Abstract This article considers the Cauchy problem for compressible Euler system in R {bf{R}} with damping, in which the coefficient depends on the space variable. Assuming the initial density has a small perturbation around a constant state and both the small perturbation and the small initial velocity field are compact supported, finite-time blow-up result will be established. This result reveals the fact that if the space-dependent damping coefficient decays fast enough in the far field (belongs to L 1 ( R ) {L}^{1}left({bf{R}}) ), then the damping is non-effective to the long-time behavior of the solution.
{"title":"Blow-up for compressible Euler system with space-dependent damping in 1-D","authors":"Jinbo Geng, Ning-An Lai, Manwai Yuen, Jiang Zhou","doi":"10.1515/anona-2022-0304","DOIUrl":"https://doi.org/10.1515/anona-2022-0304","url":null,"abstract":"Abstract This article considers the Cauchy problem for compressible Euler system in R {bf{R}} with damping, in which the coefficient depends on the space variable. Assuming the initial density has a small perturbation around a constant state and both the small perturbation and the small initial velocity field are compact supported, finite-time blow-up result will be established. This result reveals the fact that if the space-dependent damping coefficient decays fast enough in the far field (belongs to L 1 ( R ) {L}^{1}left({bf{R}}) ), then the damping is non-effective to the long-time behavior of the solution.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42971196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In the present work, we find the Lie point symmetries of the Ricci flow on an n -dimensional manifold, and we introduce a method in order to reutilize these symmetries to obtain the Lie point symmetries of particular metrics. We apply this method to retrieve the Lie point symmetries of the Einstein equations (seen as a “static” Ricci flow) and of some particular types of metrics of interest, such as, on warped products of manifolds. Finally, we use the symmetries found to obtain invariant solutions of the Ricci flow for the particular families of metrics considered.
{"title":"Symmetries of Ricci flows","authors":"Enrique López, Stylianos Dimas, Yuri Bozhkov","doi":"10.1515/anona-2023-0106","DOIUrl":"https://doi.org/10.1515/anona-2023-0106","url":null,"abstract":"Abstract In the present work, we find the Lie point symmetries of the Ricci flow on an n -dimensional manifold, and we introduce a method in order to reutilize these symmetries to obtain the Lie point symmetries of particular metrics. We apply this method to retrieve the Lie point symmetries of the Einstein equations (seen as a “static” Ricci flow) and of some particular types of metrics of interest, such as, on warped products of manifolds. Finally, we use the symmetries found to obtain invariant solutions of the Ricci flow for the particular families of metrics considered.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135954347","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this work, we consider the topological gradient method to deal with an inverse problem associated with three-dimensional Stokes equations. The problem consists in detecting unknown point forces acting on fluid from measurements on the boundary of the domain. We present an asymptotic expansion of the considered cost function using the topological sensitivity analysis method. A detection algorithm is then presented using the developed formula. Some numerical tests are presented to show the efficiency of the presented algorithm.
{"title":"On the topological gradient method for an inverse problem resolution","authors":"Mohamed Abdelwahed, Nejmeddine Chorfi","doi":"10.1515/anona-2023-0109","DOIUrl":"https://doi.org/10.1515/anona-2023-0109","url":null,"abstract":"Abstract In this work, we consider the topological gradient method to deal with an inverse problem associated with three-dimensional Stokes equations. The problem consists in detecting unknown point forces acting on fluid from measurements on the boundary of the domain. We present an asymptotic expansion of the considered cost function using the topological sensitivity analysis method. A detection algorithm is then presented using the developed formula. Some numerical tests are presented to show the efficiency of the presented algorithm.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136304599","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In the present work, we establish a blow-up criterion for viscoelastic wave equations with nonlinear damping, logarithmic source, delay in the velocity, and acoustic boundary conditions. Due to the nonlinear damping term, we cannot apply the concavity method introduced by Levine. Thus, we use the energy method to show that the solution with negative initial energy blows up after finite time. Furthermore, we investigate the upper and lower bounds of the blow-up time.
