SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3412-3429, June 2024. Abstract. We prove the existence of an open set minimizing the first Dirichlet eigenvalue of an elliptic operator with bounded, measurable coefficients, over all open sets of a given measure. Our proof is based on a free boundary approach: we characterize the eigenfunction on the optimal set as the minimizer of a penalized functional, and derive openness of the optimal set as a consequence of a Hölder estimate for the eigenfunction. We also prove that the optimal eigenfunction grows at most linearly from the free boundary, i.e., it is Lipschitz continuous at free boundary points.
{"title":"Optimal Domains for Elliptic Eigenvalue Problems with Rough Coefficients","authors":"Stanley Snelson, Eduardo V. Teixeira","doi":"10.1137/22m1523820","DOIUrl":"https://doi.org/10.1137/22m1523820","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3412-3429, June 2024. <br/> Abstract. We prove the existence of an open set minimizing the first Dirichlet eigenvalue of an elliptic operator with bounded, measurable coefficients, over all open sets of a given measure. Our proof is based on a free boundary approach: we characterize the eigenfunction on the optimal set as the minimizer of a penalized functional, and derive openness of the optimal set as a consequence of a Hölder estimate for the eigenfunction. We also prove that the optimal eigenfunction grows at most linearly from the free boundary, i.e., it is Lipschitz continuous at free boundary points.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936324","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}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3386-3411, June 2024. Abstract. Pedestrian evacuation in a corridor can de described mathematically by different variants of the model introduced by R. L. Hughes [Transp. Res. Part B Methodol., 36 (2002), pp. 507–535]. We identify a class of such models for which existence of a solution is obtained via a topological fixed point argument. In these models, the dynamics of the pedestrian density [math] (governed by a discontinuous-flux Lighthill, Whitham, and Richards model [math]) is coupled to the computation of a Lipschitz continuous “turning curve” [math]. We illustrate this construction by several examples, including the Hughes model with affine cost (a variant of the original problem that is encompassed in the framework of El-Khatib, Goatin, and Rosini [Z. Angew. Math. Phys., 64 (2013), pp. 223–251]. Existence holds either with open-end boundary conditions or with boundary conditions corresponding to panic behavior with capacity drop at exits. Other examples put forward versions of the Hughes model with inertial dynamics of the turning curve and with general costs.
SIAM 数学分析期刊》，第 56 卷，第 3 期，第 3386-3411 页，2024 年 6 月。 摘要。走廊中的行人疏散可以用 R. L. Hughes [Transp. Res. Part B Methodol.我们确定了一类这样的模型，通过拓扑定点论证可以得到解的存在性。在这些模型中，行人密度的动态[数学]（由非连续流动的莱特希尔、惠瑟姆和理查兹模型[数学]控制）与利普斯奇兹连续 "转弯曲线 "的计算[数学]相耦合。我们用几个例子来说明这一构造，包括具有仿射成本的休斯模型（El-Khatib、Goatin 和 Rosini [Z. Angew. Math. Phys.在开端边界条件或与出口处容量下降的恐慌行为相对应的边界条件下，存在性都是成立的。其他例子提出了具有转弯曲线惯性动力学和一般成本的休斯模型版本。
{"title":"Existence of Solutions for a Class of One-Dimensional Models of Pedestrian Evacuations","authors":"Boris Andreianov, Theo Girard","doi":"10.1137/23m1550256","DOIUrl":"https://doi.org/10.1137/23m1550256","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3386-3411, June 2024. <br/> Abstract. Pedestrian evacuation in a corridor can de described mathematically by different variants of the model introduced by R. L. Hughes [Transp. Res. Part B Methodol., 36 (2002), pp. 507–535]. We identify a class of such models for which existence of a solution is obtained via a topological fixed point argument. In these models, the dynamics of the pedestrian density [math] (governed by a discontinuous-flux Lighthill, Whitham, and Richards model [math]) is coupled to the computation of a Lipschitz continuous “turning curve” [math]. We illustrate this construction by several examples, including the Hughes model with affine cost (a variant of the original problem that is encompassed in the framework of El-Khatib, Goatin, and Rosini [Z. Angew. Math. Phys., 64 (2013), pp. 223–251]. Existence holds either with open-end boundary conditions or with boundary conditions corresponding to panic behavior with capacity drop at exits. Other examples put forward versions of the Hughes model with inertial dynamics of the turning curve and with general costs.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140882741","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}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3110-3143, June 2024. Abstract. We give a new proof of the scattering below the ground state energy level for a class of nonlinear Schrödinger equations (NLS) with mass-energy intercritical competing nonlinearities. Specifically, the NLS has a focusing leading order nonlinearity with a defocusing perturbation. Our strategy combines interaction Morawetz estimates à la Dodson–Murphy and a new crucial bound for the Pohozaev functional of localized functions, which is essential to overcome the lack of a monotonicity condition. Furthermore, we give the rate of blow-up for symmetric solutions.
