Pub Date : 2025-02-05DOI: 10.1016/j.jde.2025.01.090
Pavol Quittner , Philippe Souplet
We establish Liouville type theorems in the whole space and in a half-space for parabolic problems without scale invariance. To this end, we employ two methods, respectively based on the corresponding elliptic Liouville type theorems and energy estimates for suitably rescaled problems, and on reduction to a scalar equation by proportionality of components.
We then give applications of known and new Liouville type theorems to universal singularity and decay estimates for non scale invariant parabolic equations and systems involving superlinear nonlinearities with regular variation. To this end, we adapt methods from [33] to parabolic problems.
{"title":"Liouville theorems and universal estimates for superlinear parabolic problems without scale invariance","authors":"Pavol Quittner , Philippe Souplet","doi":"10.1016/j.jde.2025.01.090","DOIUrl":"10.1016/j.jde.2025.01.090","url":null,"abstract":"<div><div>We establish Liouville type theorems in the whole space and in a half-space for parabolic problems without scale invariance. To this end, we employ two methods, respectively based on the corresponding elliptic Liouville type theorems and energy estimates for suitably rescaled problems, and on reduction to a scalar equation by proportionality of components.</div><div>We then give applications of known and new Liouville type theorems to universal singularity and decay estimates for non scale invariant parabolic equations and systems involving superlinear nonlinearities with regular variation. To this end, we adapt methods from <span><span>[33]</span></span> to parabolic problems.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"427 ","pages":"Pages 404-441"},"PeriodicalIF":2.4,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-05DOI: 10.1016/j.jde.2025.01.087
Lvqiao Liu , Hao Wang
We study the regularization effect for the non-cutoff Vlasov-Poisson-Boltzmann system with hard potentials in the perturbation setting. Inspired by the techniques developed in Chen-Li-Xu [13], we prove the analytic regularity in both space and velocity variables when the initial data has mild regularity.
{"title":"Analytic smoothing effect for the non-cutoff Vlasov-Poisson-Boltzmann system with hard potentials","authors":"Lvqiao Liu , Hao Wang","doi":"10.1016/j.jde.2025.01.087","DOIUrl":"10.1016/j.jde.2025.01.087","url":null,"abstract":"<div><div>We study the regularization effect for the non-cutoff Vlasov-Poisson-Boltzmann system with hard potentials in the perturbation setting. Inspired by the techniques developed in Chen-Li-Xu <span><span>[13]</span></span>, we prove the analytic regularity in both space and velocity variables when the initial data has mild regularity.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"427 ","pages":"Pages 442-477"},"PeriodicalIF":2.4,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-05DOI: 10.1016/j.jde.2025.02.004
Yaqi Liang, Xiong Li
In this paper we consider the non-periodic pendulum equation , where and are not required to be periodic in t. Under natural assumptions, the existence of infinitely many bounded solutions is established, furthermore, it is shown that, for any given unbounded solution x, there is a solution which is bounded and such that the half power of energies of and x remain close on a time interval of length with small enough. In the end, a specific is constructed to illustrate the existence of the unbounded solution for the equation under this ; moreover, the long-time closeness result also holds under this .
{"title":"Long-time stability estimates for the non-periodic pendulum equation","authors":"Yaqi Liang, Xiong Li","doi":"10.1016/j.jde.2025.02.004","DOIUrl":"10.1016/j.jde.2025.02.004","url":null,"abstract":"<div><div>In this paper we consider the non-periodic pendulum equation <span><math><mover><mrow><mi>x</mi></mrow><mrow><mo>¨</mo></mrow></mover><mo>+</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>p</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> and <span><math><mi>p</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> are not required to be periodic in <em>t</em>. Under natural assumptions, the existence of infinitely many bounded solutions is established, furthermore, it is shown that, for any given unbounded solution <em>x</em>, there is a solution <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>ε</mi></mrow></msup></math></span> which is bounded and such that the half power of energies of <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>ε</mi></mrow></msup></math></span> and <em>x</em> remain close on a time interval of length <span><math><msup><mrow><mi>ε</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> with <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span> small enough. In the end, a specific <span><math><mi>p</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> is constructed to illustrate the existence of the unbounded solution for the equation under this <span><math><mi>p</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span>; moreover, the long-time closeness result also holds under this <span><math><mi>p</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"425 ","pages":"Pages 805-845"},"PeriodicalIF":2.4,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143319073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-05DOI: 10.1016/j.jde.2025.01.089
Yonghui Zhou , Xiaowan Li , Shuguan Ji
In this paper, we prove that the existence of globally conservative weak solutions for a class of two-component nonlinear dispersive wave system of the equations beyond wave breaking. We first introduce a new set of independent and dependent variables in connection with smooth solutions, and transform the system into an equivalent semi-linear system. We then establish the global existence of solutions for the semi-linear system via the standard theory of ordinary differential equations. Finally, by the inverse transformation method, we prove the existence of the globally conservative weak solution for the original system.