{"title":"Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions","authors":"Sun‐Hye Park","doi":"10.1515/anona-2022-0310","DOIUrl":"https://doi.org/10.1515/anona-2022-0310","url":null,"abstract":"Abstract In the present work, we establish a blow-up criterion for viscoelastic wave equations with nonlinear damping, logarithmic source, delay in the velocity, and acoustic boundary conditions. Due to the nonlinear damping term, we cannot apply the concavity method introduced by Levine. Thus, we use the energy method to show that the solution with negative initial energy blows up after finite time. Furthermore, we investigate the upper and lower bounds of the blow-up time.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46601371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We present a survey concerning the convergence, as the viscosity goes to zero, of the solutions to the three-dimensional evolutionary Navier-Stokes equations to solutions of the Euler equations. After considering the Cauchy problem, particular attention is given to the convergence under Navier slip-type boundary conditions. We show that, in the presence of flat boundaries (typically, the half-space case), convergence holds, uniformly in time, with respect to the initial data’s norm. In spite of this result (and of a similar result for arbitrary two-dimensional domains), strong inviscid limit results are proved to be false in general domains, in correspondence to a very large family of smooth initial data. In Section 6, we present a result in this direction.
{"title":"A survey on some vanishing viscosity limit results","authors":"H. Beirão da Veiga, F. Crispo","doi":"10.1515/anona-2022-0309","DOIUrl":"https://doi.org/10.1515/anona-2022-0309","url":null,"abstract":"Abstract We present a survey concerning the convergence, as the viscosity goes to zero, of the solutions to the three-dimensional evolutionary Navier-Stokes equations to solutions of the Euler equations. After considering the Cauchy problem, particular attention is given to the convergence under Navier slip-type boundary conditions. We show that, in the presence of flat boundaries (typically, the half-space case), convergence holds, uniformly in time, with respect to the initial data’s norm. In spite of this result (and of a similar result for arbitrary two-dimensional domains), strong inviscid limit results are proved to be false in general domains, in correspondence to a very large family of smooth initial data. In Section 6, we present a result in this direction.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48168710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this article, we study the asymptotic behaviour at infinity for viscosity solutions to a singular Monge-Ampère equation in half space from affine geometry. In particular, we extend the Liouville theorem for smooth solutions to the case of viscosity solutions by a completely different method from the smooth case.
{"title":"Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation","authors":"H. Jian, Xianduo Wang","doi":"10.1515/anona-2022-0284","DOIUrl":"https://doi.org/10.1515/anona-2022-0284","url":null,"abstract":"Abstract In this article, we study the asymptotic behaviour at infinity for viscosity solutions to a singular Monge-Ampère equation in half space from affine geometry. In particular, we extend the Liouville theorem for smooth solutions to the case of viscosity solutions by a completely different method from the smooth case.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47584886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider analytic families of planar vector fields depending analytically on the parameters in Λ Lambda that guarantee the existence of a (may be degenerate and with characteristic directions) monodromic singularity. We characterize the structure of the asymptotic Dulac series of the Poincaré map associated to the singularity when the family possesses a Puiseux inverse integrating factor in terms of its multiplicity and index. This characterization is only valid in a restricted monodromic parameter space Λ Λ ∗ Lambda backslash {Lambda }^{ast } associated to the nonexistence of local curves with zero angular speed. As a byproduct, we are able to study the center-focus problem (under the assumption of the existence of some Cauchy principal values) in very degenerated cases where no other tools are available. We illustrate the theory with several nontrivial examples.
