Annals of PdePub Date : 2023-03-15DOI: 10.1007/s40818-023-00149-6
John Anderson
{"title":"Global stability for a nonlinear system of anisotropic wave equations","authors":"John Anderson","doi":"10.1007/s40818-023-00149-6","DOIUrl":"10.1007/s40818-023-00149-6","url":null,"abstract":"<div><p>In this paper, we initiate the study of global stability for anisotropic systems of quasilinear wave equations. Equations of this kind arise naturally in the study of crystal optics, and they exhibit birefringence. We introduce a physical space strategy based on bilinear energy estimates that allows us to prove decay for the nonlinear problem. This uses decay for the homogeneous wave equation as a black box. The proof also requires us to interface this strategy with the vector field method and take advantage of the scaling vector field. A careful analysis of the spacetime geometry of the interaction between waves is necessary in the proof.</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":"9 1","pages":""},"PeriodicalIF":2.8,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50483976","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}
Annals of PdePub Date : 2023-02-07DOI: 10.1007/s40818-022-00144-3
Yan Guo, Mahir Hadzic, Juhi Jang
{"title":"Naked Singularities in the Einstein-Euler System","authors":"Yan Guo, Mahir Hadzic, Juhi Jang","doi":"10.1007/s40818-022-00144-3","DOIUrl":"10.1007/s40818-022-00144-3","url":null,"abstract":"<div><p>In 1990, based on numerical and formal asymptotic analysis, Ori and Piran predicted the existence of selfsimilar spacetimes, called relativistic Larson-Penston solutions, that can be suitably flattened to obtain examples of spacetimes that dynamically form naked singularities from smooth initial data, and solve the radially symmetric Einstein-Euler system. Despite its importance, a rigorous proof of the existence of such spacetimes has remained elusive, in part due to the complications associated with the analysis across the so-called sonic hypersurface. We provide a rigorous mathematical proof. Our strategy is based on a delicate study of nonlinear invariances associated with the underlying non-autonomous dynamical system to which the problem reduces after a selfsimilar reduction. Key technical ingredients are a monotonicity lemma tailored to the problem, an ad hoc shooting method developed to construct a solution connecting the sonic hypersurface to the so-called Friedmann solution, and a nonlinear argument to construct the maximal analytic extension of the solution. Finally, we reformulate the problem in double-null gauge to flatten the selfsimilar profile and thus obtain an asymptotically flat spacetime with an isolated naked singularity.</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":"9 1","pages":""},"PeriodicalIF":2.8,"publicationDate":"2023-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40818-022-00144-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9275785","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}
Annals of PdePub Date : 2022-12-17DOI: 10.1007/s40818-022-00142-5
Benjamin Dodson
{"title":"A Determination of the Blowup Solutions to the Focusing NLS with Mass Equal to the Mass of the Soliton","authors":"Benjamin Dodson","doi":"10.1007/s40818-022-00142-5","DOIUrl":"10.1007/s40818-022-00142-5","url":null,"abstract":"<div><p>In this paper we prove rigidity for blowup solutions to the focusing, mass-critical nonlinear Schrödinger equation in dimensions <span>(2 le d le 15)</span> with mass equal to the mass of the soliton. We prove that the only such solutions are the solitons and the pseudoconformal transformation of the solitons. We show that this implies a Liouville result for the nonlinear Schrödinger equation.\u0000</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":"9 1","pages":""},"PeriodicalIF":2.8,"publicationDate":"2022-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40818-022-00142-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50489466","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}
Annals of PdePub Date : 2022-12-12DOI: 10.1007/s40818-022-00128-3
Kenjiro Ishizuka, Kenji Nakanishi
{"title":"Global Dynamics Around 2-Solitons for the Nonlinear Damped Klein-Gordon Equations","authors":"Kenjiro Ishizuka, Kenji Nakanishi","doi":"10.1007/s40818-022-00128-3","DOIUrl":"10.1007/s40818-022-00128-3","url":null,"abstract":"<div><p>Global behavior of solutions is studied for the nonlinear Klein-Gordon equation with a focusing power nonlinearity and a damping term in the energy space on the Euclidean space. We give a complete classification of solutions into 5 types of global behavior for all initial data in a small neighborhood of each superposition of two ground states (2-solitons) with the opposite signs and sufficient spatial distance. The neighborhood contains, for each sign of the ground state, the manifold with codimension one in the energy space, consisting of solutions that converge to the ground state at time infinity. The two manifolds are joined at their boundary by the manifold with codimension two of solutions that are asymptotic to 2-solitons moving away from each other. The connected union of these three manifolds separates the rest of the neighborhood into the open set of global decaying solutions and that of blow-up.</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":"9 1","pages":""},"PeriodicalIF":2.8,"publicationDate":"2022-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40818-022-00128-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50474404","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}