{"title":"Globally conservative weak solutions for a class of two-component nonlinear dispersive wave equations beyond wave breaking","authors":"Yonghui Zhou , Xiaowan Li , Shuguan Ji","doi":"10.1016/j.jde.2025.01.089","DOIUrl":"10.1016/j.jde.2025.01.089","url":null,"abstract":"<div><div>In this paper, we prove that the existence of globally conservative weak solutions for a class of two-component nonlinear dispersive wave system of the equations beyond wave breaking. We first introduce a new set of independent and dependent variables in connection with smooth solutions, and transform the system into an equivalent semi-linear system. We then establish the global existence of solutions for the semi-linear system via the standard theory of ordinary differential equations. Finally, by the inverse transformation method, we prove the existence of the globally conservative weak solution for the original system.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"427 ","pages":"Pages 538-559"},"PeriodicalIF":2.4,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143104316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-05DOI: 10.1016/j.jde.2025.01.088
Christoph Fischbacher , Jonathan Stanfill
The purpose of this paper is to study nonnegative self-adjoint extensions associated with singular Sturm–Liouville expressions with strictly positive minimal operators. We provide a full characterization of all possible nonnegative self-adjoint extensions of the minimal operator in terms of generalized boundary values, as well as a parameterization of all nonnegative extensions when fixing a boundary condition at one endpoint. In addition, we investigate problems where the coefficient functions are symmetric about the midpoint of a finite interval, illustrating how every self-adjoint operator of this form is unitarily equivalent to the direct sum of two self-adjoint operators restricted to half of the interval. We also extend these result to symmetric two interval problems. We then apply our previous results to parameterize all nonnegative extensions of operators with symmetric coefficient functions. We end with an example of an operator with a symmetric Bessel-type potential (i.e., symmetric confining potential) and an application to integral inequalities.
{"title":"Nonnegative extensions of Sturm–Liouville operators with an application to problems with symmetric coefficient functions","authors":"Christoph Fischbacher , Jonathan Stanfill","doi":"10.1016/j.jde.2025.01.088","DOIUrl":"10.1016/j.jde.2025.01.088","url":null,"abstract":"<div><div>The purpose of this paper is to study nonnegative self-adjoint extensions associated with singular Sturm–Liouville expressions with strictly positive minimal operators. We provide a full characterization of all possible nonnegative self-adjoint extensions of the minimal operator in terms of generalized boundary values, as well as a parameterization of all nonnegative extensions when fixing a boundary condition at one endpoint. In addition, we investigate problems where the coefficient functions are symmetric about the midpoint of a finite interval, illustrating how every self-adjoint operator of this form is unitarily equivalent to the direct sum of two self-adjoint operators restricted to half of the interval. We also extend these result to symmetric two interval problems. We then apply our previous results to parameterize all nonnegative extensions of operators with symmetric coefficient functions. We end with an example of an operator with a symmetric Bessel-type potential (i.e., symmetric confining potential) and an application to integral inequalities.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"427 ","pages":"Pages 560-600"},"PeriodicalIF":2.4,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143170958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-05DOI: 10.1016/j.jde.2025.02.010
Nicolae Lupa , Kenneth J. Palmer , Liviu Horia Popescu
We prove that under some appropriate conditions, exponential dichotomy of an evolution family is implied by a weaker concept of structural stability. We extend precedent results, which are only valid in the case of differential equations with bounded operators. In addition, we establish new characterizations of exponential stability and exponential expansiveness, in terms of topological equivalence. Our proofs are new even in the particular case of differential equations.
