Communications on Pure and Applied Mathematics最新文献

{"title":"Long-time correlations for a hard-sphere gas at equilibrium","authors":"Thierry Bodineau,&nbsp;Isabelle Gallagher,&nbsp;Laure Saint-Raymond,&nbsp;Sergio Simonella","doi":"10.1002/cpa.22120","DOIUrl":"10.1002/cpa.22120","url":null,"abstract":"<p>It has been known since Lanford that the dynamics of a hard-sphere gas is described in the low density limit by the Boltzmann equation, at least for short times. The classical strategy of proof fails for longer times, even close to equilibrium. In this paper, we introduce a weak convergence method coupled with a sampling argument to prove that the covariance of the fluctuation field around equilibrium is governed by the linearized Boltzmann equation globally in time (including in diffusive regimes). This method is much more robust and simpler than the one devised in Bodineau et al which was specific to the 2D case.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48424260","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}

{"title":"Generalization of waving-plate theory to multiple interacting swimmers","authors":"Peter J. Baddoo,&nbsp;Nicholas J. Moore,&nbsp;Anand U. Oza,&nbsp;Darren G. Crowdy","doi":"10.1002/cpa.22113","DOIUrl":"10.1002/cpa.22113","url":null,"abstract":"<p>Early research in aerodynamics and biological propulsion was dramatically advanced by the analytical solutions of Theodorsen, von Kármán, Wu and others. While these classical solutions apply only to isolated swimmers, the flow interactions between multiple swimmers are relevant to many practical applications, including the schooling and flocking of animal collectives. In this work, we derive a class of solutions that describe the hydrodynamic interactions between an arbitrary number of swimmers in a two-dimensional inviscid fluid. Our approach is rooted in multiply-connected complex analysis and exploits several recent results. Specifically, the transcendental (Schottky–Klein) prime function serves as the basic building block to construct the appropriate conformal maps and leading-edge-suction functions, which allows us to solve the modified Schwarz problem that arises. As such, our solutions generalize classical thin aerofoil theory, specifically Wu's waving-plate analysis, to the case of multiple swimmers. For the case of a pair of interacting swimmers, we develop an efficient numerical implementation that allows rapid computations of the forces on each swimmer. We investigate flow-mediated equilibria and find excellent agreement between our new solutions and previously reported experimental results. Our solutions recover and unify disparate results in the literature, thereby opening the door for future studies into the interactions between multiple swimmers.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43968704","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}

{"title":"Reflective prolate-spheroidal operators and the adelic Grassmannian","authors":"W. Riley Casper,&nbsp;F. Alberto Grünbaum,&nbsp;Milen Yakimov,&nbsp;Ignacio Zurrián","doi":"10.1002/cpa.22118","DOIUrl":"10.1002/cpa.22118","url":null,"abstract":"<p>Beginning with the work of Landau, Pollak and Slepian in the 1960s on time-band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory, and integrable systems. Previously, such pairs were constructed by ad hoc methods, which essentially worked because a commuting operator of low order could be found by a direct calculation. We describe a general approach to these problems that proves that every point <i>W</i> of Wilson's infinite dimensional adelic Grassmannian <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>Gr</mi>\u0000 <mi>ad</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$operatorname{{mathrm{Gr}^{mathrm{ad}}}}$</annotation>\u0000 </semantics></math> gives rise to an integral operator <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>T</mi>\u0000 <mi>W</mi>\u0000 </msub>\u0000 <annotation>$T_W$</annotation>\u0000 </semantics></math>, acting on <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Γ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$L^2(Gamma )$</annotation>\u0000 </semantics></math> for a contour <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 <mo>⊂</mo>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <annotation>$Gamma subset mathbb {C}$</annotation>\u0000 </semantics></math>, which reflects a differential operator with rational coefficients <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>R</mi>\u0000 <mo>(</mo>\u0000 <mi>z</mi>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>∂</mi>\u0000 <mi>z</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$R(z, partial _z)$</annotation>\u0000 </semantics></math> in the sense that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>R</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mo>−</mo>\u0000 <mi>z</mi>\u0000 <mo>,</mo>\u0000 <mo>−</mo>\u0000 <msub>\u0000 <mi>∂</mi>\u0000 <mi>z</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>∘</mo>\u0000 <msub>\u0000 <mi>T</mi>\u0000 <mi>W</mi>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <msub>\u0000 <mi>T</m","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49479160","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}

