The study of spherically symmetric motion is important for the theory of explosion waves. In this paper, we construct rigorously self-similar solutions to the Riemann problem of the spherically symmetric Euler equations for general equations of state. We used the assumption of self-similarity to reduce the spherically symmetric Euler equations to a system of nonlinear ordinary differential equations, from which we obtain detailed structures of solutions besides their existence.
We construct spherically symmetric solutions to the Einstein-Euler equations, which contains a positive cosmological constant, say, the Einstein-Euler-de Sitter equations. We assume a realistic barotropic equation of state. Equilibria of the spherically symmetric Einstein-Euler-de Sitter equations are given by the Tolman-Oppenheimer-Volkoff-de Sitter equation. We can construct solutions near time periodic linearized solutions around the equilibrium. The Cauchy problem around the equilibrium can be solved. This work can be considered as a trial of the generalization of the previous work on the …
By introducing a new averaged quantity with a fast decay weight to perform Sideris's argument (Commun Math Phys, 1985) developed for the Euler Equations, we extend the formation of singularities of classical solution to the 3D Euler Equations established in Sideris (1985) and Makino et al. (Jpn J Appl Math, 1986) for the initial data with compactly supported disturbances to the spherically symmetric solution with general initial data in Sobolev space. Moreover, we also prove the formation of singularities of the spherically symmetric solutions to the 3D Euler-Poisson Equations, but remove the …
In this note, we prove that the solutions obtained to the spherically symmetric Euler equations in the recent works [2, 3] are weak solutions of the multi-dimensional compressible Euler equations. This follows from new uniform estimates made on the artificial viscosity approximations up to the origin, removing previous restrictions on the admissible test functions and ruling out formation of an artificial boundary layer at the origin. The uniform estimates may be of independent interest as concerns the possible rate of blow-up of the density and velocity at the origin for spherically symmetric…
We construct spherically symmetric solutions to the Einstein-Euler equations, which give models of gaseous stars in the framework of the general theory of relativity. We assume a realistic barotropic equation of state. Equilibria of the spherically symmetric Einstein-Euler equations are given by the Tolman-Oppenheimer-Volkoff equations, and time periodic solutions around the equilibrium of the linearized equations can be considered. Our aim is to find true solutions near these time-periodic approximations. Solutions satisfying so called physical boundary condition at the free boundary with the…
Global stability of the spherically symmetric nonisentropic compressible Euler equations with positive density around global-in-time background affine solutions is shown in the presence of free vacuum boundaries. Vacuum is achieved despite a non-vanishing density by considering a negatively unbounded entropy and we use a novel weighted energy method whereby the exponential of the entropy will act as a changing weight to handle the degeneracy of the vacuum boundary. Spherical symmetry introduces a coordinate singularity near the origin for which we adapt a method developed for the Euler-Poisson…
The hydrodynamic evolution of self-gravitating gaseous stars is governed by the Euler-Poisson equations. We study the structure of the linear approximation of barotropic perturbations around spherically symmetric equilibria based on functional analytic tools. In contrast to folklore, we show that the spectrum of the linearized operator for general perturbations is not of the Sturm-Liouville type unless the perturbations are restricted. In particular, we prove that it is of the Sturm-Liouville type for irrotational perturbations.
We consider spherically symmetric motions of inviscid compressible gas surrounding a solid ball under the gravity of the core. Equilibria touch the vacuum with finite radii, and the linearized equation around one of the equilibria has time-periodic solutions. To justify the linearization, we should construct true solutions for which this time-periodic solution plus the equilibrium is the first approximation. But this leads us to difficulty caused by singularities at the free boundary touching the vacuum. We solve this problem by the Nash-Moser theorem.
We study spherically symmetric motions of a gaseous star governed by the Euler-Poisson equations. Equilibria are given as solutions of the Lane-Emden equations, and the linearized equation around one of these equilibria admits time-periodic solutions. To justify the linearization, we should construct true solutions of the nonlinear evolution equation for which this time-periodic solutions plus the equilibrium is the first approximation. To do so, we apply the Nash-Moser theory. We presents a partial result under some assumptions on the adiabatic exponent of the gas, and propose an open problem…
We develop a method that works in general product Riemannian manifold to decompose the three-dimensional steady full compressible Euler system, which is of elliptic-hyperbolic composite-mixed type for subsonic flows. The method is applied to show stability of spherically symmetric subsonic flows and transonic shocks in space $\mathbb{R}^3$ under multidimensional perturbations of boundary conditions.
