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Mahir Hadzic

Publications and source records attributed to Mahir Hadzic.

29 records · Page 2Linked to original sources
arXiv2016

Nonlinear stability of expanding star solutions in the radially-symmetric mass-critical Euler-Poisson system

We prove nonlinear stability of compactly supported expanding star-solutions of the mass-critical gravitational Euler-Poisson system. These special solutions were discovered by Goldreich and Weber in 1980. The expanding rate of such solutions can be either self-similar or non-self-similar (linear), and we treat both types. An important outcome of our stability results is the existence of a new class of global-in-time radially symmetric solutions, which are not homogeneous and therefore not encompassed by the existing works. Using Lagrangian coordinates we reformulate the associated free-boundary problem as a degenerate quasilinear wave equation on a compact spatial domain. The problem is mass-critical with respect to an invariant rescaling and the analysis is carried out in similarity variables.

math.AP↗
arXiv2016

Well-posedness for the classical Stefan problem and the zero surface tension limit

We develop a framework for a unified treatment of well-posedness for the Stefan problem with or without surface tension. In the absence of surface tension, we establish well-posedness in Sobolev spaces for the classical Stefan problem. We introduce a new velocity variable which extends the velocity of the moving free-boundary into the interior domain. The equation satisfied by this velocity is used for the analysis in place of the heat equation satisfied by the temperature. Solutions to the classical Stefan problem are then constructed as the limit of solutions to a carefully chosen sequence of approximations to the velocity equation, in which the moving free-boundary is regularized and the boundary condition is modified in a such a way as to preserve the basic nonlinear structure of the original problem. With our methodology, we simultaneously find the required stability condition for well-posedness and obtain new estimates for the regularity of the moving free-boundary. Finally, we prove that solutions of the Stefan problem with positive surface tension $σ$ converge to solutions of the classical Stefan problem as $σ\to 0$.

math.AP↗
arXiv2015

A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary

We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has played an important role in prior work on the non-relativistic compressible Euler equations. Our main result is the derivation, relative to Lagrangian (also known as co-moving) coordinates, of local-in-time a priori estimates for the solution. The solution features a fluid-vacuum boundary, transported by the fluid four-velocity, along which the hyperbolicity of the equations degenerates. In this context, the relativistic Euler equations are equivalent to a degenerate quasilinear hyperbolic wave-map-like system that cannot be treated using standard energy methods.

math.AP↗
arXiv2013

The Global Future Stability of the FLRW Solutions to the Dust-Einstein System with a Positive Cosmological Constant

We study small perturbations of the well-known family of Friedman-Lema\^ıtre-Robertson-Walker (FLRW) solutions to the dust-Einstein system with a positive cosmological constant in the case that the spacelike Cauchy hypersurfaces are diffeomorphic to T^3. These solutions model a quiet pressureless fluid in a dynamic spacetime undergoing accelerated expansion. We show that the FLRW solutions are nonlinearly globally future-stable under small perturbations of their initial data. Our analysis takes place relative to a harmonic-type coordinate system, in which the cosmological constant results in the presence of dissipative terms in the evolution equations. Our result extends the results of [38,44,42], where analogous results were proved for the Euler-Einstein system under the equations of state p = c_s^2 ρ, 0<c_s^2 <= 1/3. The dust-Einstein system is the Euler-Einstein system with c_s=0. The main difficulty that we overcome is that the energy density of the dust loses one degree of differentiability compared to the cases 0 < c_s^2 <= 1/3. Because the dust-Einstein equations are coupled, this loss of differentiability introduces new obstacles for deriving estimates for the top-order derivatives of all solution variables. To resolve this difficulty, we commute the equations with a well-chosen differential operator and derive a collection of elliptic estimates that complement the energy estimates of [38,44]. An important feature of our analysis is that we are able to close our estimates even though the top-order derivatives of all solution variables can grow much more rapidly than in the cases 0<c_s^2 <= 1/3. Our results apply in particular to small compact perturbations of the vanishing dust state.

math.AP↗
arXiv2012

Stability for the spherically symmetric Einstein-Vlasov system - a coercivity estimate

The stability of static solutions of the spherically symmetric, asymptotically flat Einstein-Vlasov system is studied using a Hamiltonian approach based on energy-Casimir functionals. The main result is a coercivity estimate for the quadratic part of the expansion of the natural energy-Casimir functional about an isotropic steady state. The estimate shows in a quantified way that this quadratic part is positive definite on a class of linearly dynamically accessible perturbations, provided the particle distribution of the steady state is a strictly decreasing function of the particle energy and provided the steady state is not too relativistic. This should be an essential step in a fully non-linear stability analysis for the Einstein-Vlasov system. In the present paper it is exploited for obtaining a linearized stability result.

math-ph↗
arXiv2006

Global existence and nonlinear stability for the relativistic Vlasov-Poisson system in the gravitational case

As is well known from the work of R. Glassey} and J. Schaeffer, the main energy estimates which are used in global existence results for the gravitational Vlasov-Poisson system do not apply to the relativistic version of this system, and smooth solutions to the initial value problem with spherically symmetric initial data of negative energy blow up in finite time. For similar reasons the variational techniques by which Y. Guo and G. Rein obtained nonlinear stability results for the Vlasov-Poisson system do not apply in the relativistic situation. In the present paper a direct, non-variational approach is used to prove nonlinear stability of certain steady states of the relativistic Vlasov-Poisson system against spherically symmetric, dynamically accessible perturbations. The resulting stability estimates imply that smooth solutions with spherically symmetric initial data which are sufficiently close to the stable steady states exist globally in time.

math-ph↗
arXiv2006

A constraint variational problem arising in stellar dynamics

We use the compactness result of A. Burchard and Y. Guo (cf. \cite{BuGu}) to analyze the reduced 'energy' functional arising naturally in the stability analysis of steady states of the Vlasov-Poisson system (cf. \cite{SaSo} and \cite{Ha}). We consider the associated variational problem and present a new proof that puts it in the general framework for tackling the variational problems of this type, given by Y. Guo and G. Rein (cf. \cite{Re1} and \cite{Re2}).

math-ph↗