{"title":"Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges","authors":"Seung-Yeop Lee,&nbsp;Meng Yang","doi":"10.1002/cpa.22122","DOIUrl":"10.1002/cpa.22122","url":null,"abstract":"<p>We consider the orthogonal polynomial <i>pn</i>(<i>z</i>) with respect to the planar measure supported on the whole complex plane\u0000\u0000 </p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45547879","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}

{"title":"Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations","authors":"Jacob Bedrossian,&nbsp;Roberta Bianchini,&nbsp;Michele Coti Zelati,&nbsp;Michele Dolce","doi":"10.1002/cpa.22123","DOIUrl":"10.1002/cpa.22123","url":null,"abstract":"<p>We investigate the long-time properties of the two-dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size ε. Under the classical Miles-Howard stability condition on the Richardson number, we prove that the system experiences a shear-buoyancy instability: the density variation and velocity undergo an <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>O</mi>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>t</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$O(t^{-1/2})$</annotation>\u0000 </semantics></math> inviscid damping while the vorticity and density gradient grow as <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>O</mi>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>t</mi>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$O(t^{1/2})$</annotation>\u0000 </semantics></math>. The result holds at least until the natural, nonlinear timescale <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mo>≈</mo>\u0000 <msup>\u0000 <mi>ε</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$t approx varepsilon ^{-2}$</annotation>\u0000 </semantics></math>. Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles-Howard spectral stability condition; (B) a variation of the Fourier time-dependent energy method introduced for the inviscid, homogeneous Couette flow problem developed on a toy model adapted to the Boussinesq equations, that is, tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22123","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47454564","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}

{"title":"Detecting the birth and death of finite-time coherent sets","authors":"Gary Froyland,&nbsp;Péter Koltai","doi":"10.1002/cpa.22115","DOIUrl":"10.1002/cpa.22115","url":null,"abstract":"<p>Finite-time coherent sets (FTCSs) are distinguished regions of phase space that resist mixing with the surrounding space for some finite period of time; physical manifestations include eddies and vortices in the ocean and atmosphere, respectively. The boundaries of FTCSs are examples of Lagrangian coherent structures (LCSs). The selection of the time duration over which FTCS and LCS computations are made in practice is crucial to their success. If this time is longer than the lifetime of coherence of individual objects then existing methods will fail to detect the shorter-lived coherence. It is of clear practical interest to determine the full lifetime of coherent objects, but in complicated practical situations, for example a field of ocean eddies with varying lifetimes, this is impossible with existing approaches. Moreover, determining the timing of emergence and destruction of coherent sets is of significant scientific interest. In this work we introduce new constructions to address these issues. The key components are an inflated dynamic Laplace operator and the concept of semi-material FTCSs. We make strong mathematical connections between the inflated dynamic Laplacian and the standard dynamic Laplacian, showing that the latter arises as a limit of the former. The spectrum and eigenfunctions of the inflated dynamic Laplacian directly provide information on the number, lifetimes, and evolution of coherent sets.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22115","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42196980","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}

{"title":"On the stabilizing effect of rotation in the 3d Euler equations","authors":"Yan Guo,&nbsp;Chunyan Huang,&nbsp;Benoit Pausader,&nbsp;Klaus Widmayer","doi":"10.1002/cpa.22107","DOIUrl":"10.1002/cpa.22107","url":null,"abstract":"<p>While it is well known that constant rotation induces linear dispersive effects in various fluid models, we study here its effect on long time nonlinear dynamics in the inviscid setting. More precisely, we investigate stability in the 3d rotating Euler equations in <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <annotation>$mathbb {R}^3$</annotation>\u0000 </semantics></math> with a <i>fixed</i> speed of rotation. We show that for any <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>M</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$mathcal {M}&gt; 0$</annotation>\u0000 </semantics></math>, axisymmetric initial data of sufficiently small size ε lead to solutions that exist for a long time at least <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>ε</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mi>M</mi>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$varepsilon ^{-mathcal {M}}$</annotation>\u0000 </semantics></math> and disperse. This is a manifestation of the stabilizing effect of rotation, regardless of its speed. To achieve this we develop an anisotropic framework that naturally builds on the available symmetries. This allows for a precise quantification and control of the geometry of nonlinear interactions, while at the same time giving enough information to obtain dispersive decay via adapted linear dispersive estimates.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22107","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41248523","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}

