Pub Date : 2020-01-09DOI: 10.4310/acta.2023.v230.n2.a2
A. Ionescu, H. Jia
We prove nonlinear asymptotic stability of a large class of monotonic shear flows among solutions of the 2D Euler equations in the channel $mathbb{T}times[0,1]$. More precisely, we consider shear flows $(b(y),0)$ given by a function $b$ which is Gevrey smooth, strictly increasing, and linear outside a compact subset of the interval $(0,1)$ (to avoid boundary contributions which are incompatible with inviscid damping). We also assume that the associated linearized operator satisfies a suitable spectral condition, which is needed to prove linear inviscid damping. Under these assumptions, we show that if $u$ is a solution which is a small and Gevrey smooth perturbation of such a shear flow $(b(y),0)$ at time $t=0$, then the velocity field $u$ converges strongly to a nearby shear flow as the time goes to infinity. This is the first nonlinear asymptotic stability result for Euler equations around general steady solutions for which the linearized flow cannot be explicitly solved.
{"title":"Nonlinear inviscid damping near monotonic shear flows","authors":"A. Ionescu, H. Jia","doi":"10.4310/acta.2023.v230.n2.a2","DOIUrl":"https://doi.org/10.4310/acta.2023.v230.n2.a2","url":null,"abstract":"We prove nonlinear asymptotic stability of a large class of monotonic shear flows among solutions of the 2D Euler equations in the channel $mathbb{T}times[0,1]$. More precisely, we consider shear flows $(b(y),0)$ given by a function $b$ which is Gevrey smooth, strictly increasing, and linear outside a compact subset of the interval $(0,1)$ (to avoid boundary contributions which are incompatible with inviscid damping). We also assume that the associated linearized operator satisfies a suitable spectral condition, which is needed to prove linear inviscid damping. \u0000Under these assumptions, we show that if $u$ is a solution which is a small and Gevrey smooth perturbation of such a shear flow $(b(y),0)$ at time $t=0$, then the velocity field $u$ converges strongly to a nearby shear flow as the time goes to infinity. This is the first nonlinear asymptotic stability result for Euler equations around general steady solutions for which the linearized flow cannot be explicitly solved.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2020-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44346605","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 : 2020-01-01DOI: 10.4467/20843828am.19.002.12110
A. Edigarian
. M. A. Selby [ 8–10 ] and, independently, N. Sibony [ 11 ] proved that on the complex plane c -completeness is equivalent to c -finitely com-pactness. Their proofs are quite similar and are based on [ 4 ]. We give more refined equivalent conditions and, along the way, simplify the proofs.
． M. A. Selby[8-10]和N. Sibony[11]分别证明了在复平面上c -完备性等价于c -有限紧性。它们的证明非常相似，都基于[4]。我们给出了更精细的等价条件，同时简化了证明。
{"title":"Carathéodory completeness on the plane","authors":"A. Edigarian","doi":"10.4467/20843828am.19.002.12110","DOIUrl":"https://doi.org/10.4467/20843828am.19.002.12110","url":null,"abstract":". M. A. Selby [ 8–10 ] and, independently, N. Sibony [ 11 ] proved that on the complex plane c -completeness is equivalent to c -finitely com-pactness. Their proofs are quite similar and are based on [ 4 ]. We give more refined equivalent conditions and, along the way, simplify the proofs.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"1 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70986158","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 : 2019-12-17DOI: 10.4467/20843828am.19.001.12109
Zahra Barqsouz, S. O. Faramarzi
We show some results about local homology modules when they are in a Serre subcategory of the category of R-modules. For an ideal a of R, we also define the concept of the condition C on a Serre category, which seems dual to the condition Ca in Melkersson [1]. As a main result we show that for an Artinian R-module M and any Serre subcategory S of the category of R-modules and a non-negative integer s, HomR(R/a,H a s(M)) ∈ S if Hi (M) ∈ S for all i > s.
{"title":"Local homology and Serre categories","authors":"Zahra Barqsouz, S. O. Faramarzi","doi":"10.4467/20843828am.19.001.12109","DOIUrl":"https://doi.org/10.4467/20843828am.19.001.12109","url":null,"abstract":"We show some results about local homology modules when they are in a Serre subcategory of the category of R-modules. For an ideal a of R, we also define the concept of the condition C on a Serre category, which seems dual to the condition Ca in Melkersson [1]. As a main result we show that for an Artinian R-module M and any Serre subcategory S of the category of R-modules and a non-negative integer s, HomR(R/a,H a s(M)) ∈ S if Hi (M) ∈ S for all i > s.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43502407","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 : 2019-12-16DOI: 10.4310/acta.2023.v230.n1.a1
Thomas Duyckaerts, C. Kenig, F. Merle
Consider the energy-critical focusing wave equation in odd space dimension $Ngeq 3$. The equation has a nonzero radial stationary solution $W$, which is unique up to scaling and sign change. In this paper we prove that any radial, bounded in the energy norm solution of the equation behaves asymptotically as a sum of modulated $W$s, decoupled by the scaling, and a radiation term. The proof essentially boils down to the fact that the equation does not have purely nonradiative multisoliton solutions. The proof overcomes the fundamental obstruction for the extension of the 3D case (treated in our previous work, Cambridge Journal of Mathematics 2013, arXiv:1204.0031) by reducing the study of a multisoliton solution to a finite dimensional system of ordinary differential equations on the modulation parameters. The key ingredient of the proof is to show that this system of equations creates some radiation, contradicting the existence of pure multisolitons.