{"title":"Scattering for Nonradial 3D NLS with Combined Nonlinearities: The Interaction Morawetz Approach","authors":"Jacopo Bellazzini, Van Duong Dinh, Luigi Forcella","doi":"10.1137/23m1559063","DOIUrl":"https://doi.org/10.1137/23m1559063","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3110-3143, June 2024. <br/> Abstract. We give a new proof of the scattering below the ground state energy level for a class of nonlinear Schrödinger equations (NLS) with mass-energy intercritical competing nonlinearities. Specifically, the NLS has a focusing leading order nonlinearity with a defocusing perturbation. Our strategy combines interaction Morawetz estimates à la Dodson–Murphy and a new crucial bound for the Pohozaev functional of localized functions, which is essential to overcome the lack of a monotonicity condition. Furthermore, we give the rate of blow-up for symmetric solutions.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140828112","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}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3275-3325, June 2024. Abstract. We study the asymptotic stability of a front connecting two unstable states. Such a structure typically appears when the stable state behind a Fisher–Kolmogorov–Petrovskii–Piskunov front destabilizes when going through an essential Turing bifurcation, giving rise to oscillating patterns. Despite the instability of both end-states, we obtain for the first time stability of such a structure against suitably localized perturbations, with algebraic temporal decay [math]. To deal with the instability behind the front, we simultaneously control the error in two different norms. In the first norm, enhanced diffusive decay is obtained at a linear level through pointwise resolvent estimates. In the second norm, better suited for nonlinear analysis, we show that the error stays bounded in time by use of mode filters.
{"title":"Nonlinear Convective Stability of a Critical Pulled Front Undergoing a Turing Bifurcation at Its Back: A Case Study","authors":"Louis Garénaux","doi":"10.1137/21m1451038","DOIUrl":"https://doi.org/10.1137/21m1451038","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3275-3325, June 2024. <br/> Abstract. We study the asymptotic stability of a front connecting two unstable states. Such a structure typically appears when the stable state behind a Fisher–Kolmogorov–Petrovskii–Piskunov front destabilizes when going through an essential Turing bifurcation, giving rise to oscillating patterns. Despite the instability of both end-states, we obtain for the first time stability of such a structure against suitably localized perturbations, with algebraic temporal decay [math]. To deal with the instability behind the front, we simultaneously control the error in two different norms. In the first norm, enhanced diffusive decay is obtained at a linear level through pointwise resolvent estimates. In the second norm, better suited for nonlinear analysis, we show that the error stays bounded in time by use of mode filters.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140828137","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}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3326-3356, June 2024. Abstract. In this article, we study a Mayer optimal control problem on the space of Borel probability measures over a compact Riemannian manifold [math]. This is motivated by certain situations where a central planner of a deterministic controlled system has only imperfect information on the initial state of the system. The lack of information here is very specific. It is described by a Borel probability measure along which the initial state is distributed. We define a new notion of viscosity in this space by taking test functions that are directionally differentiable and can be written as a difference of two semiconvex functions. With this choice of test functions, we extend the notion of viscosity to Hamilton–Jacobi–Bellman equations in Wasserstein spaces and we establish that the value function is the unique viscosity solution of a Hamilton–Jacobi–Bellman equation in the Wasserstein space over [math].
{"title":"Deterministic Optimal Control on Riemannian Manifolds Under Probability Knowledge of the Initial Condition","authors":"Frédéric Jean, Othmane Jerhaoui, Hasnaa Zidani","doi":"10.1137/23m1575251","DOIUrl":"https://doi.org/10.1137/23m1575251","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3326-3356, June 2024. <br/> Abstract. In this article, we study a Mayer optimal control problem on the space of Borel probability measures over a compact Riemannian manifold [math]. This is motivated by certain situations where a central planner of a deterministic controlled system has only imperfect information on the initial state of the system. The lack of information here is very specific. It is described by a Borel probability measure along which the initial state is distributed. We define a new notion of viscosity in this space by taking test functions that are directionally differentiable and can be written as a difference of two semiconvex functions. With this choice of test functions, we extend the notion of viscosity to Hamilton–Jacobi–Bellman equations in Wasserstein spaces and we establish that the value function is the unique viscosity solution of a Hamilton–Jacobi–Bellman equation in the Wasserstein space over [math].","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140828107","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}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3357-3385, June 2024. Abstract. The quasi-geostrohpic (QG) equation has been used to capture the asymptotic dynamics of the rotating stratified Boussinesq flows in the regime of strong stratification and rapid rotation. In this paper, we establish the invalidity of such approximation when the rotation-stratification ratio is either fixed to be unity or tends to unity sufficiently slowly in the asymptotic regime: the difference between the rotating stratified Boussinesq flow and the corresponding QG flow remains strictly away from zero, independently of the intensities of rotation and stratification. In contrast, we also show that the convergence occurs when the rotation-stratification ratio is fixed to be a number other than unity or converges to unity sufficiently fast. As a corollary, we compute a lower bound of the convergence rate, which blows up as the rotation-stratification ratio goes to unity.
