Pub Date : 2025-02-24DOI: 10.1016/j.matpur.2025.103690
Karl Kunisch , Gengsheng Wang , Huaiqiang Yu
A quantitative frequency-domain condition related to the exponential stabilizability for infinite-dimensional linear control systems is presented. It is proven that this condition is necessary and sufficient for the stabilizability of special systems, while it is a necessary condition for the stabilizability in general. Applications are provided.
{"title":"Frequency-domain criterion on the stabilizability for infinite-dimensional linear control systems","authors":"Karl Kunisch , Gengsheng Wang , Huaiqiang Yu","doi":"10.1016/j.matpur.2025.103690","DOIUrl":"10.1016/j.matpur.2025.103690","url":null,"abstract":"<div><div>A quantitative frequency-domain condition related to the exponential stabilizability for infinite-dimensional linear control systems is presented. It is proven that this condition is necessary and sufficient for the stabilizability of special systems, while it is a necessary condition for the stabilizability in general. Applications are provided.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"196 ","pages":"Article 103690"},"PeriodicalIF":2.1,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143507535","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}
Pub Date : 2025-02-24DOI: 10.1016/j.matpur.2025.103694
Kyoung-Seog Lee , Han-Bom Moon
We show that the derived categories of symmetric products of a curve are embedded into the derived categories of the moduli spaces of vector bundles of large ranks on the curve. It supports a prediction of the existence of a semiorthogonal decomposition of the derived category of the moduli space, expected by a motivic computation. As an application, we show that all Jacobian varieties, symmetric products of curves, and all principally polarized abelian varieties of dimension at most three, are Fano visitors. We also obtain similar results for motives.
{"title":"Derived categories of symmetric products and moduli spaces of vector bundles on a curve","authors":"Kyoung-Seog Lee , Han-Bom Moon","doi":"10.1016/j.matpur.2025.103694","DOIUrl":"10.1016/j.matpur.2025.103694","url":null,"abstract":"<div><div>We show that the derived categories of symmetric products of a curve are embedded into the derived categories of the moduli spaces of vector bundles of large ranks on the curve. It supports a prediction of the existence of a semiorthogonal decomposition of the derived category of the moduli space, expected by a motivic computation. As an application, we show that all Jacobian varieties, symmetric products of curves, and all principally polarized abelian varieties of dimension at most three, are Fano visitors. We also obtain similar results for motives.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"197 ","pages":"Article 103694"},"PeriodicalIF":2.1,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509395","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}
Pub Date : 2025-02-24DOI: 10.1016/j.matpur.2025.103692
Thomas Alazard , Jeremy L. Marzuola , Jian Wang
Motivated by numerically modeling surface waves for inviscid Euler equations, we analyze linear models for damped water waves and establish decay properties for the energy for sufficiently regular initial configurations. Our findings give the explicit decay rates for the energy, but do not address reflection/transmission of waves at the interface of the damping. Still for a subset of the models considered, this represents the first result proving the decay of the energy of the surface wave models.
{"title":"Damping for fractional wave equations and applications to water waves","authors":"Thomas Alazard , Jeremy L. Marzuola , Jian Wang","doi":"10.1016/j.matpur.2025.103692","DOIUrl":"10.1016/j.matpur.2025.103692","url":null,"abstract":"<div><div>Motivated by numerically modeling surface waves for inviscid Euler equations, we analyze linear models for damped water waves and establish decay properties for the energy for sufficiently regular initial configurations. Our findings give the explicit decay rates for the energy, but do not address reflection/transmission of waves at the interface of the damping. Still for a subset of the models considered, this represents the first result proving the decay of the energy of the surface wave models.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"196 ","pages":"Article 103692"},"PeriodicalIF":2.1,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529175","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}
Pub Date : 2025-02-24DOI: 10.1016/j.matpur.2025.103686
Huai-Dong Cao , Junming Xie
This is a sequel to our paper [24], in which we investigated the geometry of 4-dimensional gradient shrinking Ricci solitons with half positive (nonnegative) isotropic curvature. In this paper, we mainly focus on 4-dimensional gradient steady Ricci solitons with nonnegative isotropic curvature (WPIC) or half nonnegative isotropic curvature (half WPIC). In particular, for 4D complete ancient solutions with WPIC, we are able to prove the 2-nonnegativity of the Ricci curvature and bound the curvature tensor Rm by . For 4D gradient steady solitons with WPIC, we obtain a classification result. We also give a partial classification of 4D gradient steady Ricci solitons with half WPIC. Moreover, we obtain a preliminary classification result for 4D complete gradient expanding Ricci solitons with WPIC. Finally, motivated by the recent work [59], we improve our earlier results in [24] on 4D gradient shrinking Ricci solitons with half PIC or half WPIC, and also provide a characterization of complete gradient Kähler-Ricci shrinkers in complex dimension two among 4-dimensional gradient Ricci shrinkers.