We prove local in time Strichartz estimates for the Dirac equation on spherically symmetric manifolds. As an application, we give a result of local well-posedness for some nonlinear models.
In the paper, we consider the free boundary value problem to 3D spherically symmetric compressible isentropic Navier-Stokes-Poisson equations for self-gravitating gaseous stars with $γ$-law pressure density function for $6/5 <γ\leq 4/3$. For stress free boundary condition and zero flow density continuously across the free boundary, the global existence of spherically symmetric weak solutions is shown, and the regularity and long time behavior of global solution are investigated for spherically symmetric initial data with the total mass smaller than a critical mass.
In this note, by constructing suitable approximate solutions, we prove the existence of global weak solutions to the compressible Navier-Stokes equations with density-dependent viscosity coefficients in the whole space $\mathbb{R}^N$, $N\geq2$ (or exterior domain), when the initial data are spherically symmetric. In particular, we prove the existence of spherically symmetric solutions to the Saint-Venant model for shallow water in the whole space (or exterior domain).
We study Wave Maps from R^{2+1} to the hyperbolic plane with smooth compactly supported initial data which are close to smooth spherically symmetric ones with respect to some H^{1+μ}, μ>0. We show that such Wave Maps don't develop singularities and stay close to the Wave Map extending the spherically symmetric data with respect to all H^{1+δ}, δ<μ_{0}(μ). We obtain a similar result for Wave Maps whose initial data are close to geodesic ones. This generalizes a theorem of Sideris for this context.
We study the evolution of a self-gravitating compressible fluid in spherical symmetry and we prove the existence of weak solutions with bounded variation for the Einstein-Euler equations of general relativity. We formulate the initial value problem in Eddington-Finkelstein coordinates and prescribe spherically symmetric data on a characteristic initial hypersurface. We introduce here a broad class of initial data which contain no trapped surfaces, and we then prove that their Cauchy development contains trapped surfaces. We therefore establish the formation of trapped surfaces in weak solution…
We establish global regularity for the logarithmically energy-supercritical wave equation $\Box u = u^5 \log(2+u^2)$ in three spatial dimensions for spherically symmetric initial data, by modifying an argument of Ginibre, Soffer and Velo \cite{gsv} for the energy-critical equation. This example demonstrates that critical regularity arguments can penetrate very slightly into the supercritical regime.
We are concerned with globally defined entropy solutions to the Euler equations for compressible fluid flows in transonic nozzles with general cross-sectional areas. Such nozzles include the de Laval nozzles and other more general nozzles whose cross-sectional area functions are allowed at the nozzle ends to be either zero (closed ends) or infinity (unbounded ends). To achieve this, in this paper, we develop a vanishing viscosity method to construct globally defined approximate solutions and then establish essential uniform estimates in weighted $L^p$ norms for the whole range of physical adia…
We obtain lower bounds for the first Laplacian eigenvalues of geodesic balls of spherically symmetric manifolds. These lower bounds are only $C^{0}$ dependent on the metric coefficients.
We construct a one-parameter family of static and spherically symmetric solutions to the Einstein-Vlasov system bifurcating from the Schwarzschild spacetime. The constructed solutions have the property that the spatial support of the matter is a finite, spherically symmetric shell located away from the black hole. Our proof is mostly based on the analysis of the set of trapped timelike geodesics and of the effective potential energy for static spacetimes close to Schwarzschild. This provides an alternative approach to the construction of static solutions to the Einstein-Vlasov system in a neig…
We prove that for large enough data, the life span of smooth solutions to the Cauchy problem for the following two quasilinear hyperbolic systems is finite: (1) equations of relativistic compressible fluid dynamics, (2) equations of plasma dynamics (Euler-Maxwell equations with background charge) in the spherically symmetric case.
We disprove a conjecture in Density Functional Theory, relative to multimarginal optimal transport maps with Coulomb cost. We also provide examples of maps satisfying optimality conditions for special classes of data.
Certain steady states of the Vlasov-Poisson system can be characterized as minimizers of an energy-Casimir functional, and this fact implies a nonlinear stability property of such steady states. In previous investigations by Y. Guo and the author stability was obtained only with respect to spherically symmetric perturbations. In the present investigation we show how to remove this unphysical restriction.
In this paper we study spherically symmetric monopoles, which are critical points for the Yang-Mills-Higgs functional over a disk in 3 dimensions, with prescribed degree and covariant constant at the boundary. This is a 3-dimensional gauge-theory generalization of the Ginzburg-Landau model in 2 dimensions.
