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Stable singularity formation for the inviscid primitive equations 无粘原始方程的稳定奇点形成
IF 1.9 1区 数学
Annales De L Institut Henri Poincare-Analyse Non Lineaire Pub Date : 2021-12-17 DOI: 10.4171/aihpc/87
Charles Collot, S. Ibrahim, Quyuan Lin
{"title":"Stable singularity formation for the inviscid primitive equations","authors":"Charles Collot, S. Ibrahim, Quyuan Lin","doi":"10.4171/aihpc/87","DOIUrl":"https://doi.org/10.4171/aihpc/87","url":null,"abstract":"The primitive equations (PEs) model large scale dynamics of the oceans and the atmosphere. While it is by now well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces, and that there are solutions to the inviscid PEs (also called the hydrostatic Euler equations) that develop singularities in finite time, the qualitative description of the blowup still remains undiscovered. In this paper, we provide a full description of two blowup mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic in time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameters family of non-smooth profiles, and is stable only by smoother perturbations.","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80990052","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}
引用次数: 5
Stability of the density patches problem with vacuum for incompressible inhomogeneous viscous flows 不可压缩非均匀粘性流的真空密度补丁问题的稳定性
IF 1.9 1区 数学
Annales De L Institut Henri Poincare-Analyse Non Lineaire Pub Date : 2021-12-13 DOI: 10.4171/aihpc/83
R. Danchin, P. Mucha, Tomasz Piasecki
{"title":"Stability of the density patches problem with vacuum for incompressible inhomogeneous viscous flows","authors":"R. Danchin, P. Mucha, Tomasz Piasecki","doi":"10.4171/aihpc/83","DOIUrl":"https://doi.org/10.4171/aihpc/83","url":null,"abstract":"We consider the inhomogeneous incompressible Navier-Stokes system in a smooth two or three dimensional bounded domain, in the case where the initial density is only bounded. Existence and uniqueness for such initial data was shown recently in [10], but the stability issue was left open. After observing that the solutions constructed in [10] have exponential decay, a result of independent interest, we prove the stability with respect to initial data, first in Lagrangian coordinates, and then in the Eulerian frame. We actually obtain stability in $L_2({mathbb R}_+;H^1(Omega))$ for the velocity and in a negative Sobolev space for the density. Let us underline that, as opposed to prior works, in case of vacuum, our stability estimates are not weighted by the initial densities. Hence, our result applies in particular to the classical density patches problem, where the density is a characteristic function.","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88489346","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}
引用次数: 2
Local minimality of $mathbb{R}^N$-valued and $mathbb{S}^N$-valued Ginzburg–Landau vortex solutions in the unit ball $B^N$ 单位球$B^N$中$mathbb{R}^N$值和$mathbb{S}^N$值Ginzburg-Landau涡解的局部极小性
IF 1.9 1区 数学
Annales De L Institut Henri Poincare-Analyse Non Lineaire Pub Date : 2021-11-15 DOI: 10.4171/aihpc/84
R. Ignat, Luc Nguyen
{"title":"Local minimality of $mathbb{R}^N$-valued and $mathbb{S}^N$-valued Ginzburg–Landau vortex solutions in the unit ball $B^N$","authors":"R. Ignat, Luc Nguyen","doi":"10.4171/aihpc/84","DOIUrl":"https://doi.org/10.4171/aihpc/84","url":null,"abstract":"We study the existence, uniqueness and minimality of critical points of the form $m_{varepsilon,eta}(x) = (f_{varepsilon,eta}(|x|)frac{x}{|x|}, g_{varepsilon,eta}(|x|))$ of the functional [ E_{varepsilon,eta}[m] = int_{B^N} Big[frac{1}{2} |nabla m|^2 + frac{1}{2varepsilon^2} (1 - |m|^2)^2 + frac{1}{2eta^2} m_{N+1}^2Big],dx ] for $m=(m_1, dots, m_N, m_{N+1}) in H^1(B^N,mathbb{R}^{N+1})$ with $m(x) = (x,0)$ on $partial B^N$. We establish a necessary and sufficient condition on the dimension $N$ and the parameters $varepsilon$ and $eta$ for the existence of an escaping vortex solution $(f_{varepsilon,eta}, g_{varepsilon,eta})$ with $g_{varepsilon,eta}>0$. We also establish its uniqueness and local minimality. In the limiting case $eta = 0$, we prove the local minimality of the degree-one vortex solution for the Ginzburg-Landau (GL) energy for every $varepsilon>0$ and $N geq 2$. Similarly, when $varepsilon = 0$, we prove the local minimality of the degree-one escaping vortex solution to an $mathbb{S}^N$-valued GL model arising in micromagnetics for every $eta>0$ and $2 leq N leq 6$.","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87508332","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}
引用次数: 0
A normalized solitary wave solution of the Maxwell-Dirac equations 麦克斯韦-狄拉克方程的归一化孤波解
IF 1.9 1区 数学
Annales De L Institut Henri Poincare-Analyse Non Lineaire Pub Date : 2021-11-01 DOI: 10.1016/j.anihpc.2020.12.006
Margherita Nolasco