这是我们论文[24]的续集，在[24]中，我们研究了具有半正（非负）各向同性曲率的四维梯度收缩Ricci孤子的几何。本文主要研究具有非负各向同性曲率（WPIC）或半非负各向同性曲率（half non - anisotropic curvature， WPIC）的四维梯度稳定Ricci孤子。特别是对于具有WPIC的4D完全古解，我们证明了Ricci曲率的2-非负性，并将曲率张量Rm限定为|Rm|≤R。对于具有WPIC的四维梯度稳定孤子，我们得到了一个分类结果。给出了具有半WPIC的四维梯度稳定Ricci孤子的部分分类。此外，我们还利用WPIC获得了4D完全梯度展开Ricci孤子的初步分类结果。最后，在最近工作[59]的激励下，我们改进了[24]中关于半PIC或半WPIC的4D梯度收缩Ricci孤子的早期结果，并在4维梯度Ricci收缩子中给出了复二维完全梯度Kähler-Ricci收缩子的表征。
{"title":"Four-dimensional gradient Ricci solitons with (half) nonnegative isotropic curvature","authors":"Huai-Dong Cao , Junming Xie","doi":"10.1016/j.matpur.2025.103686","DOIUrl":"10.1016/j.matpur.2025.103686","url":null,"abstract":"<div><div>This is a sequel to our paper <span><span>[24]</span></span>, in which we investigated the geometry of 4-dimensional gradient shrinking Ricci solitons with half positive (nonnegative) isotropic curvature. In this paper, we mainly focus on 4-dimensional gradient steady Ricci solitons with nonnegative isotropic curvature (WPIC) or half nonnegative isotropic curvature (half WPIC). In particular, for 4D complete <em>ancient solutions</em> with WPIC, we are able to prove the 2-nonnegativity of the Ricci curvature and bound the curvature tensor <em>Rm</em> by <span><math><mo>|</mo><mi>R</mi><mi>m</mi><mo>|</mo><mo>≤</mo><mi>R</mi></math></span>. For 4D gradient steady solitons with WPIC, we obtain a classification result. We also give a partial classification of 4D gradient steady Ricci solitons with half WPIC. Moreover, we obtain a preliminary classification result for 4D complete gradient <em>expanding Ricci solitons</em> with WPIC. Finally, motivated by the recent work <span><span>[59]</span></span>, we improve our earlier results in <span><span>[24]</span></span> on 4D gradient <em>shrinking Ricci solitons</em> with half PIC or half WPIC, and also provide a characterization of complete gradient Kähler-Ricci shrinkers in complex dimension two among 4-dimensional gradient Ricci shrinkers.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"197 ","pages":"Article 103686"},"PeriodicalIF":2.1,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511264","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}
Pub Date : 2025-02-21DOI: 10.1016/j.matpur.2025.103689
Nuno Costa Dias , Franz Luef , João Nuno Prata
We revisit the uncertainty principle from the point of view suggested by A. Wigderson and Y. Wigderson. This approach is based on a primary uncertainty principle from which one can derive several inequalities expressing the impossibility of a simultaneous sharp localization in time and frequency. Moreover, it requires no specific properties of the Fourier transform and can therefore be easily applied to all operators satisfying the primary uncertainty principle. A. Wigderson and Y. Wigderson also suggested many generalizations to higher dimensions and stated several conjectures which we address in the present paper. We argue that we have to consider a more general primary uncertainty principle to prove the results suggested by the authors. As a by-product we obtain some new inequalities akin to the Cowling-Price uncertainty principle, a generalization of the Heisenberg uncertainty principle, and derive the entropic uncertainty principle from the primary uncertainty principles.
