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

{"title":"Homotopy properties of the complex of frames of a unitary space","authors":"Kevin I. Piterman,&nbsp;Volkmar Welker","doi":"10.1112/jlms.12978","DOIUrl":"https://doi.org/10.1112/jlms.12978","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>V</mi>\u0000 <annotation>$V$</annotation>\u0000 </semantics></math> be a finite-dimensional vector space equipped with a nondegenerate Hermitian form over a field <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>${mathbb {K}}$</annotation>\u0000 </semantics></math>. Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>(</mo>\u0000 <mi>V</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${mathcal {G}}(V)$</annotation>\u0000 </semantics></math> be the graph with vertex set the one-dimensional nondegenerate subspaces of <span></span><math>\u0000 <semantics>\u0000 <mi>V</mi>\u0000 <annotation>$V$</annotation>\u0000 </semantics></math> and adjacency relation given by orthogonality. We give a complete description of when <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>(</mo>\u0000 <mi>V</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${mathcal {G}}(V)$</annotation>\u0000 </semantics></math> is connected in terms of the dimension of <span></span><math>\u0000 <semantics>\u0000 <mi>V</mi>\u0000 <annotation>$V$</annotation>\u0000 </semantics></math> and the size of the ground field <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>${mathbb {K}}$</annotation>\u0000 </semantics></math>. Furthermore, we prove that if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>dim</mo>\u0000 <mo>(</mo>\u0000 <mi>V</mi>\u0000 <mo>)</mo>\u0000 <mo>&gt;</mo>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation>$dim (V) &amp;gt; 4$</annotation>\u0000 </semantics></math>, then the clique complex <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 <mo>(</mo>\u0000 <mi>V</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${mathcal {F}}(V)$</annotation>\u0000 </semantics></math> of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>(</mo>\u0000 <mi>V</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${mathcal {G}}(V)$</annotation>\u0000 </semantics></math> is simply connected. For finite fields <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>${mathbb {K}}$</annotation>\u0000 </semantics></math>, we also compute the eigenvalues of the adjacency matrix of <span></span><math>\u0000 <semantics>\u0000","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142050539","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":"On finitely generated Engel branch groups","authors":"J. Moritz Petschick","doi":"10.1112/jlms.12980","DOIUrl":"https://doi.org/10.1112/jlms.12980","url":null,"abstract":"<p>We construct finitely generated Engel branch groups, answering a question of Fernández-Alcober, Noce and Tracey on the existence of such objects. In particular, the groups constructed are not nilpotent, yielding the second known class of examples of finitely generated non-nilpotent Engel groups following a construction by Golod from 1969. To do so, we exhibit groups acting on rooted trees with growing valency on which word lengths of elements are contracting very quickly under section maps. Our methods apply in principle to a wider class of iterated identities, of which the Engel words are a special case.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12980","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142050540","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":"Global classical solutions to a multidimensional radiation hydrodynamics model with symmetry and large initial data","authors":"Jing Wei,&nbsp;Minyi Zhang,&nbsp;Changjiang Zhu","doi":"10.1112/jlms.12973","DOIUrl":"https://doi.org/10.1112/jlms.12973","url":null,"abstract":"<p>As a first stage to study the global large solutions of the radiation hydrodynamics model with viscosity and thermal conductivity in the high-dimensional space, we study the problems in high dimensions with some symmetry, such as the spherically or cylindrically symmetric solutions. Specifically, we will study the global classical large solutions to the radiation hydrodynamics model with spherically or cylindrically symmetric initial data. The key point is to obtain the strict positive lower and upper bounds of the density <span></span><math>\u0000 <semantics>\u0000 <mi>ρ</mi>\u0000 <annotation>$rho$</annotation>\u0000 </semantics></math> and the lower bound of the temperature <span></span><math>\u0000 <semantics>\u0000 <mi>θ</mi>\u0000 <annotation>$theta$</annotation>\u0000 </semantics></math>. Compared with the Navier–Stokes equations, these estimates in the present paper are more complicated due to the influence of the radiation. To overcome the difficulties caused by the radiation, we construct a pointwise estimate between the radiative heat flux <span></span><math>\u0000 <semantics>\u0000 <mi>q</mi>\u0000 <annotation>$q$</annotation>\u0000 </semantics></math> and the temperature <span></span><math>\u0000 <semantics>\u0000 <mi>θ</mi>\u0000 <annotation>$theta$</annotation>\u0000 </semantics></math> by studying the boundary value problem of the corresponding ordinary differential equation. And we consider a general heat conductivity: <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>κ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>ρ</mi>\u0000 <mo>,</mo>\u0000 <mi>θ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>⩾</mo>\u0000 <mi>C</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>+</mo>\u0000 <msup>\u0000 <mi>θ</mi>\u0000 <mi>β</mi>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$kappa (rho,theta)geqslant C(1+theta ^beta)$</annotation>\u0000 </semantics></math> if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ρ</mi>\u0000 <mo>⩽</mo>\u0000 <msub>\u0000 <mi>ρ</mi>\u0000 <mo>+</mo>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$rho leqslant rho _+$</annotation>\u0000 </semantics></math>; <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>κ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>ρ</mi>\u0000 <mo>,</mo>\u0000 <mi>θ</mi>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142045288","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":"Almost sure bounds for a weighted Steinhaus random multiplicative function","authors":"Seth Hardy","doi":"10.1112/jlms.12979","DOIUrl":"https://doi.org/10.1112/jlms.12979","url":null,"abstract":"<p>We obtain almost sure bounds for the weighted sum <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mo>∑</mo>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>⩽</mo>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mfrac>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <msqrt>\u0000 <mi>n</mi>\u0000 </msqrt>\u0000 </mfrac>\u0000 </mrow>\u0000 <annotation>$sum _{n leqslant t} frac{f(n)}{sqrt {n}}$</annotation>\u0000 </semantics></math>, where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$f(n)$</annotation>\u0000 </semantics></math> is a Steinhaus random multiplicative function. Specifically, we obtain the bounds predicted by exponentiating the law of the iterated logarithm, giving sharp upper and lower bounds.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12979","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142045289","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":"Construction of multi-bubble blow-up solutions to the \u0000 \u0000 \u0000 L\u0000 2\u0000 \u0000 $L^2$\u0000 -critical half-wave equation","authors":"Daomin Cao,&nbsp;Yiming Su,&nbsp;Deng Zhang","doi":"10.1112/jlms.12974","DOIUrl":"https://doi.org/10.1112/jlms.12974","url":null,"abstract":"<p>This paper concerns the bubbling phenomena for the <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$L^2$</annotation>\u0000 </semantics></math>-critical half-wave equation in dimension one. Given arbitrarily finitely many distinct singularities, we construct blow-up solutions concentrating exactly at these singularities. This provides the first examples of multi-bubble solutions for the half-wave equation. In particular, the solutions exhibit the mass quantization property. Our proof strategy draws upon the modulation method in Krieger, Lenzmann and Raphaël [Arch. Ration. Mech. Anal. 209 (2013), no. 1, 61–129] for the single-bubble case, and explores the localization techniques in Cao, Su and Zhang [Arch. Ration. Mech. Anal. 247 (2023), no. 1, Paper No. 4] and Röckner, Su and Zhang [Trans. Amer. Math. Soc., 377 (2024), no. 1, 517–588] for bubbling solutions to non-linear Schrödinger equations (NLS). However, unlike the single-bubble or NLS cases, different bubbles exhibit the strongest interactions in dimension one. In order to get sharp estimates to control these interactions, as well as non-local effects on localization functions, we utilize the Carlderón estimate and the integration representation formula of the half-wave operator, and find that there exists a narrow room between the orders <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>t</mi>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mo>+</mo>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$|t|^{2+}$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>t</mi>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <mo>−</mo>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$|t|^{3-}$</annotation>\u0000 </semantics></math> for the remainder in the geometrical decomposition. Based on this, a novel bootstrap scheme is introduced to address the multi-bubble non-local structure.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142021777","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":"Extension of planar Hölder homeomorphisms","authors":"Stanislav Hencl,&nbsp;Aleksis Koski","doi":"10.1112/jlms.12970","DOIUrl":"https://doi.org/10.1112/jlms.12970","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>∈</mo>\u0000 <mo>(</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$alpha in (0,1)$</annotation>\u0000 </semantics></math>. We show that any <span></span><math>\u0000 <semantics>\u0000 <mi>α</mi>\u0000 <annotation>$alpha$</annotation>\u0000 </semantics></math>-Hölder homeomorphism from the unit circle in the plane to the plane can be extended to an <span></span><math>\u0000 <semantics>\u0000 <mi>α</mi>\u0000 <annotation>$alpha$</annotation>\u0000 </semantics></math>-Hölder homeomorphism from the whole unit disc.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141994154","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":"Special cubic zeros and the dual variety","authors":"Victor Y. Wang","doi":"10.1112/jlms.12975","DOIUrl":"https://doi.org/10.1112/jlms.12975","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$F$</annotation>\u0000 </semantics></math> be a diagonal cubic form over <span></span><math>\u0000 <semantics>\u0000 <mi>Z</mi>\u0000 <annotation>$mathbb {Z}$</annotation>\u0000 </semantics></math> in six variables. From the dual variety in the delta method of Duke–Friedlander–Iwaniec and Heath-Brown, we unconditionally extract a weighted count of certain special integral zeros of <span></span><math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$F$</annotation>\u0000 </semantics></math> in regions of diameter <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 <mo>→</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$X rightarrow infty$</annotation>\u0000 </semantics></math>. Heath-Brown did the same in four variables, but our analysis differs and captures some novel features. We also put forth an axiomatic framework for more general <span></span><math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$F$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12975","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141986091","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":"Curvature varifolds with orthogonal boundary","authors":"Ernst Kuwert,&nbsp;Marius Müller","doi":"10.1112/jlms.12976","DOIUrl":"https://doi.org/10.1112/jlms.12976","url":null,"abstract":"<p>We consider the class <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>S</mi>\u0000 <mo>⊥</mo>\u0000 <mi>m</mi>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Ω</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>${bf S}^m_perp (Omega)$</annotation>\u0000 </semantics></math> of <span></span><math>\u0000 <semantics>\u0000 <mi>m</mi>\u0000 <annotation>$m$</annotation>\u0000 </semantics></math>-dimensional surfaces in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mover>\u0000 <mi>Ω</mi>\u0000 <mo>¯</mo>\u0000 </mover>\u0000 <mo>⊂</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$overline{Omega } subset {mathbb {R}}^n$</annotation>\u0000 </semantics></math> that intersect <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 <mo>=</mo>\u0000 <mi>∂</mi>\u0000 <mi>Ω</mi>\u0000 </mrow>\u0000 <annotation>$S = partial Omega$</annotation>\u0000 </semantics></math> orthogonally along the boundary. A piece of an affine <span></span><math>\u0000 <semantics>\u0000 <mi>m</mi>\u0000 <annotation>$m$</annotation>\u0000 </semantics></math>-plane in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>S</mi>\u0000 <mo>⊥</mo>\u0000 <mi>m</mi>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Ω</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>${bf S}^m_perp (Omega)$</annotation>\u0000 </semantics></math> is called an orthogonal slice. We prove estimates for the area by the <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>p</mi>\u0000 </msup>\u0000 <annotation>$L^p$</annotation>\u0000 </semantics></math> integral of the second fundamental form in three cases: first, when <span></span><math>\u0000 <semantics>\u0000 <mi>Ω</mi>\u0000 <annotation>$Omega$</annotation>\u0000 </semantics></math> admits no orthogonal slices, second for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12976","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141986092","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":"Asymptotic behavior of Laplacian eigenvalues of subspace inclusion graphs","authors":"Alan Lew","doi":"10.1112/jlms.12972","DOIUrl":"https://doi.org/10.1112/jlms.12972","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mtext>Fl</mtext>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$text{Fl}_{n,q}$</annotation>\u0000 </semantics></math> be the simplicial complex whose vertices are the nontrivial subspaces of <span></span><math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>F</mi>\u0000 <mi>q</mi>\u0000 <mi>n</mi>\u0000 </msubsup>\u0000 <annotation>$mathbb {F}_q^n$</annotation>\u0000 </semantics></math> and whose simplices correspond to families of subspaces forming a flag. Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>Δ</mi>\u0000 <mi>k</mi>\u0000 <mo>+</mo>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mtext>Fl</mtext>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$Delta ^{+}_k(text{Fl}_{n,q})$</annotation>\u0000 </semantics></math> be the <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math>-dimensional weighted upper Laplacian on <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mtext>Fl</mtext>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$ text{Fl}_{n,q}$</annotation>\u0000 </semantics></math>. The spectrum of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>Δ</mi>\u0000 <mi>k</mi>\u0000 <mo>+</mo>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mtext>Fl</mtext>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$Delta ^{+}_k","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12972","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141980291","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":"Transient asymptotics of the modified Camassa–Holm equation","authors":"Taiyang Xu,&nbsp;Yiling Yang,&nbsp;Lun Zhang","doi":"10.1112/jlms.12967","DOIUrl":"https://doi.org/10.1112/jlms.12967","url":null,"abstract":"<p>We investigate long time asymptotics of the modified Camassa–Holm equation in three transition zones under a nonzero background. The first transition zone lies between the soliton region and the first oscillatory region, the second one lies between the second oscillatory region and the fast decay region, and possibly, the third one, namely, the collisionless shock region, that bridges the first transition region and the first oscillatory region. Under a low regularity condition on the initial data, we obtain Painlevé-type asymptotic formulae in the first two transition regions, while the transient asymptotics in the third region involves the Jacobi theta function. We establish our results by performing a <span></span><math>\u0000 <semantics>\u0000 <mover>\u0000 <mi>∂</mi>\u0000 <mo>¯</mo>\u0000 </mover>\u0000 <annotation>$bar{partial }$</annotation>\u0000 </semantics></math> nonlinear steepest descent analysis to the associated Riemann–Hilbert problem.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141966483","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}