, non-local scalar conservation law where the effective velocity vector depends on the pheromone field in a small region around each point, and thus, on the solution itself. After presenting and motivating the problem, we present some numerical simulations of a closely related problem, and then prove a well-posedness and stability result for the conservation law.
{"title":"A non-local scalar conservation law describing navigation processes","authors":"Paulo Amorim, F. Berthelin, T. Goudon","doi":"10.1142/S0219891620500265","DOIUrl":"https://doi.org/10.1142/S0219891620500265","url":null,"abstract":"We consider a non-local scalar conservation law in two space dimensions which arises as the formal hydrodynamic limit of a Fokker–Planck equation. This Fokker–Planck equation is, in turn, the kinetic description of an individual-based model describing the navigation of self-propelled particles in a pheromone landscape. The pheromone may be linked to the agent distribution itself, leading to a nonlinear, non-local scalar conservation law where the effective velocity vector depends on the pheromone field in a small region around each point, and thus, on the solution itself. After presenting and motivating the problem, we present some numerical simulations of a closely related problem, and then prove a well-posedness and stability result for the conservation law.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47748280","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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