{"title":"Structural stability of evolution families in finite-dimensional spaces","authors":"Nicolae Lupa , Kenneth J. Palmer , Liviu Horia Popescu","doi":"10.1016/j.jde.2025.02.010","DOIUrl":"10.1016/j.jde.2025.02.010","url":null,"abstract":"<div><div>We prove that under some appropriate conditions, exponential dichotomy of an evolution family is implied by a weaker concept of structural stability. We extend precedent results, which are only valid in the case of differential equations with bounded operators. In addition, we establish new characterizations of exponential stability and exponential expansiveness, in terms of topological equivalence. Our proofs are new even in the particular case of differential equations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"424 ","pages":"Pages 792-814"},"PeriodicalIF":2.4,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143260671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-03DOI: 10.1016/j.jde.2025.01.081
Dongfeng Zhang, Junxiang Xu
<div><div>In this paper we consider the following nonlinear quasi-periodic system near an elliptic equilibrium point<span><span><span><math><mover><mrow><mi>x</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>=</mo><mo>(</mo><mi>A</mi><mo>+</mo><mi>ϵ</mi><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo><mo>)</mo><mi>x</mi><mo>+</mo><mi>ϵ</mi><mi>g</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo><mo>+</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo><mo>,</mo><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo></math></span></span></span> where <em>A</em> is a <span><math><mi>d</mi><mo>×</mo><mi>d</mi></math></span> constant matrix of elliptic type, which has multiple eigenvalues, <span><math><mi>P</mi><mo>,</mo><mi>g</mi></math></span> and <em>h</em> are all analytic quasi-periodic in <em>t</em> with basic frequencies <span><math><mi>ω</mi><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><mi>α</mi><mo>)</mo></math></span>, where <em>α</em> is irrational, <span><math><mi>ϵ</mi><mi>g</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo></math></span> is a small perturbation with <em>ϵ</em> as a small parameter, <span><math><mi>h</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> as <span><math><mi>x</mi><mo>→</mo><mn>0</mn></math></span>. It is proved that for most sufficiently small <em>ϵ</em>, the system is reducible to the following form:<span><span><span><math><mover><mrow><mi>x</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>=</mo><mo>(</mo><mi>A</mi><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo><mo>)</mo><mi>x</mi><mo>+</mo><msub><mrow><mi>h</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo><mo>,</mo><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>B</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo></math></span> is a diagonal block of real functions, <span><math><msub><mrow><mi>h</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>(</mo><mi>x</mi><mo>→</mo><mn>0</mn><mo>)</mo></math></span> is a high-order term. Therefore, the system has a quasi-periodic solution with basic frequencies <span><math><mi>ω</mi><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><mi>α</mi><mo>)</mo></math></span>, such that it goes to zero when <em>ϵ</em> does. As some applications, we apply our results to <em>n</em> coupled Schrödinger equations, damped equations and quasi-periodic forced reversible systems to study the existence of
{"title":"Quasi-periodic solutions of a class of nonlinear quasi-periodic systems with Liouvillean frequencies and multiple eigenvalues","authors":"Dongfeng Zhang, Junxiang Xu","doi":"10.1016/j.jde.2025.01.081","DOIUrl":"10.1016/j.jde.2025.01.081","url":null,"abstract":"<div><div>In this paper we consider the following nonlinear quasi-periodic system near an elliptic equilibrium point<span><span><span><math><mover><mrow><mi>x</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>=</mo><mo>(</mo><mi>A</mi><mo>+</mo><mi>ϵ</mi><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo><mo>)</mo><mi>x</mi><mo>+</mo><mi>ϵ</mi><mi>g</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo><mo>+</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo><mo>,</mo><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo></math></span></span></span> where <em>A</em> is a <span><math><mi>d</mi><mo>×</mo><mi>d</mi></math></span> constant matrix of elliptic type, which has multiple eigenvalues, <span><math><mi>P</mi><mo>,</mo><mi>g</mi></math></span> and <em>h</em> are all analytic quasi-periodic in <em>t</em> with basic frequencies <span><math><mi>ω</mi><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><mi>α</mi><mo>)</mo></math></span>, where <em>α</em> is irrational, <span><math><mi>ϵ</mi><mi>g</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo></math></span> is a small perturbation with <em>ϵ</em> as a small parameter, <span><math><mi>h</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> as <span><math><mi>x</mi><mo>→</mo><mn>0</mn></math></span>. It is proved that for most sufficiently small <em>ϵ</em>, the system is reducible to the following form:<span><span><span><math><mover><mrow><mi>x</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>=</mo><mo>(</mo><mi>A</mi><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo><mo>)</mo><mi>x</mi><mo>+</mo><msub><mrow><mi>h</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo><mo>,</mo><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>B</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo></math></span> is a diagonal block of real functions, <span><math><msub><mrow><mi>h</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>(</mo><mi>x</mi><mo>→</mo><mn>0</mn><mo>)</mo></math></span> is a high-order term. Therefore, the system has a quasi-periodic solution with basic frequencies <span><math><mi>ω</mi><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><mi>α</mi><mo>)</mo></math></span>, such that it goes to zero when <em>ϵ</em> does. As some applications, we apply our results to <em>n</em> coupled Schrödinger equations, damped equations and quasi-periodic forced reversible systems to study the existence of","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"427 ","pages":"Pages 350-403"},"PeriodicalIF":2.4,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a new definition of measure-valued solutions for the two dimensional Euler equations with general pressure laws. This generalization of the traditional weak solutions can describe flow fields with properties of high concentrations on mass and momentum. We derive the intrinsic partial differential equations governing the front surface of the concentration discontinuities, which can at certain extend be considered as generalization of the classical Rankine-Hugoniot conditions for the Euler equations.
