Journal of the London Mathematical Society-Second Series最新文献

{"title":"3-Circle theorem for Willmore surface I","authors":"Yuxiang Li,&nbsp;Hao Yin","doi":"10.1112/jlms.70165","DOIUrl":"https://doi.org/10.1112/jlms.70165","url":null,"abstract":"<p>In this paper, we study the blow-up of Willmore surfaces. By using the 3-circle theorem, we prove a decay estimate of the second fundamental form along the neck region. This estimate provides a new perspective and streamlined proofs to a few key results in this field, such as the energy identity(quantization), removable singularities and gap theorem.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 5","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143883969","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}

{"title":"Uniqueness of dg-lifts via restriction to injective objects","authors":"Francesco Genovese","doi":"10.1112/jlms.70166","DOIUrl":"https://doi.org/10.1112/jlms.70166","url":null,"abstract":"<p>We prove a uniqueness result of dg-lifts for the derived pushforward and pullback functors of a flat morphism between separated Noetherian schemes, between the unbounded or bounded below derived categories of quasi-coherent sheaves. The technique is purely algebraic-categorical and involves reconstructing dg-lifts uniquely from their restrictions to the subcategories of injective objects.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 5","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143884247","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}

{"title":"Embedding groups into boundedly acyclic groups","authors":"Fan Wu,&nbsp;Xiaolei Wu,&nbsp;Mengfei Zhao,&nbsp;Zixiang Zhou","doi":"10.1112/jlms.70164","DOIUrl":"https://doi.org/10.1112/jlms.70164","url":null,"abstract":"<p>We show that the <span></span><math>\u0000 <semantics>\u0000 <mi>ϕ</mi>\u0000 <annotation>$phi$</annotation>\u0000 </semantics></math>-labeled Thompson groups and the twisted Brin–Thompson groups are boundedly acyclic. This allows us to prove several new embedding results for groups. First, every group of type <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$F_n$</annotation>\u0000 </semantics></math> embeds quasi-isometrically into a boundedly acyclic group of type <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$F_n$</annotation>\u0000 </semantics></math> that has no proper finite index subgroups. This improves a result of Bridson and a theorem of Fournier-Facio–Löh–Moraschini. Second, every group of type <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$F_n$</annotation>\u0000 </semantics></math> embeds quasi-isometrically into a 5-uniformly perfect group of type <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$F_n$</annotation>\u0000 </semantics></math>. Third, using Belk–Zaremsky's construction of twisted Brin–Thompson groups, we show that every finitely generated group embeds quasi-isometrically into a finitely generated boundedly acyclic simple group. We also partially answer some questions of Brothier and Tanushevski regarding the finiteness property of <span></span><math>\u0000 <semantics>\u0000 <mi>ϕ</mi>\u0000 <annotation>$phi$</annotation>\u0000 </semantics></math>-labeled Thompson group <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>V</mi>\u0000 <mi>ϕ</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$V_phi (G)$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mi>ϕ</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$F_phi (G)$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 5","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143879710","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}

{"title":"Minimal limit key polynomials","authors":"Enric Nart,&nbsp;Josnei Novacoski","doi":"10.1112/jlms.70162","DOIUrl":"https://doi.org/10.1112/jlms.70162","url":null,"abstract":"<p>In this paper, we extend the theory of minimal limit key polynomials of valuations on the polynomial ring <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>K</mi>\u0000 <mo>[</mo>\u0000 <mi>x</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <annotation>$K[x]$</annotation>\u0000 </semantics></math>. Minimal key polynomials are useful to describe, for instance, the defect of an extension of valued fields. We use the theory of cuts on ordered abelian groups to show that the previous results on bounded sets of key polynomials of rank one valuations extend to vertically bounded sets of key polynomials of valuations of an arbitrary rank. We also discuss properties of minimal limit key polynomials in the vertically unbounded case.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 5","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70162","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871472","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}

{"title":"Arithmetic Satake compactifications and algebraic Drinfeld modular forms","authors":"Urs Hartl,&nbsp;Chia-Fu Yu","doi":"10.1112/jlms.70082","DOIUrl":"https://doi.org/10.1112/jlms.70082","url":null,"abstract":"<p>In this article, we construct the arithmetic Satake compactification of the Drinfeld moduli schemes of arbitrary rank over the ring of integers of any global function field away from the level structure, and show that the universal family extends uniquely to a generalized Drinfeld module over the compactification. Using these and functorial properties, we define algebraic Drinfeld modular forms over more general bases and the action of the (prime-to-residue characteristic and level) Hecke algebra. The construction also furnishes many algebraic Drinfeld modular forms obtained from the coefficients of the universal family that are also Hecke eigenforms. Among them, we obtain generalized Hasse invariants that are already defined on the arithmetic Satake compactification and not only its special fiber. We use these generalized Hasse invariants to study the geometry of the special fiber. We conjecture that our Satake compactification is Cohen–Macaulay. If this is the case, we establish the Jacquet–Langlands correspondence (mod <span></span><math>\u0000 <semantics>\u0000 <mi>v</mi>\u0000 <annotation>$v$</annotation>\u0000 </semantics></math>) between Hecke eigensystems of rank <span></span><math>\u0000 <semantics>\u0000 <mi>r</mi>\u0000 <annotation>$r$</annotation>\u0000 </semantics></math> Drinfeld modular forms and those of algebraic modular forms (in the sense of Gross) attached to a compact inner form of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>GL</mi>\u0000 <mi>r</mi>\u0000 </msub>\u0000 <annotation>$mathop {rm GL}nolimits _r$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70082","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143861915","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}

