 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
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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

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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.

In this paper we introduce a constructive approach to study wellposedness of solutions to stochastic uidstructure interaction with stochastic noise. We focus on a benchmark problem in stochastic uidstructure interaction, and prove the existence of a unique weak solution in the probabilistically strong sense. The benchmark problem consists of the 2D timedependent Stokes equations describing the ow of an incompressible, viscous uid interacting with a linearly elastic membrane modeled by the 1D linear wave equation. The membrane is stochastically forced by the timedependent white noise. The uid and the structure are linearly coupled. The constructive existence proof is based on a timediscretization via an operator splitting approach. This introduces a sequence of approximate solutions, which are random variables. We show the existence of a subsequence of approximate solutions which converges, almost surely, to a weak solution in the probabilistically strong sense. The proof is based on uniform energy estimates in terms of the expectation of the energy norms, which are the backbone for a weak compactness argument giving rise to a weakly convergent subsequence of probability measures associated with the approximate solutions. Probabilistic techniques based on the Skorohod representation theorem and the GyongyKrylov lemma are then employed to obtain almost sure convergence of a subsequence of the random approximate solutions to a weak solution in the probabilistically strong sense. The result shows that the deterministic benchmark FSI model is robust to stochastic noise, even in the presence of rough white noise in time. To the best of our knowledge, this is the rst wellposedness result for stochastic uidstructure interaction.more » « less