{"title":"The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor","authors":"I. A. García, J. Giné","doi":"10.1515/anona-2022-0314","DOIUrl":"https://doi.org/10.1515/anona-2022-0314","url":null,"abstract":"Abstract We consider analytic families of planar vector fields depending analytically on the parameters in Λ Lambda that guarantee the existence of a (may be degenerate and with characteristic directions) monodromic singularity. We characterize the structure of the asymptotic Dulac series of the Poincaré map associated to the singularity when the family possesses a Puiseux inverse integrating factor in terms of its multiplicity and index. This characterization is only valid in a restricted monodromic parameter space Λ Λ ∗ Lambda backslash {Lambda }^{ast } associated to the nonexistence of local curves with zero angular speed. As a byproduct, we are able to study the center-focus problem (under the assumption of the existence of some Cauchy principal values) in very degenerated cases where no other tools are available. We illustrate the theory with several nontrivial examples.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43206351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this article, we are concerned with normalized solutions for the p p -Laplacian equation with a trapping potential and L r {L}^{r} -supercritical growth, where r = p r=p or 2 . 2. The solutions correspond to critical points of the underlying energy functional subject to the L r {L}^{r} -norm constraint, namely, ∫ R N ∣ u ∣ r d x = c {int }_{{{mathbb{R}}}^{N}}| u{| }^{r}{rm{d}}x=c for given c > 0 . cgt 0. When r = p , r=p, we show that such problem has a ground state with positive energy for c c small enough. When r = 2 , r=2, we show that such problem has at least two solutions both with positive energy, which one is a ground state and the other one is a high-energy solution.
摘要在本文中，我们讨论了具有俘获势和Lr｛L｝^｛r｝-超临界生长的p-拉普拉斯方程的归一化解，其中r＝p r＝p或2。2.这些解对应于Lr｛L｝^｛r｝-范数约束下的潜在能量泛函主体的临界点，即，对于给定的c>0，当给定c>0时，ξr NÜuÜr d x=c。cgt 0。当r＝p，r＝p时，我们证明了对于c c足够小，这样的问题具有正能量的基态。当r＝2，r＝2时，我们证明了这类问题至少有两个解都是正能量的，一个是基态，另一个是高能解。
{"title":"Normalized solutions for the p-Laplacian equation with a trapping potential","authors":"Chao Wang, Juntao Sun","doi":"10.1515/anona-2022-0291","DOIUrl":"https://doi.org/10.1515/anona-2022-0291","url":null,"abstract":"Abstract In this article, we are concerned with normalized solutions for the p p -Laplacian equation with a trapping potential and L r {L}^{r} -supercritical growth, where r = p r=p or 2 . 2. The solutions correspond to critical points of the underlying energy functional subject to the L r {L}^{r} -norm constraint, namely, ∫ R N ∣ u ∣ r d x = c {int }_{{{mathbb{R}}}^{N}}| u{| }^{r}{rm{d}}x=c for given c > 0 . cgt 0. When r = p , r=p, we show that such problem has a ground state with positive energy for c c small enough. When r = 2 , r=2, we show that such problem has at least two solutions both with positive energy, which one is a ground state and the other one is a high-energy solution.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43212309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this article, we formulate a model describing the evolution of thickness of a grounded shallow ice sheet. The thickness of the ice sheet is constrained to be nonnegative. This renders the problem under consideration an obstacle problem. A rigorous analysis shows that the model is thus governed by a set of variational inequalities that involve nonlinearities in the time derivative and in the elliptic term, and that it admits solutions, whose existence is established by means of a semi-discrete scheme and the penalty method.
{"title":"On the dynamics of grounded shallow ice sheets: Modeling and analysis","authors":"Paolo Piersanti, R. Temam","doi":"10.1515/anona-2022-0280","DOIUrl":"https://doi.org/10.1515/anona-2022-0280","url":null,"abstract":"Abstract In this article, we formulate a model describing the evolution of thickness of a grounded shallow ice sheet. The thickness of the ice sheet is constrained to be nonnegative. This renders the problem under consideration an obstacle problem. A rigorous analysis shows that the model is thus governed by a set of variational inequalities that involve nonlinearities in the time derivative and in the elliptic term, and that it admits solutions, whose existence is established by means of a semi-discrete scheme and the penalty method.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47943948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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