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Nastasia Grubic

Publications and source records attributed to Nastasia Grubic.

7 recordsLinked to original sources
arXiv2023

Finite-time singularity formation for angled-crested water waves

We show that the water waves system is locally wellposed in weighted Sobolev spaces which allow for interfaces with corners. No symmetry assumptions are required. These singular points are not rigid: if the initial interface exhibits a corner, it remains a corner but generically its angle changes. Using a characterization of the asymptotic behavior of the fluid near a corner that follows from our a priori energy estimates, we show the existence of initial data in these spaces for which the fluid becomes singular in finite time.

math.AP↗
arXiv2021

Self-intersecting interfaces for stationary solutions of the two-fluid Euler equations

We prove that there are stationary solutions to the 2D incompressible free boundary Euler equations with two fluids, possibly with a small gravity constant, that feature a splash singularity. More precisely, in the solutions we construct the interface is a $C^{2,α}$ smooth curve that intersects itself at one point, and the vorticity density on the interface is of class $C^α$. The proof consists in perturbing Crapper's family of formal stationary solutions with one fluid, so the crux is to introduce a small but positive second-fluid density. To do so, we use a novel set of weighted estimates for self-intersecting interfaces that squeeze an incompressible fluid. These estimates will also be applied to interface evolution problems in a forthcoming paper.

math.AP↗
arXiv2017

On the existence of stationary splash singularities for the Euler equations

In this paper we discuss the existence of stationary incompressible fluids with splash singularities. Specifically, we show that there are stationary solutions to the Euler equations with two fluids whose interfaces are arbitrarily close to a splash, and that there are stationary water waves with splash singularities.

math.AP↗
arXiv2014

On the area of the symmetry orbits in weakly regular Einstein-Euler spacetimes with Gowdy symmetry

This paper establishes novel bounds for Gowdy-symmetric Einstein-Euler spacetimes and completes the analysis, initiated by LeFloch and Rendall, of the global areal foliation for these spacetimes. We thus consider the initial value problem for the Einstein-Euler equations under the assumption of Gowdy symmetry. We establish that, for the maximal Cauchy development of future contracting initial data, the area of the group orbits approaches zero toward the future. This property holds as one approaches the future boundary of the spacetime, provided a geometry invariant associated with the Gowdy symmetry property is initially non-vanishing. Our condition is sharp within the class of spatially homogeneous spacetimes.

gr-qc↗
arXiv2014

The equations of elastostatics in a Riemannian manifold

To begin with, we identify the equations of elastostatics in a Riemannian manifold, which generalize those of classical elasticity in the three-dimensional Euclidean space. Our approach relies on the principle of least energy, which asserts that the deformation of the elastic body arising in response to given loads minimizes over a specific set of admissible deformations the total energy of the elastic body, defined as the difference between the strain energy and the potential of the loads. Assuming that the strain energy is a function of the metric tensor field induced by the deformation, we first derive the principle of virtual work and the associated nonlinear boundary value problem of nonlinear elasticity from the expression of the total energy of the elastic body. We then show that this boundary value problem possesses a solution if the loads are sufficiently small (in a sense we specify).

math.AP↗
arXiv2012

Weakly regular Einstein-Euler spacetimes with Gowdy symmetry. The global areal foliation

We consider weakly regular Gowdy-symmetric spacetimes on T3 satisfying the Einstein-Euler equations of general relativity, and we solve the initial value problem when the initial data set has bounded variation, only, so that the corresponding spacetime may contain impulsive gravitational waves as well as shock waves. By analyzing, both, future expanding and future contracting spacetimes, we establish the existence of a global foliation by spacelike hypersurfaces so that the time function coincides with the area of the surfaces of symmetry and asymptotically approaches infinity in the expanding case and zero in the contracting case. More precisely, the latter property in the contracting case holds provided the mass density does not exceed a certain threshold, which is a natural assumption since certain exceptional data with sufficiently large mass density are known to give rise to a Cauchy horizon, on which the area function attains a positive value. An earlier result by LeFloch and Rendall assumed a different class of weak regularity and did not determine the range of the area function in the contracting case. Our method of proof is based on a version of the random choice scheme adapted to the Einstein equations for the symmetry and regularity class under consideration. We also analyze the Einstein constraint equations under weak regularity.

gr-qc↗