{"title":"Nonconvergence of the Rotating Stratified Flows Toward the Quasi-Geostrophic Dynamics","authors":"Min Jun Jo, Junha Kim, Jihoon Lee","doi":"10.1137/23m1559130","DOIUrl":"https://doi.org/10.1137/23m1559130","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3357-3385, June 2024. <br/>Abstract. The quasi-geostrohpic (QG) equation has been used to capture the asymptotic dynamics of the rotating stratified Boussinesq flows in the regime of strong stratification and rapid rotation. In this paper, we establish the invalidity of such approximation when the rotation-stratification ratio is either fixed to be unity or tends to unity sufficiently slowly in the asymptotic regime: the difference between the rotating stratified Boussinesq flow and the corresponding QG flow remains strictly away from zero, independently of the intensities of rotation and stratification. In contrast, we also show that the convergence occurs when the rotation-stratification ratio is fixed to be a number other than unity or converges to unity sufficiently fast. As a corollary, we compute a lower bound of the convergence rate, which blows up as the rotation-stratification ratio goes to unity.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140828133","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}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3082-3109, June 2024. Abstract. In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing nonlinear Schrödinger equation with various (deterministic and random) initial data. We show that nonlinear solutions enjoy the same decay rate as the linear ones. The regularity assumption on the initial data is much lower than in previous results (see [C. Fan and Z. Zhao, Discrete Contin. Dyn. Syst., 41 (2021), pp. 3973–3984] and the references therein), and, moreover, we quantify the decay, which is another novelty of this work. Furthermore, we show that the (physical) randomization of the initial data can be used to replace the [math]-data assumption (see [C. Fan and Z. Zhao, Proc. Amer. Math. Soc., 151 (2023), pp. 2527–2542] for the necessity of the [math]-data assumption).
SIAM 数学分析期刊》，第 56 卷，第 3 期，第 3082-3109 页，2024 年 6 月。 摘要。本文讨论了具有各种（确定性和随机）初始数据的三维立方离焦非线性薛定谔方程解的定量（点式）衰减估计。我们证明，非线性解与线性解具有相同的衰减率。对初始数据的正则性假设比以前的结果（见 [C. Fan and Z. Zhao, Discrete Contin. Dyn. Syst.此外，我们还证明了初始数据的（物理）随机化可以用来取代[math]-data 假设（关于[math]-data 假设的必要性，请参见[C. Fan and Z. Zhao, Proc. Amer. Math. Soc., 151 (2023), pp.）
{"title":"On Decaying Properties of Nonlinear Schrödinger Equations","authors":"Chenjie Fan, Gigliola Staffilani, Zehua Zhao","doi":"10.1137/23m1557544","DOIUrl":"https://doi.org/10.1137/23m1557544","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3082-3109, June 2024. <br/> Abstract. In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing nonlinear Schrödinger equation with various (deterministic and random) initial data. We show that nonlinear solutions enjoy the same decay rate as the linear ones. The regularity assumption on the initial data is much lower than in previous results (see [C. Fan and Z. Zhao, Discrete Contin. Dyn. Syst., 41 (2021), pp. 3973–3984] and the references therein), and, moreover, we quantify the decay, which is another novelty of this work. Furthermore, we show that the (physical) randomization of the initial data can be used to replace the [math]-data assumption (see [C. Fan and Z. Zhao, Proc. Amer. Math. Soc., 151 (2023), pp. 2527–2542] for the necessity of the [math]-data assumption).","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140828138","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}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3203-3251, June 2024. Abstract. We consider a one-dimensional, trapped, focusing Bose gas where [math] bosons interact with each other via both a two-body interaction potential of the form [math] and an attractive three-body interaction potential of the form [math], where [math], [math], [math], [math], and [math]. The system is stable either for any [math] as long as [math] —the critical strength of the one-dimensional focusing quintic nonlinear Schrödinger (NLS) equation— or for [math] when [math]. In the former case, fixing [math], we prove that in the mean-field limit the many-body system exhibits the Bose–Einstein condensation on the cubic-quintic NLS ground states. When assuming [math] and [math] as [math], with the former convergence being slow enough and “not faster” than the latter, we prove that the ground state of the system is fully condensed on the (unique) solution to the quintic NLS equation. In the latter case, [math] fixed, we obtain the convergence of many-body energy for small [math] when [math] is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence [math] is “faster” than the slow enough convergence [math].