Annals of PdePub Date : 2022-12-12DOI: 10.1007/s40818-022-00143-4
Sanchit Chaturvedi, Cole Graham
{"title":"The Inviscid Limit of Viscous Burgers at Nondegenerate Shock Formation","authors":"Sanchit Chaturvedi, Cole Graham","doi":"10.1007/s40818-022-00143-4","DOIUrl":"10.1007/s40818-022-00143-4","url":null,"abstract":"<div><p>We study the vanishing viscosity limit of the one-dimensional Burgers equation near nondegenerate shock formation. We develop a matched asymptotic expansion that describes small-viscosity solutions to arbitrary order up to the moment the first shock forms. The inner part of this expansion has a novel structure based on a fractional spacetime Taylor series for the inviscid solution. We obtain sharp vanishing viscosity rates in a variety of norms, including <span>(L^infty )</span>. Comparable prior results break down in the vicinity of shock formation. We partially fill this gap.</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":"9 1","pages":""},"PeriodicalIF":2.8,"publicationDate":"2022-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50474399","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}
Annals of PdePub Date : 2022-11-19DOI: 10.1007/s40818-022-00141-6
Tristan Buckmaster, Theodore D. Drivas, Steve Shkoller, Vlad Vicol
{"title":"Simultaneous Development of Shocks and Cusps for 2D Euler with Azimuthal Symmetry from Smooth Data","authors":"Tristan Buckmaster, Theodore D. Drivas, Steve Shkoller, Vlad Vicol","doi":"10.1007/s40818-022-00141-6","DOIUrl":"10.1007/s40818-022-00141-6","url":null,"abstract":"<div><p>A fundamental question in fluid dynamics concerns the formation of discontinuous shock waves from smooth initial data. We prove that from smooth initial data, smooth solutions to the 2d Euler equations in azimuthal symmetry form a first singularity, the so-called <span>(C^{frac{1}{3}} )</span> <i>pre-shock</i>. The solution in the vicinity of this pre-shock is shown to have a fractional series expansion with coefficients computed from the data. Using this precise description of the pre-shock, we prove that a <i>discontinuous shock</i> instantaneously develops after the pre-shock. This <i>regular shock solution</i> is shown to be unique in a class of entropy solutions with azimuthal symmetry and regularity determined by the pre-shock expansion. Simultaneous to the development of the shock front, two other characteristic surfaces of cusp-type singularities emerge from the pre-shock. These surfaces have been termed <i>weak discontinuities</i> by Landau & Lifschitz [12, Chapter IX, §96], who conjectured some type of singular behavior of derivatives along such surfaces. We prove that along the slowest surface, all fluid variables except the entropy have <span>(C^{1, {frac{1}{2}} })</span> one-sided cusps from the shock side, and that the normal velocity is decreasing in the direction of its motion; we thus term this surface a <i>weak rarefaction wave</i>. Along the surface moving with the fluid velocity, density and entropy form <span>(C^{1, {frac{1}{2}} })</span> one-sided cusps while the pressure and normal velocity remain <span>(C^2)</span>; as such, we term this surface a <i>weak contact discontinuity</i>.</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":"8 2","pages":""},"PeriodicalIF":2.8,"publicationDate":"2022-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40818-022-00141-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50497863","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}
Annals of PdePub Date : 2022-11-15DOI: 10.1007/s40818-022-00139-0
Siyuan Ma, Lin Zhang
{"title":"Price’s Law for Spin Fields on a Schwarzschild Background","authors":"Siyuan Ma, Lin Zhang","doi":"10.1007/s40818-022-00139-0","DOIUrl":"10.1007/s40818-022-00139-0","url":null,"abstract":"<div><p>In this work, we derive the globally precise late-time asymptotics for the spin-<span>({mathfrak {s}})</span> fields on a Schwarzschild background, including the scalar field <span>(({mathfrak {s}}=0))</span>, the Maxwell field <span>(({mathfrak {s}}=pm 1))</span> and the linearized gravity <span>(({mathfrak {s}}=pm 2))</span>. The conjectured Price’s law in the physics literature which predicts the sharp rates of decay of the spin <span>(s=pm {mathfrak {s}})</span> components towards the future null infinity as well as in a compact region is shown. Further, we confirm the heuristic claim by Barack and Ori that the spin <span>(+1, +2)</span> components have an extra power of decay at the event horizon than the conjectured Price’s law. The asymptotics are derived via a unified, detailed analysis of the Teukolsky master equation that is satisfied by all these components.</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":"8 2","pages":""},"PeriodicalIF":2.8,"publicationDate":"2022-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40818-022-00139-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50485680","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}