{"title":"Soliton resolution for the radial critical wave equation in all odd space dimensions","authors":"Thomas Duyckaerts, C. Kenig, F. Merle","doi":"10.4310/acta.2023.v230.n1.a1","DOIUrl":"https://doi.org/10.4310/acta.2023.v230.n1.a1","url":null,"abstract":"Consider the energy-critical focusing wave equation in odd space dimension $Ngeq 3$. The equation has a nonzero radial stationary solution $W$, which is unique up to scaling and sign change. In this paper we prove that any radial, bounded in the energy norm solution of the equation behaves asymptotically as a sum of modulated $W$s, decoupled by the scaling, and a radiation term. \u0000The proof essentially boils down to the fact that the equation does not have purely nonradiative multisoliton solutions. The proof overcomes the fundamental obstruction for the extension of the 3D case (treated in our previous work, Cambridge Journal of Mathematics 2013, arXiv:1204.0031) by reducing the study of a multisoliton solution to a finite dimensional system of ordinary differential equations on the modulation parameters. The key ingredient of the proof is to show that this system of equations creates some radiation, contradicting the existence of pure multisolitons.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2019-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46563093","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 : 2019-07-22DOI: 10.4310/acta.2020.v224.n2.a1
X. Cabré, A. Figalli, Xavier Ros-Oton, J. Serra
In this paper we prove the following long-standing conjecture: stable solutions to semilinear elliptic equations are bounded (and thus smooth) in dimension $n leq 9$. This result, that was only known to be true for $nleq4$, is optimal: $log(1/|x|^2)$ is a $W^{1,2}$ singular stable solution for $ngeq10$. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension $n leq 9$, stable solutions are bounded in terms only of their $L^1$ norm, independently of the nonlinearity. In addition, in every dimension we establish a higher integrability result for the gradient and optimal integrability results for the solution in Morrey spaces. As one can see by a series of classical examples, all our results are sharp. Furthermore, as a corollary we obtain that extremal solutions of Gelfand problems are $W^{1,2}$ in every dimension and they are smooth in dimension $n leq 9$. This answers to two famous open problems posed by Brezis and Brezis-Vazquez.
{"title":"Stable solutions to semilinear elliptic equations are smooth up to dimension $9$","authors":"X. Cabré, A. Figalli, Xavier Ros-Oton, J. Serra","doi":"10.4310/acta.2020.v224.n2.a1","DOIUrl":"https://doi.org/10.4310/acta.2020.v224.n2.a1","url":null,"abstract":"In this paper we prove the following long-standing conjecture: stable solutions to semilinear elliptic equations are bounded (and thus smooth) in dimension $n leq 9$. \u0000This result, that was only known to be true for $nleq4$, is optimal: $log(1/|x|^2)$ is a $W^{1,2}$ singular stable solution for $ngeq10$. \u0000The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension $n leq 9$, stable solutions are bounded in terms only of their $L^1$ norm, independently of the nonlinearity. In addition, in every dimension we establish a higher integrability result for the gradient and optimal integrability results for the solution in Morrey spaces. \u0000As one can see by a series of classical examples, all our results are sharp. Furthermore, as a corollary we obtain that extremal solutions of Gelfand problems are $W^{1,2}$ in every dimension and they are smooth in dimension $n leq 9$. This answers to two famous open problems posed by Brezis and Brezis-Vazquez.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2019-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47959434","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 : 2019-07-01DOI: 10.4310/ACTA.2020.V225.N2.A1
J. Bell, J. Diller, Mattias Jonsson
We give an example of a dominant rational selfmap of the projective plane whose dynamical degree is a transcendental number.
