Our result is a construction of infinitely many radial selfsimilar implosion profiles for the gravitational Euler–Poisson system. The problem can be expressed as solving a system of nonautonomous nonlinear ODEs. The first rigorous existence result for a nontrivial solution to these ODEs is due to Guo et al. (Commun Math Phys 386(3):1551–1601, 2021), in which they construct a solution found numerically by Larson (Mon Not R Astron Soc 145(3):271–295, 1969) and Penston (Mon Not R Astron Soc 144(4):425–448, 1969) independently. The solutions we construct belong to a different regime and correspond to a strict subset of the family of profiles discovered numerically by Hunter (Astrophys J 218:834, 1977). Our proof adapts a technique developed by Collot et al. (Mem Am Math Soc 260(1255):v+97, 2019), in which they study blowup for a family of energysupercritical focusing semilinear heat equations. In our case, the quasilinearity presents complications, most severely near the sonic point where the system degenerates.
 NSFPAR ID:
 10419967
 Date Published:
 Journal Name:
 Annals of PDE
 Volume:
 9
 Issue:
 1
 ISSN:
 25245317
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract 
In the previous works [RSR19, SR22] we have introduced a new type of selfsimilarity for the Einstein vacuum equations characterized by the fact that the homothetic vector field may be spacelike on the past light cone of the singularity. In this work we give a systematic treatment of this new selfsimilarity. In particular, we provide geometric characterizations of spacetimes admitting the new symmetry and show the existence and uniqueness of formal expansions around the past null cone of the singularity which may be considered analogues of the wellknown Fefferman–Graham expansions. In combination with results from [RSR19] our analysis will show that the twisted selfsimilar solutions are sufficiently general to describe all possible asymptotic behaviors for spacetimes in the small data regime which are selfsimilar and whose homothetic vector field is everywhere spacelike on an initial spacelike hypersurface. We present an application of this later fact to the understanding of the global structure of Fefferman–Graham spacetimes and the naked singularities of [RSR19, SR22]. Lastly, we observe that by an amalgamation of the techniques from [RSR18, RSR19], one may associate true solutions to the Einstein vacuum equations to each of our formal expansions in a suitable region of spacetime.more » « less

We prove the existence and uniqueness of a solution of a C^{0}Interior Penalty Discontinuous Galerkin (C^{0}IPDG) method for the numerical solution of a fourth‐order total variation flow problem that has been developed in part I of the paper. The proof relies on a nonlinear version of the Lax‐Milgram Lemma. It requires to establish that the nonlinear operator associated with the C^{0}IPDG approximation is Lipschitz continuous and strongly monotone on bounded sets of the underlying finite element space.

We prove the existence of a weak solution to a fluidstructure interaction (FSI) problem between the flow of an incompressible, viscous fluid modeled by the NavierStokes equations, and a poroviscoelastic medium modeled by the Biot equations. The two are nonlinearly coupled over an interface with mass and elastic energy, modeled by a reticular plate equation, which is transparent to fluid flow. The existence proof is constructive, consisting of two steps. First, the existence of a weak solution to a regularized problem is shown. Next, a weakclassical consistency result is obtained, showing that the weak solution to the regularized problem converges, as the regularization parameter approaches zero, to a classical solution to the original problem, when such a classical solution exists. While the assumptions in the first step only require the Biot medium to be poroelastic, the second step requires additional regularity, namely, that the Biot medium is poroviscoelastic. This is the first weak solution existence result for an FSI problem with nonlinear coupling involving a Biot model for poro(visco)elastic media.more » « less

Abstract We study the mean curvature flow in 3dimensional null hypersurfaces. In a spacetime a hypersurface is called null, if its induced metric is degenerate. The speed of the mean curvature flow of spacelike surfaces in a null hypersurface is the projection of the codimensiontwo mean curvature vector onto the null hypersurface. We impose fairly mild conditions on the null hypersurface. Then for an outer untrapped initial surface, a condition which resembles the meanconvexity of a surface in Euclidean space, we prove that the mean curvature flow exists for all times and converges smoothly to a marginally outer trapped surface (MOTS). As an application we obtain the existence of a global foliation of the past of an outermost MOTS, provided the null hypersurface admits an untrapped foliation asymptotically.