Pub Date : 2025-01-01Epub Date: 2025-02-18DOI: 10.1007/s00030-025-01037-7
Justin Forlano, Guopeng Li, Tengfei Zhao
In this paper, we establish the unconditional deep-water limit of the intermediate long wave equation (ILW) to the Benjamin-Ono equation (BO) in low-regularity Sobolev spaces on both the real line and the circle. Our main tool is new unconditional uniqueness results for ILW in when on the line and on the circle, where . Here, we adapt the strategy of Moşincat-Pilod (Pure Appl Anal 5:285-322, 2023) for BO to the setting of ILW by viewing ILW as a perturbation of BO and making use of the smoothing property of the perturbation term.
{"title":"Unconditional deep-water limit of the intermediate long wave equation in low-regularity.","authors":"Justin Forlano, Guopeng Li, Tengfei Zhao","doi":"10.1007/s00030-025-01037-7","DOIUrl":"https://doi.org/10.1007/s00030-025-01037-7","url":null,"abstract":"<p><p>In this paper, we establish the unconditional deep-water limit of the intermediate long wave equation (ILW) to the Benjamin-Ono equation (BO) in low-regularity Sobolev spaces on both the real line and the circle. Our main tool is new unconditional uniqueness results for ILW in <math><msup><mi>H</mi> <mi>s</mi></msup> </math> when <math> <mrow><msub><mi>s</mi> <mn>0</mn></msub> <mo><</mo> <mi>s</mi> <mo>≤</mo> <mfrac><mn>1</mn> <mn>4</mn></mfrac> </mrow> </math> on the line and <math> <mrow><msub><mi>s</mi> <mn>0</mn></msub> <mo><</mo> <mi>s</mi> <mo><</mo> <mfrac><mn>1</mn> <mn>2</mn></mfrac> </mrow> </math> on the circle, where <math> <mrow><msub><mi>s</mi> <mn>0</mn></msub> <mo>=</mo> <mn>3</mn> <mo>-</mo> <msqrt><mrow><mn>33</mn> <mo>/</mo> <mn>4</mn></mrow> </msqrt> <mo>≈</mo> <mn>0.1277</mn></mrow> </math> . Here, we adapt the strategy of Moşincat-Pilod (Pure Appl Anal 5:285-322, 2023) for BO to the setting of ILW by viewing ILW as a perturbation of BO and making use of the smoothing property of the perturbation term.</p>","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":"32 2","pages":"28"},"PeriodicalIF":1.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11836242/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143469789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-01Epub Date: 2024-08-06DOI: 10.1007/s00030-024-00985-w
Juan Yang, Jeff Morgan, Bao Quoc Tang
The global existence and boundedness of solutions to quasi-linear reaction-diffusion systems are investigated. The system arises from compartmental models describing the spread of infectious diseases proposed in Viguerie et al. (Appl Math Lett 111:106617, 2021); Viguerie et al. (Comput Mech 66(5):1131-1152, 2020), where the diffusion rate is assumed to depend on the total population, leading to quasilinear diffusion with possible degeneracy. The mathematical analysis of this model has been addressed recently in Auricchio et al. (Math Methods Appl Sci 46:12529-12548, 2023) where it was essentially assumed that all sub-populations diffuse at the same rate, which yields a positive lower bound of the total population, thus removing the degeneracy. In this work, we remove this assumption completely and show the global existence and boundedness of solutions by exploiting a recently developed -energy method. Our approach is applicable to a larger class of systems and is sufficiently robust to allow model variants and different boundary conditions.