{"title":"Borel fields and measured fields of Polish spaces, Banach spaces, von Neumann algebras, and \u0000 \u0000 \u0000 C\u0000 ∗\u0000 \u0000 ${rm C}^*$\u0000 -algebras","authors":"Stefaan Vaes,&nbsp;Lise Wouters","doi":"10.1112/jlms.70159","DOIUrl":"https://doi.org/10.1112/jlms.70159","url":null,"abstract":"&lt;p&gt;Several recent articles in operator algebras make a nontrivial use of the theory of measurable fields of von Neumann algebras &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$(M_x)_{x in X}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and related structures. This includes the associated field &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mo&gt;Aut&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$(operatorname{Aut}M_x)_{x in X}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of automorphism groups and more general measurable fields of Polish groups with actions on Polish spaces. Nevertheless, a fully rigorous and at the same time sufficiently broad and flexible theory of such Borel fields and measurable fields is not available in the literature. We fill this gap in this paper and include a few counterexamples to illustrate the subtlety: for instance, for a Borel field &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$(M_x)_{x in X}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of von Neumann algebras, the field of Polish groups &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mo&gt;Aut&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$(operatorname{Aut}M_x)_{x in X}$&lt;/annotation&gt;\u0000 &lt;/semanti","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143852780","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}

{"title":"Hodge and Frobenius colevels of algebraic varieties","authors":"Daqing Wan,&nbsp;Dingxin Zhang","doi":"10.1112/jlms.70160","DOIUrl":"https://doi.org/10.1112/jlms.70160","url":null,"abstract":"<p>We provide new improved lower bounds for the Hodge and Frobenius colevels of algebraic varieties (over <span></span><math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$mathbb {C}$</annotation>\u0000 </semantics></math> or over a finite field) in all cohomological degrees. These bounds are expressed in terms of the dimension of the variety and multi-degrees of its defining equations. Our results lead to an enhanced positive answer to a question raised by Esnault and the first author.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143852775","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}

{"title":"Derived smooth induction with applications","authors":"Peter Schneider,&nbsp;Claus Sorensen","doi":"10.1112/jlms.70157","DOIUrl":"https://doi.org/10.1112/jlms.70157","url":null,"abstract":"<p>In natural characteristic, smooth induction from an open subgroup does not always give an exact functor. In this article we initiate a study of the right derived functors, and we give applications to the nonexistence of projective representations and duality.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143840622","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}

{"title":"Stability of smooth multisolitons for the two-component Camassa–Holm system","authors":"Zhi-Jia Wu,&nbsp;Shou-Fu Tian,&nbsp;Yue Liu,&nbsp;Zhong Wang","doi":"10.1112/jlms.70158","DOIUrl":"https://doi.org/10.1112/jlms.70158","url":null,"abstract":"<p>In this work, we study the stability of exact smooth multisolitons for the two-component Camassa–Holm system, a completely integrable model for unidirectional shallow-water waves. By analyzing the spectrum of the linearized system and using a suitable Lyapunov functional, we show that these smooth multisolitons, as nonisolated constrained minimizers, are dynamically stable under small perturbations in an appropriate Sobolev space. A key part of our analysis involves employing the recursion operator from the bi-Hamiltonian structure and introducing a new transformation to diagonalize the nondiagonal matrix form in the second variation of the Lyapunov functional. This approach is essential due to the significant differences compared to the classical Camassa–Holm equation.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143840928","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}

{"title":"The three limits of the hydrostatic approximation","authors":"Ken Furukawa,&nbsp;Yoshikazu Giga,&nbsp;Matthias Hieber,&nbsp;Amru Hussein,&nbsp;Takahito Kashiwabara,&nbsp;Marc Wrona","doi":"10.1112/jlms.70130","DOIUrl":"https://doi.org/10.1112/jlms.70130","url":null,"abstract":"&lt;p&gt;The primitive equations are derived from the 3D Navier–Stokes equations by the hydrostatic approximation. Formally, assuming an &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;annotation&gt;$varepsilon$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-thin domain and anisotropic viscosities with vertical viscosity &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ν&lt;/mi&gt;\u0000 &lt;mi&gt;z&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mi&gt;O&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mi&gt;γ&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$nu _z=mathcal {O}(varepsilon ^gamma)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;γ&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$gamma =2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, one obtains the primitive equations with full viscosity as &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$varepsilon rightarrow 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Here, we take two more limit equations into consideration: For &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;γ&lt;/mi&gt;\u0000 &lt;mo&gt;&lt;&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$gamma &lt;2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; the 2D Navier–Stokes equations are obtained. For &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;γ&lt;/mi&gt;\u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$gamma &gt;2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; the primitive equations with only horizontal viscosity &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Δ&lt;/mi&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$-Delta _H$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; as &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70130","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143840868","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}