{"title":"A normalized solitary wave solution of the Maxwell-Dirac equations","authors":"Margherita Nolasco","doi":"10.1016/j.anihpc.2020.12.006","DOIUrl":"10.1016/j.anihpc.2020.12.006","url":null,"abstract":"<div><p>We prove the existence of a <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span><span>-normalized solitary wave solution for the Maxwell-Dirac equations in (3+1)-Minkowski space. In addition, for the Coulomb-Dirac model, describing fermions with attractive Coulomb interactions in the mean-field limit, we prove the existence of the (positive) energy minimizer.</span></p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.12.006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74435857","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}
引用次数: 5
Stability of planar rarefaction waves under general viscosity perturbation of the isentropic Euler system 等熵欧拉系统一般粘度扰动下平面稀疏波的稳定性
IF 1.9 1区 数学
Annales De L Institut Henri Poincare-Analyse Non Lineaire Pub Date : 2021-11-01 DOI: 10.1016/j.anihpc.2021.01.001
Eduard Feireisl , Antonín Novotný
{"title":"Stability of planar rarefaction waves under general viscosity perturbation of the isentropic Euler system","authors":"Eduard Feireisl ,&nbsp;Antonín Novotný","doi":"10.1016/j.anihpc.2021.01.001","DOIUrl":"10.1016/j.anihpc.2021.01.001","url":null,"abstract":"<div><p>We consider the vanishing viscosity limit for a model of a general non-Newtonian compressible fluid in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, <span><math><mi>d</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></math></span><span>. We suppose that the initial data<span><span> approach a profile determined by the Riemann data generating a planar rarefaction wave for the isentropic Euler system. Under these circumstances the associated sequence of dissipative solutions approaches the corresponding rarefaction wave strongly in the energy norm in the vanishing viscosity limit. The result covers the particular case of a linearly </span>viscous fluid governed by the Navier–Stokes system.</span></span></p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2021.01.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86086653","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}
引用次数: 5
Polyhomogénéité des métriques compatibles avec une structure de Lie à l'infini le long du flot de Ricci 与沿里奇流的无限李结构相容的度量的多同质性
IF 1.9 1区 数学
Annales De L Institut Henri Poincare-Analyse Non Lineaire Pub Date : 2021-11-01 DOI: 10.1016/j.anihpc.2021.01.003
Mahdi Ammar
{"title":"Polyhomogénéité des métriques compatibles avec une structure de Lie à l'infini le long du flot de Ricci","authors":"Mahdi Ammar","doi":"10.1016/j.anihpc.2021.01.003","DOIUrl":"10.1016/j.anihpc.2021.01.003","url":null,"abstract":"<div><p>Le long du flot de Ricci, on étudie la polyhomogénéité des métriques pour des variétés riemanniennes non-compactes ayant « une structure de Lie fibrée à l'infini », c'est-à-dire une classe de structures de Lie à l'infini qui induit dans un sens précis des structures de fibrés sur les bords d'une certaine compactification par une variété à coins. Lorsque cette compactification est une variété à bord, cette classe de métriques contient notamment les b-métriques de Melrose, les métriques à bord fibré de Mazzeo-Melrose et les métriques edge de Mazzeo. On montre alors que la polyhomogénéité à l'infini des métriques compatibles avec une structure de Lie fibrée à l'infini est préservée localement par le flot de Ricci-DeTurck. Si la métrique initiale est asymptotiquement Einstein, on obtient la polyhomogénéité des métriques tant que le flot existe. De plus, si la métrique initiale est « lisse jusqu'au bord », alors il en sera de même pour les solutions du flot de Ricci normalisé et du flot de Ricci-DeTurck.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2021.01.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74658964","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}
引用次数: 3
Global C∞ regularity of the steady Prandtl equation with favorable pressure gradient 具有有利压力梯度的稳定Prandtl方程的全局C∞正则性
IF 1.9 1区 数学
Annales De L Institut Henri Poincare-Analyse Non Lineaire Pub Date : 2021-11-01 DOI: 10.1016/j.anihpc.2021.02.007
Yue Wang , Zhifei Zhang
{"title":"Global C∞ regularity of the steady Prandtl equation with favorable pressure gradient","authors":"Yue Wang ,&nbsp;Zhifei Zhang","doi":"10.1016/j.anihpc.2021.02.007","DOIUrl":"10.1016/j.anihpc.2021.02.007","url":null,"abstract":"<div><p>In the case of <span><em>favorable </em><em>pressure gradient</em></span>, Oleinik obtained the <em>global-in-x</em> solutions to the steady Prandtl equations with <em>low regularity</em> (see Oleinik and Samokhin <span>[9]</span>, P.21, Theorem 2.1.1). Due to the degeneracy of the equation near the boundary, the question of higher regularity of Oleinik's solutions remains open. See the <em>local-in-x</em> higher regularity established by Guo and Iyer <span>[5]</span>. In this paper, we prove that Oleinik's solutions are smooth up to the boundary <span><math><mi>y</mi><mo>=</mo><mn>0</mn></math></span> for any <span><math><mi>x</mi><mo>&gt;</mo><mn>0</mn></math></span>, using further maximum principle techniques. Moreover, since Oleinik only assumed low regularity on the data prescribed at <span><math><mi>x</mi><mo>=</mo><mn>0</mn></math></span>, our result implies instant smoothness (in the steady case, <span><math><mi>x</mi><mo>=</mo><mn>0</mn></math></span> is often considered as initial time).</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2021.02.007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80599900","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}