研究了准线性反应扩散系统解的全局存在性和有界性。该系统源于 Viguerie 等人（Appl Math Lett 111:106617, 2021）和 Viguerie 等人（Comput Mech 66(5):1131-1152, 2020）提出的描述传染病传播的分区模型，其中假定扩散率取决于总人口，从而导致可能存在退化的准线性扩散。最近，Auricchio 等人（Math Methods Appl Sci 46:12529-12548, 2023）对这一模型进行了数学分析，主要假设所有子种群的扩散速率相同，从而得到总种群的正下限，从而消除了退化现象。在这项工作中，我们完全取消了这一假设，并利用最近开发的 L p 能量方法证明了解的全局存在性和有界性。我们的方法适用于更大类别的系统，并具有足够的鲁棒性，允许模型变体和不同的边界条件。
{"title":"On quasi-linear reaction diffusion systems arising from compartmental SEIR models.","authors":"Juan Yang, Jeff Morgan, Bao Quoc Tang","doi":"10.1007/s00030-024-00985-w","DOIUrl":"10.1007/s00030-024-00985-w","url":null,"abstract":"<p><p>The global existence and boundedness of solutions to quasi-linear reaction-diffusion systems are investigated. The system arises from compartmental models describing the spread of infectious diseases proposed in Viguerie et al. (Appl Math Lett 111:106617, 2021); Viguerie et al. (Comput Mech 66(5):1131-1152, 2020), where the diffusion rate is assumed to depend on the total population, leading to quasilinear diffusion with possible degeneracy. The mathematical analysis of this model has been addressed recently in Auricchio et al. (Math Methods Appl Sci 46:12529-12548, 2023) where it was essentially assumed that all sub-populations diffuse at the same rate, which yields a positive lower bound of the total population, thus removing the degeneracy. In this work, we remove this assumption completely and show the global existence and boundedness of solutions by exploiting a recently developed <math><msup><mi>L</mi> <mi>p</mi></msup> </math> -energy method. Our approach is applicable to a larger class of systems and is sufficiently robust to allow model variants and different boundary conditions.</p>","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":"31 5","pages":"98"},"PeriodicalIF":1.1,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11303479/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141908124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-04DOI: 10.1007/s00030-023-00889-1
Sébastien Court
Abstract The purpose of this paper is to model mathematically mechanical aspects of cardiac tissues. The latter constitute an elastic domain whose total volume remains constant. The time deformation of the heart tissue is modeled with the elastodynamics equations dealing with the displacement field as main unknown. These equations are coupled with a pressure whose variations characterize the heart beat. This pressure variable corresponds to a Lagrange multiplier associated with the so-called global injectivity condition. We derive the corresponding coupled system with nonhomogeneous boundary conditions where the pressure variable appears. For mathematical convenience a damping term is added, and for a given class of strain energies we prove the existence of local-in-time solutions in the context of the $$L^p$$ Lp -parabolic maximal regularity.
摘要本文的目的是对心脏组织的力学方面进行数学建模。后者构成了一个总积保持恒定的弹性域。采用以位移场为主要未知量的弹性动力学方程来模拟心脏组织的时间变形。这些方程式与压力相结合，压力的变化是心跳的特征。这个压力变量对应于与所谓的全局注入条件相关的拉格朗日乘数。导出了具有压力变量的非齐次边界条件下的耦合系统。为了数学上的方便，我们增加了一个阻尼项，并且对于给定的应变能，我们证明了在$$L^p$$ L p -抛物极大正则性条件下局部解的存在性。
{"title":"A damped elastodynamics system under the global injectivity condition: local wellposedness in $$L^p$$-spaces","authors":"Sébastien Court","doi":"10.1007/s00030-023-00889-1","DOIUrl":"https://doi.org/10.1007/s00030-023-00889-1","url":null,"abstract":"Abstract The purpose of this paper is to model mathematically mechanical aspects of cardiac tissues. The latter constitute an elastic domain whose total volume remains constant. The time deformation of the heart tissue is modeled with the elastodynamics equations dealing with the displacement field as main unknown. These equations are coupled with a pressure whose variations characterize the heart beat. This pressure variable corresponds to a Lagrange multiplier associated with the so-called global injectivity condition. We derive the corresponding coupled system with nonhomogeneous boundary conditions where the pressure variable appears. For mathematical convenience a damping term is added, and for a given class of strain energies we prove the existence of local-in-time solutions in the context of the $$L^p$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -parabolic maximal regularity.","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":"4 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135774232","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 : 2023-10-26DOI: 10.1007/s00030-023-00891-7
Davide Barilari, Karen Habermann
Abstract We construct and study the intrinsic sub-Laplacian, defined outside the set of characteristic points, for a smooth hypersurface embedded in a contact sub-Riemannian manifold. We prove that, away from characteristic points, the intrinsic sub-Laplacian arises as the limit of Laplace–Beltrami operators built by means of Riemannian approximations to the sub-Riemannian structure using the Reeb vector field. We carefully analyse three families of model cases for this setting obtained by considering canonical hypersurfaces embedded in model spaces for contact sub-Riemannian manifolds. In these model cases, we show that the intrinsic sub-Laplacian is stochastically complete and in particular, that the stochastic process induced by the intrinsic sub-Laplacian almost surely does not hit characteristic points.
