{"title":"Existence of multi-dimensional contact discontinuities for the ideal compressible magnetohydrodynamics","authors":"Yanjin Wang, Zhouping Xin","doi":"10.1002/cpa.22148","DOIUrl":"10.1002/cpa.22148","url":null,"abstract":"<p>We establish the local existence and uniqueness of multi-dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic systems of conservation laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversely, which lead to a two-phase free boundary problem where the pressure, velocity and magnetic field are continuous across the interface whereas the entropy and density may have jumps. To overcome the difficulties of possible nonlinear Rayleigh–Taylor instability and loss of derivatives, here we use crucially the Lagrangian formulation and Cauchy's celebrated integral (1815) for the magnetic field. These motivate us to define two special good unknowns; one enables us to capture the boundary regularizing effect of the transversal magnetic field on the flow map, and the other one allows us to get around the troublesome boundary integrals due to the transversality of the magnetic field. In particular, our result removes the additional assumption of the Rayleigh–Taylor sign condition required by Morando, Trakhinin and Trebeschi (<i>J. Differ. Equ</i>. <b>258</b> (2015), no. 7, 2531–2571; <i>Arch. Ration. Mech. Anal</i>. <b>228</b> (2018), no. 2, 697–742) and holds for both 2D and 3D and hence gives a complete answer to the two open questions raised therein. Moreover, there is <i>no loss of derivatives</i> in our well-posedness theory. The solution is constructed as <i>the inviscid limit</i> of solutions to some suitably-chosen nonlinear approximate problems for the two-phase compressible viscous non-resistive MHD.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43933569","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":"Hearing the shape of ancient noncollapsed flows in \u0000 \u0000 \u0000 R\u0000 4\u0000 \u0000 $mathbb {R}^{4}$","authors":"Wenkui Du, Robert Haslhofer","doi":"10.1002/cpa.22140","DOIUrl":"10.1002/cpa.22140","url":null,"abstract":"We consider ancient noncollapsed mean curvature flows in R4$mathbb {R}^4$ whose tangent flow at −∞$-infty$ is a bubble‐sheet. We carry out a fine spectral analysis for the bubble‐sheet function u that measures the deviation of the renormalized flow from the round cylinder R2×S1(2)$mathbb {R}^2 times S^1(sqrt {2})$ and prove that for τ→−∞$tau rightarrow -infty$ we have the fine asymptotics u(y,θ,τ)=(y⊤Qy−2tr(Q))/|τ|+o(|τ|−1)$u(y,theta ,tau )= (y^top Qy -2textrm {tr}(Q))/|tau | + o(|tau |^{-1})$ , where Q=Q(τ)$Q=Q(tau )$ is a symmetric 2 × 2‐matrix whose eigenvalues are quantized to be either 0 or −1/8$-1/sqrt {8}$ . This naturally breaks up the classification problem for general ancient noncollapsed flows in R4$mathbb {R}^4$ into three cases depending on the rank of Q. In the case rk(Q)=0$mathrm{rk}(Q)=0$ , generalizing a prior result of Choi, Hershkovits and the second author, we prove that the flow is either a round shrinking cylinder or R×$mathbb {R}times$ 2d‐bowl. In the case rk(Q)=1$mathrm{rk}(Q)=1$ , under the additional assumption that the flow either splits off a line or is self‐similarly translating, as a consequence of recent work by Angenent, Brendle, Choi, Daskalopoulos, Hershkovits, Sesum and the second author we show that the flow must be R×$mathbb {R}times$ 2d‐oval or belongs to the one‐parameter family of 3d oval‐bowls constructed by Hoffman‐Ilmanen‐Martin‐White, respectively. Finally, in the case rk(Q)=2$mathrm{rk}(Q)=2$ we show that the flow is compact and SO(2)‐symmetric and for τ→−∞$tau rightarrow -infty$ has the same sharp asymptotics as the O(2) × O(2)‐symmetric ancient ovals constructed by Hershkovits and the second author. The full classification problem will be addressed in subsequent papers based on the results of the present paper.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45668603","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}