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})

On melting and freezing for the 2d radial Stefan problem

We consider the two dimensional free boundary Stefan problem describing the evolution of a spherically symmetric ice ball $\{r\leq \lambda(t)\}$. We revisit the pioneering analysis of [20] and prove the existence in the radial class of finite time melting regimes $$\lambda(t)=\left\{\begin{array}{ll} (T-t)^{1/2}e^{-\frac{\sqrt{2}}{2}\sqrt{|\ln(T-t)|}+O(1)}\\ (c+o(1))\frac{(T-t)^{\frac{k+1}{2}}}{|\ln (T-t)|^{\frac{k+1}{2k}}}, \ \ k\in \Bbb N^*\end{array}\right. \quad\text{ as } t\to T$$ which respectively correspond to the fundamental stable melting rate, and a sequence of codimension $k\in \Bbb N^*$ excited regimes. Our analysis fully revisits a related construction for the harmonic heat flow in [42] by introducing a new and canonical functional framework for the study of type II (i.e. non self similar) blow up. We also show a deep duality between the construction of the melting regimes and the derivation of a discrete sequence of global-in-time freezing regimes $$\lambda_\infty - \lambda(t)\sim\left\{\begin{array}{ll} \frac{1}{\log t}\\ \frac{1}{t^{k}(\log t)^{2}}, \ \ k\in \Bbb N^*\end{array}\right. \quad\text{ as } t\to +\infty$$ which correspond respectively to the fundamental stable freezing rate, and excited regimes which are codimension $k$ stable.Comment: 70 pages, a few references added and typos correcte

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.Comment: 38 page

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 $\sigma$ converge to solutions of the classical Stefan problem as $\sigma \to 0$.Comment: Various typos corrected and references adde

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\^{\i}tre-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 \rho, 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.Comment: In the latest version, we added a few references and corrected some typo

Stability and instability of self-gravitating relativistic matter distributions

We consider steady state solutions of the massive, asymptotically flat, spherically symmetric Einstein-Vlasov system, i.e., relativistic models of galaxies or globular clusters, and steady state solutions of the Einstein-Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift $\kappa>0$. We prove their linear instability when $\kappa$ is sufficiently large, i.e., when they are strongly relativistic, and that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in the 1960s by Zel'dovich \& Podurets (for the Einstein-Vlasov system) and by Harrison, Thorne, Wakano, \& Wheeler (for the Einstein-Euler system). Our results are in sharp contrast to the corresponding non-relativistic, Newtonian setting. We carry out a careful analysis of the linearized dynamics around the above steady states and prove an exponential trichotomy result and the corresponding index theorems for the stable/unstable invariant spaces. Finally, in the case of the Einstein-Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.Comment: 92 pages; several proofs are revised and some previous errors correcte