{"title":"Intrinsic sub-Laplacian for hypersurface in a contact sub-Riemannian manifold","authors":"Davide Barilari, Karen Habermann","doi":"10.1007/s00030-023-00891-7","DOIUrl":"https://doi.org/10.1007/s00030-023-00891-7","url":null,"abstract":"Abstract We construct and study the intrinsic sub-Laplacian, defined outside the set of characteristic points, for a smooth hypersurface embedded in a contact sub-Riemannian manifold. We prove that, away from characteristic points, the intrinsic sub-Laplacian arises as the limit of Laplace–Beltrami operators built by means of Riemannian approximations to the sub-Riemannian structure using the Reeb vector field. We carefully analyse three families of model cases for this setting obtained by considering canonical hypersurfaces embedded in model spaces for contact sub-Riemannian manifolds. In these model cases, we show that the intrinsic sub-Laplacian is stochastically complete and in particular, that the stochastic process induced by the intrinsic sub-Laplacian almost surely does not hit characteristic points.","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":"20 1-2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134907020","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 : 2023-10-20DOI: 10.1007/s00030-023-00890-8
Matt Holzer, Matthew Kearney, Samuel Molseed, Katie Tuttle, David Wigginton
{"title":"Pushed fronts in a Fisher–KPP–Burgers system using geometric desingularization","authors":"Matt Holzer, Matthew Kearney, Samuel Molseed, Katie Tuttle, David Wigginton","doi":"10.1007/s00030-023-00890-8","DOIUrl":"https://doi.org/10.1007/s00030-023-00890-8","url":null,"abstract":"","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135616056","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 : 2023-10-19DOI: 10.1007/s00030-023-00892-6
Delia Schiera
{"title":"A family of nonlocal degenerate operators: maximum principles and related properties","authors":"Delia Schiera","doi":"10.1007/s00030-023-00892-6","DOIUrl":"https://doi.org/10.1007/s00030-023-00892-6","url":null,"abstract":"","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135730517","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 : 2023-10-06DOI: 10.1007/s00030-023-00888-2
Dimitrios Gazoulis
Abstract We will prove that solutions of the Allen–Cahn equations that satisfy the equipartition of the energy can be transformed into solutions of the Euler equations with constant pressure. As a consequence, we obtain De Giorgi type results, that is, the level sets of entire solutions are hyperplanes. Also, we will determine the structure of solutions of the Allen–Cahn system in two dimensions that satisfy the equipartition. In addition, we apply the Leray projection on the Allen–Cahn system and provide some explicit entire solutions. Finally, we obtain some examples of smooth entire solutions of the Euler equations. For specific type of initial conditions, some of these solutions can be extended to the Navier–Stokes equations. The motivation of this paper is to find a transformation that relates the solutions of the Allen–Cahn equations to solutions of the minimal surface equation of one dimension less. We prove this result for equipartitioned solutions in dimension three.
{"title":"A Relation of the Allen–Cahn equations and the Euler equations and applications of the equipartition","authors":"Dimitrios Gazoulis","doi":"10.1007/s00030-023-00888-2","DOIUrl":"https://doi.org/10.1007/s00030-023-00888-2","url":null,"abstract":"Abstract We will prove that solutions of the Allen–Cahn equations that satisfy the equipartition of the energy can be transformed into solutions of the Euler equations with constant pressure. As a consequence, we obtain De Giorgi type results, that is, the level sets of entire solutions are hyperplanes. Also, we will determine the structure of solutions of the Allen–Cahn system in two dimensions that satisfy the equipartition. In addition, we apply the Leray projection on the Allen–Cahn system and provide some explicit entire solutions. Finally, we obtain some examples of smooth entire solutions of the Euler equations. For specific type of initial conditions, some of these solutions can be extended to the Navier–Stokes equations. The motivation of this paper is to find a transformation that relates the solutions of the Allen–Cahn equations to solutions of the minimal surface equation of one dimension less. We prove this result for equipartitioned solutions in dimension three.","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135347489","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 : 2023-10-04DOI: 10.1007/s00030-023-00877-5
Ugo Bessi
{"title":"Harmonic embeddings of the Stretched Sierpinski Gasket","authors":"Ugo Bessi","doi":"10.1007/s00030-023-00877-5","DOIUrl":"https://doi.org/10.1007/s00030-023-00877-5","url":null,"abstract":"","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135591708","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 : 2023-10-03DOI: 10.1007/s00030-023-00886-4
Eleonora Cinti, Francesca Colasuonno
Abstract We establish a priori $$L^infty $$ L∞ -estimates for non-negative solutions of a semilinear nonlocal Neumann problem. As a consequence of these estimates, we get non-existence of non-constant solutions under suitable assumptions on the diffusion coefficient and on the nonlinearity. Moreover, we prove an existence result for radial, radially non-decreasing solutions in the case of a possible supercritical nonlinearity, extending to the case $$00<s≤1/2 the analysis started in [7].
{"title":"Existence and non-existence results for a semilinear fractional Neumann problem","authors":"Eleonora Cinti, Francesca Colasuonno","doi":"10.1007/s00030-023-00886-4","DOIUrl":"https://doi.org/10.1007/s00030-023-00886-4","url":null,"abstract":"Abstract We establish a priori $$L^infty $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> -estimates for non-negative solutions of a semilinear nonlocal Neumann problem. As a consequence of these estimates, we get non-existence of non-constant solutions under suitable assumptions on the diffusion coefficient and on the nonlinearity. Moreover, we prove an existence result for radial, radially non-decreasing solutions in the case of a possible supercritical nonlinearity, extending to the case $$0<sle 1/2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>s</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> the analysis started in [7].","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135695578","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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