{"title":"On Wigdersons' approach to the uncertainty principle","authors":"Nuno Costa Dias , Franz Luef , João Nuno Prata","doi":"10.1016/j.matpur.2025.103689","DOIUrl":"10.1016/j.matpur.2025.103689","url":null,"abstract":"<div><div>We revisit the uncertainty principle from the point of view suggested by A. Wigderson and Y. Wigderson. This approach is based on a primary uncertainty principle from which one can derive several inequalities expressing the impossibility of a simultaneous sharp localization in time and frequency. Moreover, it requires no specific properties of the Fourier transform and can therefore be easily applied to all operators satisfying the primary uncertainty principle. A. Wigderson and Y. Wigderson also suggested many generalizations to higher dimensions and stated several conjectures which we address in the present paper. We argue that we have to consider a more general primary uncertainty principle to prove the results suggested by the authors. As a by-product we obtain some new inequalities akin to the Cowling-Price uncertainty principle, a generalization of the Heisenberg uncertainty principle, and derive the entropic uncertainty principle from the primary uncertainty principles.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"198 ","pages":"Article 103689"},"PeriodicalIF":2.1,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534555","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}
Pub Date : 2025-02-21DOI: 10.1016/j.matpur.2025.103685
L. Huysmans , Edriss S. Titi
We study the vanishing viscosity/diffusivity limit for the transport of a passive scalar by a bounded, divergence-free vector field . This is described by the Cauchy problem to the PDE , or with viscosity , to the PDE . In the first part of this work, we construct a bounded, divergence-free vector field for which, for any non-constant initial datum, the viscous solutions along different subsequences of the vanishing viscosity limit converge to different solutions to the inviscid problem. In the second part, we construct another bounded, divergence-free vector field for which, for every initial datum, the vanishing viscosity limit of solutions exists, is unique, and converges to an inviscid solution; however, when the initial datum is not constant, this inviscid limit is physically inadmissible due to increasing energy/entropy.
{"title":"Non-uniqueness & inadmissibility of the vanishing viscosity limit of the passive scalar transport equation","authors":"L. Huysmans , Edriss S. Titi","doi":"10.1016/j.matpur.2025.103685","DOIUrl":"10.1016/j.matpur.2025.103685","url":null,"abstract":"<div><div>We study the vanishing viscosity/diffusivity limit for the transport of a passive scalar <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><mi>R</mi></math></span> by a bounded, divergence-free vector field <span><math><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. This is described by the Cauchy problem to the PDE <span><math><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mi>f</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>, or with viscosity <span><math><mi>ν</mi><mo>></mo><mn>0</mn></math></span>, to the PDE <span><math><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mi>f</mi><mo>)</mo><mo>−</mo><mi>ν</mi><mi>Δ</mi><mi>f</mi><mo>=</mo><mn>0</mn></math></span>. In the first part of this work, we construct a bounded, divergence-free vector field <span><math><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> for which, for any non-constant initial datum, the viscous solutions along different subsequences of the vanishing viscosity limit converge to different solutions to the inviscid problem. In the second part, we construct another bounded, divergence-free vector field <span><math><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> for which, for every initial datum, the vanishing viscosity limit of solutions exists, is unique, and converges to an inviscid solution; however, when the initial datum is not constant, this inviscid limit is physically inadmissible due to increasing energy/entropy.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"198 ","pages":"Article 103685"},"PeriodicalIF":2.1,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.matpur.2024.103633
Aldo Pratelli , Giorgio Saracco
We prove a lower bound for the Cheeger constant of a cylinder , where Ω is an open and bounded set. As a consequence, we obtain existence of minimizers for the shape functional defined as the ratio between the first Dirichlet eigenvalue of the p-Laplacian and the p-th power of the Cheeger constant, within the class of bounded convex sets in any . This positively solves open conjectures raised by Parini (J. Convex Anal. (2017)) and by Briani–Buttazzo–Prinari (Ann. Mat. Pura Appl. (2023)).
{"title":"Cylindrical estimates for the Cheeger constant and applications","authors":"Aldo Pratelli , Giorgio Saracco","doi":"10.1016/j.matpur.2024.103633","DOIUrl":"10.1016/j.matpur.2024.103633","url":null,"abstract":"<div><div>We prove a lower bound for the Cheeger constant of a cylinder <span><math><mi>Ω</mi><mo>×</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>L</mi><mo>)</mo></math></span>, where Ω is an open and bounded set. As a consequence, we obtain existence of minimizers for the shape functional defined as the ratio between the first Dirichlet eigenvalue of the <em>p</em>-Laplacian and the <em>p</em>-th power of the Cheeger constant, within the class of bounded convex sets in any <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. This positively solves open conjectures raised by Parini (<em>J. Convex Anal.</em> (2017)) and by Briani–Buttazzo–Prinari (<em>Ann. Mat. Pura Appl.</em> (2023)).</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"194 ","pages":"Article 103633"},"PeriodicalIF":2.1,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143146799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a scale-invariant, semi-global existence result and a trapped surface formation result in the context of coupled Einstein-Yang-Mills theory, without symmetry assumptions. More precisely, we prove a scale-invariant semi-global existence theorem and show that the focusing of the gravitational and/or chromoelectric-chromomagnetic waves could lead to the formation of a trapped surface. Adopting the signature for decay rates approach introduced in [1], we develop a novel gauge (and scale) invariant hierarchy of non-linear estimates for the Yang-Mills curvature which, together with the estimates for the gravitational degrees of freedom, yields the desired semi-global existence result. Once semi-global existence has been established, the formation of a trapped surface follows from a standard ODE argument.