{"title":"Polynomial-time universality and limitations of deep learning","authors":"Emmanuel Abbe,&nbsp;Colin Sandon","doi":"10.1002/cpa.22121","DOIUrl":"10.1002/cpa.22121","url":null,"abstract":"<p>The goal of this paper is to characterize function distributions that general neural networks trained by descent algorithms (GD/SGD), can or cannot learn in polytime. The results are: (1) The paradigm of general neural networks trained by SGD is poly-time universal: any function distribution that can be learned from samples in polytime can also be learned by a poly-size neural net trained by SGD with polynomial parameters. In particular, this can be achieved despite polynomial noise on the gradients, implying a separation result between SGD-based deep learning and statistical query algorithms, as the latter are not comparably universal due to cases like parities. This also shows that deep learning does not suffer from the limitations of shallow networks. (2) The paper further gives a lower-bound on the generalization error of descent algorithms, which relies on two quantities: the cross-predictability, an average-case quantity related to the statistical dimension, and the null-flow, a quantity specific to descent algorithms. The lower-bound implies in particular that for functions of low enough cross-predictability, the above robust universality breaks down once the gradients are averaged over too many samples (as in perfect GD) rather than fewer (as in SGD). (3) Finally, it is shown that if larger amounts of noise are added on the initialization and on the gradients, then SGD is no longer comparably universal due again to distributions having low enough cross-predictability.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22121","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43124195","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}

{"title":"Compressible Navier-Stokes equations with ripped density","authors":"Raphaël Danchin,&nbsp;Piotr BogusŁaw Mucha","doi":"10.1002/cpa.22116","DOIUrl":"10.1002/cpa.22116","url":null,"abstract":"<p>We are concerned with the Cauchy problem for the two-dimensional compressible Navier-Stokes equations  supplemented with general <i>H</i>¹ initial velocity and bounded initial density not necessarily strictly positive: it may be the characteristic function of any set, for instance. In the perfect gas case, we establish global-in-time existence and uniqueness, provided the volume (bulk) viscosity coefficient is large enough. For more general pressure laws (like e.g., <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mi>ρ</mi>\u0000 <mi>γ</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$P=rho ^gamma$</annotation>\u0000 </semantics></math> with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>γ</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$gamma &gt;1$</annotation>\u0000 </semantics></math>), we still get global existence, but uniqueness remains an open question. As a by-product of our results, we give a rigorous justification of the convergence to the inhomogeneous incompressible Navier-Stokes equations when the bulk viscosity tends to infinity. In the three-dimensional case, similar results are proved for short time without restriction on the viscosity, and for large time if the initial velocity field is small enough.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41561666","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}

{"title":"Giant component for the supercritical level-set percolation of the Gaussian free field on regular expander graphs","authors":"Jiří Černý","doi":"10.1002/cpa.22112","DOIUrl":"10.1002/cpa.22112","url":null,"abstract":"<p>We consider the zero-average Gaussian free field on a certain class of finite <i>d</i>-regular graphs for fixed <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>≥</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$dge 3$</annotation>\u0000 </semantics></math>. This class includes <i>d</i>-regular expanders of large girth and typical realisations of random <i>d</i>-regular graphs. We show that the level set of the zero-average Gaussian free field above level <i>h</i> has a giant component in the whole supercritical phase, that is for all <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>h</mi>\u0000 <mo>&lt;</mo>\u0000 <msub>\u0000 <mi>h</mi>\u0000 <mi>★</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$h&lt;h_star$</annotation>\u0000 </semantics></math>, with probability tending to one as the size of the graphs tends to infinity. In addition, we show that this component is unique. This significantly improves the result of [4], where it was shown that a linear fraction of vertices is in mesoscopic components if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>h</mi>\u0000 <mo>&lt;</mo>\u0000 <msub>\u0000 <mi>h</mi>\u0000 <mi>★</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$h&lt;h_star$</annotation>\u0000 </semantics></math>, and together with the description of the subcritical phase from [4] establishes a fully-fledged percolation phase transition for the model.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49449835","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}