我们给出了一个射影平面的主有理自映射的例子，它的动力度是一个超越数。
{"title":"A transcendental dynamical degree","authors":"J. Bell, J. Diller, Mattias Jonsson","doi":"10.4310/ACTA.2020.V225.N2.A1","DOIUrl":"https://doi.org/10.4310/ACTA.2020.V225.N2.A1","url":null,"abstract":"We give an example of a dominant rational selfmap of the projective plane whose dynamical degree is a transcendental number.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43577785","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 : 2019-05-30DOI: 10.4310/acta.2023.v230.n1.a2
N. Holden, Xin Sun
We consider an embedding of planar maps into an equilateral triangle $Delta$ which we call the Cardy embedding. The embedding is a discrete approximation of a conformal map based on percolation observables that are used in Smirnov's proof of Cardy's formula. Under the Cardy embedding, the planar map induces a metric and an area measure on $Delta$ and a boundary measure on $partial Delta$. We prove that for uniformly sampled triangulations, the metric and the measures converge jointly in the scaling limit to the Brownian disk conformally embedded into $Delta$ (i.e., to the $sqrt{8/3}$-Liouville quantum gravity disk). As part of our proof, we prove scaling limit results for critical site percolation on the uniform triangulations, in a quenched sense. In particular, we establish the scaling limit of the percolation crossing probability for a uniformly sampled triangulation with four boundary marked points.
{"title":"Convergence of uniform triangulations under the Cardy embedding","authors":"N. Holden, Xin Sun","doi":"10.4310/acta.2023.v230.n1.a2","DOIUrl":"https://doi.org/10.4310/acta.2023.v230.n1.a2","url":null,"abstract":"We consider an embedding of planar maps into an equilateral triangle $Delta$ which we call the Cardy embedding. The embedding is a discrete approximation of a conformal map based on percolation observables that are used in Smirnov's proof of Cardy's formula. Under the Cardy embedding, the planar map induces a metric and an area measure on $Delta$ and a boundary measure on $partial Delta$. We prove that for uniformly sampled triangulations, the metric and the measures converge jointly in the scaling limit to the Brownian disk conformally embedded into $Delta$ (i.e., to the $sqrt{8/3}$-Liouville quantum gravity disk). As part of our proof, we prove scaling limit results for critical site percolation on the uniform triangulations, in a quenched sense. In particular, we establish the scaling limit of the percolation crossing probability for a uniformly sampled triangulation with four boundary marked points.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2019-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42995138","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 : 2019-05-21DOI: 10.4310/acta.2021.v227.n1.a1
Pierre Baumann, J. Kamnitzer, A. Knutson
Using the geometric Satake correspondence, the Mirkovic-Vilonen cycles in the affine Grasssmannian give bases for representations of a semisimple group G . We prove that these bases are "perfect", i.e. compatible with the action of the Chevelley generators of the positive half of the Lie algebra g. We compute this action in terms of intersection multiplicities in the affine Grassmannian. We prove that these bases stitch together to a basis for the algebra C[N] of regular functions on the unipotent subgroup. We compute the multiplication in this MV basis using intersection multiplicities in the Beilinson-Drinfeld Grassmannian, thus proving a conjecture of Anderson. In the third part of the paper, we define a map from C[N] to a convolution algebra of measures on the dual of the Cartan subalgebra of g. We characterize this map using the universal centralizer space of G. We prove that the measure associated to an MV basis element equals the Duistermaat-Heckman measure of the corresponding MV cycle. This leads to a proof of a conjecture of Muthiah. Finally, we use the map to measures to compare the MV basis and Lusztig's dual semicanonical basis. We formulate conjectures relating the algebraic invariants of preprojective algebra modules (which underlie the dual semicanonical basis) and geometric invariants of MV cycles. In the appendix, we use these ideas to prove that the MV basis and the dual semicanonical basis do not coincide in SL_6.
{"title":"The Mirković–Vilonen basis and Duistermaat–Heckman measures","authors":"Pierre Baumann, J. Kamnitzer, A. Knutson","doi":"10.4310/acta.2021.v227.n1.a1","DOIUrl":"https://doi.org/10.4310/acta.2021.v227.n1.a1","url":null,"abstract":"Using the geometric Satake correspondence, the Mirkovic-Vilonen cycles in the affine Grasssmannian give bases for representations of a semisimple group G . We prove that these bases are \"perfect\", i.e. compatible with the action of the Chevelley generators of the positive half of the Lie algebra g. We compute this action in terms of intersection multiplicities in the affine Grassmannian. We prove that these bases stitch together to a basis for the algebra C[N] of regular functions on the unipotent subgroup. We compute the multiplication in this MV basis using intersection multiplicities in the Beilinson-Drinfeld Grassmannian, thus proving a conjecture of Anderson. In the third part of the paper, we define a map from C[N] to a convolution algebra of measures on the dual of the Cartan subalgebra of g. We characterize this map using the universal centralizer space of G. We prove that the measure associated to an MV basis element equals the Duistermaat-Heckman measure of the corresponding MV cycle. This leads to a proof of a conjecture of Muthiah. Finally, we use the map to measures to compare the MV basis and Lusztig's dual semicanonical basis. We formulate conjectures relating the algebraic invariants of preprojective algebra modules (which underlie the dual semicanonical basis) and geometric invariants of MV cycles. In the appendix, we use these ideas to prove that the MV basis and the dual semicanonical basis do not coincide in SL_6.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2019-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45929406","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：