{"title":"Formation of trapped surfaces in the Einstein-Yang-Mills system","authors":"Nikolaos Athanasiou , Puskar Mondal , Shing-Tung Yau","doi":"10.1016/j.matpur.2025.103661","DOIUrl":"10.1016/j.matpur.2025.103661","url":null,"abstract":"<div><div>We prove a scale-invariant, semi-global existence result and a trapped surface formation result in the context of coupled Einstein-Yang-Mills theory, without symmetry assumptions. More precisely, we prove a scale-invariant semi-global existence theorem and show that the focusing of the gravitational and/or chromoelectric-chromomagnetic waves could lead to the formation of a trapped surface. Adopting the signature for decay rates approach introduced in <span><span>[1]</span></span>, we develop a novel gauge (and scale) invariant hierarchy of non-linear estimates for the Yang-Mills curvature which, together with the estimates for the gravitational degrees of freedom, yields the desired semi-global existence result. Once semi-global existence has been established, the formation of a trapped surface follows from a standard ODE argument.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"194 ","pages":"Article 103661"},"PeriodicalIF":2.1,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143146801","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.matpur.2025.103660
A. Piatnitski , E. Zhizhina
We study homogenization problem for non-autonomous parabolic equations of the form with an integral convolution type operator that has a non-symmetric jump kernel which is periodic in spatial variables and stationary random in time. We show that asymptotically the spatial and temporal evolutions of the solutions are getting decoupled and can be described separately, and, under additional mixing conditions on the coefficient, the homogenized equation is a SPDE with a finite dimensional multiplicative noise.
{"title":"Homogenization of non-autonomous evolution problems for convolution type operators in randomly evolving media","authors":"A. Piatnitski , E. Zhizhina","doi":"10.1016/j.matpur.2025.103660","DOIUrl":"10.1016/j.matpur.2025.103660","url":null,"abstract":"<div><div>We study homogenization problem for non-autonomous parabolic equations of the form <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>=</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>)</mo><mi>u</mi></math></span> with an integral convolution type operator <span><math><mi>L</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> that has a non-symmetric jump kernel which is periodic in spatial variables and stationary random in time. We show that asymptotically the spatial and temporal evolutions of the solutions are getting decoupled and can be described separately, and, under additional mixing conditions on the coefficient, the homogenized equation is a SPDE with a finite dimensional multiplicative noise.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"194 ","pages":"Article 103660"},"PeriodicalIF":2.1,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143146971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.matpur.2025.103670
Yuming Qin , Xiuqing Wang
In this paper, we consider the 3D Prandtl equation in a periodic domain and prove the local existence and uniqueness of solutions by the energy method in a polynomial weighted Sobolev space. Compared to the existence and uniqueness of solutions to the classical Prandtl equations where the Crocco transform has always been used with the general outer flow , this Crocco transform is not needed here for 3D Prandtl equations. We use the skill of cancellation mechanism and construct a new unknown function to show that the existence and uniqueness of solutions to 3D Prandtl equations (cf. Masmoudi and Wong (2015) [1]) which extends from the two dimensional case in [1] to the present three dimensional case with a special structure.
本文考虑周期域上三维Prandtl方程，在多项式加权Sobolev空间中，用能量法证明了该方程解的局部存在唯一性。与经典普朗特方程解的存在唯一性相比，经典普朗特方程解在一般外流U≠常数的情况下一直使用Crocco变换，而3D普朗特方程不需要Crocco变换。我们利用消去机制的技巧，构造了一个新的未知函数，证明了三维Prandtl方程（cf. Masmoudi and Wong(2015)[1]）解的存在唯一性，从[1]的二维情况扩展到目前具有特殊结构的三维情况。
{"title":"Local existence of solutions to 3D Prandtl equations with a special structure","authors":"Yuming Qin , Xiuqing Wang","doi":"10.1016/j.matpur.2025.103670","DOIUrl":"10.1016/j.matpur.2025.103670","url":null,"abstract":"<div><div>In this paper, we consider the 3D Prandtl equation in a periodic domain and prove the local existence and uniqueness of solutions by the energy method in a polynomial weighted Sobolev space. Compared to the existence and uniqueness of solutions to the classical Prandtl equations where the Crocco transform has always been used with the general outer flow <span><math><mi>U</mi><mo>≠</mo><mtext>constant</mtext></math></span>, this Crocco transform is not needed here for 3D Prandtl equations. We use the skill of cancellation mechanism and construct a new unknown function to show that the existence and uniqueness of solutions to 3D Prandtl equations (cf. Masmoudi and Wong (2015) <span><span>[1]</span></span>) which extends from the two dimensional case in <span><span>[1]</span></span> to the present three dimensional case with a special structure.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"194 ","pages":"Article 103670"},"PeriodicalIF":2.1,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143146800","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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