We also get some new application results to singular Riemann problems of pressureless Euler equations.
{"title":"High concentration property on discontinuity in two-dimensional unsteady compressible Euler equations","authors":"Qihui Gao , Aifang Qu , Xiaozhou Yang , Hairong Yuan","doi":"10.1016/j.jde.2025.01.082","DOIUrl":"10.1016/j.jde.2025.01.082","url":null,"abstract":"<div><div>We propose a new definition of measure-valued solutions for the two dimensional Euler equations with general pressure laws. This generalization of the traditional weak solutions can describe flow fields with properties of high concentrations on mass and momentum. We derive the intrinsic partial differential equations governing the front surface of the concentration discontinuities, which can at certain extend be considered as generalization of the classical Rankine-Hugoniot conditions for the Euler equations.</div><div>We also get some new application results to singular Riemann problems of pressureless Euler equations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"427 ","pages":"Pages 194-218"},"PeriodicalIF":2.4,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-31DOI: 10.1016/j.jde.2025.01.084
Shangkun Weng , Zihao Zhang , Yan Zhou
We establish the existence and uniqueness of the transonic shock solution for steady isentropic Euler system with an external force in a rectangular cylinder under the three-dimensional perturbations for the incoming supersonic flow, the exit pressure and the external force. The external force has a stabilization effect on the transonic shocks in flat nozzles and the transonic shock is completely free, we do not require it passing through a fixed point. By utilizing the deformation-curl decomposition to decouple the hyperbolic and elliptic modes in the steady Euler system effectively and reformulating the Rankine-Hugoniot conditions, the transonic shock problem is reduced to a deformation-curl first order system for the velocity field with nonlocal terms supplementing with an unusual second order differential boundary condition on the shock front, an algebraic equation for determining the shock front and two transport equations for the Bernoulli's quantity and the first component of the vorticity.
{"title":"Structural stability of three dimensional transonic shock flows with an external force","authors":"Shangkun Weng , Zihao Zhang , Yan Zhou","doi":"10.1016/j.jde.2025.01.084","DOIUrl":"10.1016/j.jde.2025.01.084","url":null,"abstract":"<div><div>We establish the existence and uniqueness of the transonic shock solution for steady isentropic Euler system with an external force in a rectangular cylinder under the three-dimensional perturbations for the incoming supersonic flow, the exit pressure and the external force. The external force has a stabilization effect on the transonic shocks in flat nozzles and the transonic shock is completely free, we do not require it passing through a fixed point. By utilizing the deformation-curl decomposition to decouple the hyperbolic and elliptic modes in the steady Euler system effectively and reformulating the Rankine-Hugoniot conditions, the transonic shock problem is reduced to a deformation-curl first order system for the velocity field with nonlocal terms supplementing with an unusual second order differential boundary condition on the shock front, an algebraic equation for determining the shock front and two transport equations for the Bernoulli's quantity and the first component of the vorticity.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"427 ","pages":"Pages 310-349"},"PeriodicalIF":2.4,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-31DOI: 10.1016/j.jde.2025.01.086
Yuri Trakhinin
We consider an interface with surface tension that separates a perfectly conducting inviscid fluid from a vacuum. The fluid flow is governed by the equations of ideal compressible magnetohydrodynamics (MHD), while the electric and magnetic fields in vacuum satisfy the Maxwell equations. With boundary conditions on the interface this forms a nonlinear hyperbolic problem with a characteristic free boundary. For the corresponding linearized problem we derive an energy a priori estimate in a conormal Sobolev space without assuming any stability conditions on the unperturbed flow. This verifies the stabilizing effect of surface tension because, as was shown in [11], a sufficiently large vacuum electric field can make the linearized problem ill-posed for the case of zero surface tension. The main ingredients in proving the energy estimate are a suitable secondary symmetrization of the Maxwell equations in vacuum and making full use of the boundary regularity enhanced from the surface tension.
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