引用次数: 9
The Pohozaev-Schoen identity on asymptotically Euclidean manifolds: Conservation laws and their applications 渐近欧几里得流形上的Pohozaev-Schoen恒等式:守恒定律及其应用
IF 1.9 1区 数学
Annales De L Institut Henri Poincare-Analyse Non Lineaire Pub Date : 2021-11-01 DOI: 10.1016/j.anihpc.2021.01.002
R. Avalos , A. Freitas
{"title":"The Pohozaev-Schoen identity on asymptotically Euclidean manifolds: Conservation laws and their applications","authors":"R. Avalos ,&nbsp;A. Freitas","doi":"10.1016/j.anihpc.2021.01.002","DOIUrl":"10.1016/j.anihpc.2021.01.002","url":null,"abstract":"<div><p><span>The aim of this paper is to present a version of the generalized Pohozaev-Schoen identity in the context of asymptotically Euclidean manifolds. Since these kind of geometric identities have proven to be a very powerful tool when analysing different geometric problems for compact manifolds, we will present a variety of applications within this new context. Among these applications, we will show some rigidity results for asymptotically Euclidean Ricci-solitons and Codazzi-solitons. Also, we will present an almost-Schur type inequality valid in this non-compact setting which does not need restrictions on the </span>Ricci curvature. Finally, we will show how some rigidity results related with static potentials also follow from these type of conservation principles.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2021.01.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78790456","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}
引用次数: 2
Stability, well-posedness and regularity of the homogeneous Landau equation for hard potentials 硬势齐次朗道方程的稳定性、适定性和规律性
IF 1.9 1区 数学
Annales De L Institut Henri Poincare-Analyse Non Lineaire Pub Date : 2021-11-01 DOI: 10.1016/j.anihpc.2021.02.004
Nicolas Fournier , Daniel Heydecker
{"title":"Stability, well-posedness and regularity of the homogeneous Landau equation for hard potentials","authors":"Nicolas Fournier ,&nbsp;Daniel Heydecker","doi":"10.1016/j.anihpc.2021.02.004","DOIUrl":"10.1016/j.anihpc.2021.02.004","url":null,"abstract":"<div><p>We establish the well-posedness and some quantitative stability of the spatially homogeneous Landau equation for hard potentials, using some specific Monge-Kantorovich cost, assuming only that the initial condition is a probability measure with a finite moment of order <em>p</em> for some <span><math><mi>p</mi><mo>&gt;</mo><mn>2</mn></math></span>. As a consequence, we extend previous regularity results and show that all non-degenerate measure-valued solutions to the Landau equation, with a finite initial energy, immediately admit analytic densities with finite entropy. Along the way, we prove that the Landau equation instantaneously creates Gaussian moments. We also show existence of weak solutions under the only assumption of finite initial energy.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2021.02.004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72933245","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}
引用次数: 4
Global semiclassical limit from Hartree to Vlasov equation for concentrated initial data 集中初始数据从Hartree到Vlasov方程的全局半经典极限
IF 1.9 1区 数学
Annales De L Institut Henri Poincare-Analyse Non Lineaire Pub Date : 2021-11-01 DOI: 10.1016/j.anihpc.2021.01.004
Laurent Lafleche
{"title":"Global semiclassical limit from Hartree to Vlasov equation for concentrated initial data","authors":"Laurent Lafleche","doi":"10.1016/j.anihpc.2021.01.004","DOIUrl":"10.1016/j.anihpc.2021.01.004","url":null,"abstract":"<div><p>We prove a quantitative and <strong>global in time</strong><span> semiclassical limit from the Hartree to the Vlasov equation in the case of a singular interaction potential in dimension </span><span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>, including the case of a Coulomb singularity in dimension <span><math><mi>d</mi><mo>=</mo><mn>3</mn></math></span><span>. This result holds for initial data<span> concentrated enough in the sense that some space moments are initially sufficiently small. As an intermediate result, we also obtain quantitative bounds on the space and velocity moments of even order and the asymptotic behavior<span> of the spatial density due to dispersion effects, uniform in the Planck constant </span></span></span><em>ħ</em>.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2021.01.004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82727459","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}
引用次数: 17
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