{"title":"On One-Dimensional Bose Gases with Two-Body and (Critical) Attractive Three-Body Interactions","authors":"Dinh-Thi Nguyen, Julien Ricaud","doi":"10.1137/22m1535139","DOIUrl":"https://doi.org/10.1137/22m1535139","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3203-3251, June 2024. <br/> Abstract. We consider a one-dimensional, trapped, focusing Bose gas where [math] bosons interact with each other via both a two-body interaction potential of the form [math] and an attractive three-body interaction potential of the form [math], where [math], [math], [math], [math], and [math]. The system is stable either for any [math] as long as [math] —the critical strength of the one-dimensional focusing quintic nonlinear Schrödinger (NLS) equation— or for [math] when [math]. In the former case, fixing [math], we prove that in the mean-field limit the many-body system exhibits the Bose–Einstein condensation on the cubic-quintic NLS ground states. When assuming [math] and [math] as [math], with the former convergence being slow enough and “not faster” than the latter, we prove that the ground state of the system is fully condensed on the (unique) solution to the quintic NLS equation. In the latter case, [math] fixed, we obtain the convergence of many-body energy for small [math] when [math] is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence [math] is “faster” than the slow enough convergence [math].","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140828136","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}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3252-3274, June 2024. Abstract. The present paper is devoted to investigating the second-grade fluid system in a strip domain [math]. We obtain the global well-posedness result with small analytic initial datum and justify the limit strictly from the hydrostatic second-grade fluid system to the hydrostatic Navier–Stokes system.
{"title":"Global Well-Posedness and Vanishing Normal Stress Coefficients for the Hydrostatic Second-Grade Fluid Equations","authors":"Marius Paicu, Tianyuan Yu, Ning Zhu","doi":"10.1137/23m1565085","DOIUrl":"https://doi.org/10.1137/23m1565085","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3252-3274, June 2024. <br/> Abstract. The present paper is devoted to investigating the second-grade fluid system in a strip domain [math]. We obtain the global well-posedness result with small analytic initial datum and justify the limit strictly from the hydrostatic second-grade fluid system to the hydrostatic Navier–Stokes system.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140828134","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}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3144-3202, June 2024. Abstract. It is challenging to perform a multiscale analysis of mesoscopic systems exhibiting singularities at the macroscopic scale. In this paper, we study the hydrodynamic limit of the Boltzmann equation [math] toward the singular solutions of 2D incompressible Euler equations whose vorticity is unbounded: [math]. We obtain a microscopic description of the singularity through the so-called kinetic vorticity and understand its behavior in the vicinity of the macroscopic singularity. As a consequence of our new analysis, we settle affirmatively an open problem of convergence toward Lagrangian solutions of the 2D incompressible Euler equation whose vorticity is unbounded ([math] for any fixed [math]). Moreover, we prove the convergence of kinetic vorticities toward the vorticity of the Lagrangian solution of the Euler equation. In particular, we obtain the rate of convergence when the vorticity blows up moderately in [math] as [math] (a localized Yudovich class).
{"title":"Vorticity Convergence from Boltzmann to 2D Incompressible Euler Equations Below Yudovich Class","authors":"Chanwoo Kim, Joonhyun La","doi":"10.1137/23m1549857","DOIUrl":"https://doi.org/10.1137/23m1549857","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3144-3202, June 2024. <br/> Abstract. It is challenging to perform a multiscale analysis of mesoscopic systems exhibiting singularities at the macroscopic scale. In this paper, we study the hydrodynamic limit of the Boltzmann equation [math] toward the singular solutions of 2D incompressible Euler equations whose vorticity is unbounded: [math]. We obtain a microscopic description of the singularity through the so-called kinetic vorticity and understand its behavior in the vicinity of the macroscopic singularity. As a consequence of our new analysis, we settle affirmatively an open problem of convergence toward Lagrangian solutions of the 2D incompressible Euler equation whose vorticity is unbounded ([math] for any fixed [math]). Moreover, we prove the convergence of kinetic vorticities toward the vorticity of the Lagrangian solution of the Euler equation. In particular, we obtain the rate of convergence when the vorticity blows up moderately in [math] as [math] (a localized Yudovich class).","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140828110","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}
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