Annals of PdePub Date : 2022-11-13DOI: 10.1007/s40818-022-00140-7
Jiajie Chen, Thomas Y. Hou, De Huang
{"title":"Asymptotically self-similar blowup of the Hou-Luo model for the 3D Euler equations","authors":"Jiajie Chen, Thomas Y. Hou, De Huang","doi":"10.1007/s40818-022-00140-7","DOIUrl":"10.1007/s40818-022-00140-7","url":null,"abstract":"<div><p>Inspired by the numerical evidence of a potential 3D Euler singularity [54, 55], we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in [54, 55] for the 3D Euler equations with boundary. Our finite time blowup solution for the HL model and the singular solution considered in [54, 55] share some essential features, including similar blowup exponents, symmetry properties of the solution, and the sign of the solution. We use a dynamical rescaling formulation and the strategy proposed in our recent work in [11] to establish the nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the HL model with smooth initial data and finite energy will develop a focusing asymptotically self-similar singularity in finite time. Moreover the self-similar profile is unique within a small energy ball and the <span>(C^gamma )</span> norm of the density <span>(theta )</span> with <span>(gamma approx 1/3)</span> is uniformly bounded up to the singularity time.</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":"8 2","pages":""},"PeriodicalIF":2.8,"publicationDate":"2022-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50479258","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}
Annals of PdePub Date : 2022-10-20DOI: 10.1007/s40818-022-00124-7
Stefan Czimek, Olivier Graf
{"title":"The Canonical Foliation On Null Hypersurfaces in Low Regularity","authors":"Stefan Czimek, Olivier Graf","doi":"10.1007/s40818-022-00124-7","DOIUrl":"10.1007/s40818-022-00124-7","url":null,"abstract":"<div><p>Let <span>({{mathcal {H}}})</span> denote the future outgoing null hypersurface emanating from a spacelike 2-sphere <i>S</i> in a vacuum spacetime <span>(({{mathcal {M}}},textbf{g}))</span>. In this paper we study the so-called <i>canonical foliation</i> on <span>({{mathcal {H}}})</span> introduced in [13, 22] and show that the corresponding geometry is controlled locally only in terms of the initial geometry on <i>S</i> and the <span>(L^2)</span> curvature flux through <span>({{mathcal {H}}})</span>. In particular, we show that the ingoing and outgoing null expansions <span>({textrm{tr}}chi )</span> and <span>({textrm{tr}}{{{underline{chi }}}})</span> are both locally uniformly bounded. The proof of our estimates relies on a generalisation of the methods of [15,16,17] and [1, 2, 26, 32] where the geodesic foliation on null hypersurfaces <span>({{mathcal {H}}})</span> is studied. The results of this paper, while of independent interest, are essential for the proof of the spacelike-characteristic bounded <span>(L^2)</span> curvature theorem [12].</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":"8 2","pages":""},"PeriodicalIF":2.8,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50500828","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}
Annals of PdePub Date : 2022-10-20DOI: 10.1007/s40818-022-00122-9
Stefan Czimek, Olivier Graf
{"title":"The Spacelike-Characteristic Cauchy Problem of General Relativity in Low Regularity","authors":"Stefan Czimek, Olivier Graf","doi":"10.1007/s40818-022-00122-9","DOIUrl":"10.1007/s40818-022-00122-9","url":null,"abstract":"<div><p>In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. Given initial data on a maximal spacelike hypersurface <span>(Sigma simeq overline{B_1} subset {{mathbb {R}}}^3)</span> and the outgoing null hypersurface <span>({{mathcal {H}}})</span> emanating from <span>({partial }Sigma )</span>, we prove <i>a priori</i> estimates for the resulting future development in terms of low-regularity bounds on the initial data at the level of curvature in <span>(L^2)</span>. The proof uses the bounded <span>(L^2)</span> curvature theorem [22], the extension procedure for the constraint equations [12], Cheeger-Gromov theory in low regularity [13], the canonical foliation on null hypersurfaces in low regularity [15] and global elliptic estimates for spacelike maximal hypersurfaces.</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":"8 2","pages":""},"PeriodicalIF":2.8,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50500827","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}