arXiv subjects
Publications and source records attributed to Changhua Wei.
We investigate the dynamics of classical solutions to the dust fluid model under spherical symmetry in Schwarzschild spacetime. According to whether the solution will persist globally or develop a finite-time singularity, a precise classification of initial data is provided. Moreover, a detailed analysis on the exact blowup profile near the singularity is given.
This paper is concerned with the Cauchy problem of the evolutionary Faddeev model, a system that maps from the Minkowski space $\mathbb{R}^{1+3}$ to the unit sphere $\mathbb{S}^2$. The model is a system of nonlinear wave equations whose nonlinearities exhibit a null structure and include semilinear terms, quasilinear terms, and the unknowns themselves. By considering a class of large initial data (in energy norm) of the short pulse type, we prove that the evolutionary Faddeev model admits a globally smooth solution via energy estimates. The main result is achieved through the selection of appropriate multipliers that are specially adapted to the geometry of the system.
This paper is concerned with the global existence and blowup of the classical solution to the Cauchy problem of the relativistic Euler equation with $ p=0 $ in a fixed Friedmann-Lema\^ıtre-Robertson-Walker (FLRW) spacetime. The aim of this work is to study clearly the effect of the expansion rate of the spacetime on the life span of the classical solution to the pressureless fluid. Since the density and the velocity of the relativistic dust admits the same principal part, we can obtain much more accurate results by the characteristic method rather than energy estimates.
This paper is concerned with the global stability of the plane wave solutions to the relativistic string equation with non-small perturbations. Under certain decay assumptions on the plane wave, we conclude that the perturbed system admits a globally smooth solution if the perturbation along the transversal direction is sufficiently small, while the travelling direction is allowed to be large. By choosing a gauge adapted to the plane wave solution, we deduce an equivalent Euler-Lagrangian equation for the perturbation whose quasilinear structure is reflected precisely in the induced geometry of the relativistic string. It then helps to proceed a geometrically adapted and weighted energy argument for which robust estimates suffice. Moreover, due to the non-trivial background solutions, the induced metric of the relativistic string involves linear perturbations with undetermined signs, and hence a key observation is needed to guarantee that the energies associated to the multipliers are positive up to lower order terms. %rather than quadratic perturbations as in the case of trivial background solutions.
This paper is concerned with the Cauchy problem for the relativistic membrane equation (RME) embedded in $\mathbb R^{1+(1+n)}$ with $n=2,3$. We show that the RME with a class of large (in energy norm) initial data admits a global, smooth solution. The initial data are given by the short pulse type, which is introduced by Christodoulou in his work on the formation of black holes [10]. Due to the quasilinear feature of RME, we construct two multipliers adapted to the geometry of membrane and present an efficient way for proving the global existence of smooth solution to the geometric wave equation with double null structure. We also derive the asymptotic geometry of the future null infinity and find out a nonlinear (expanding) effect at infinity.
We prove the global existence of smooth solution to the relativistic string equation in a class of data that is not small. Our solution admits the feature that the right-travelling wave can be large and the left-travelling wave is sufficiently small, and vice versa. In particular, the large-size solution exists in the whole space, instead of a null strip arising from the short pulse data. This generalizes the result of Liuli-Yang-Yu (Adv. Math. 2018) to the quasilinear setting with non-small data. In addition, in our companion paper, we are able to show the global solution here can also be seen as the non-small perturbations of the plane wave solutions.
We establish the future non-linear stability of Friedmann-Lema\^ıtre-Robertson-Walker (FLRW) solutions to the Einstein-Euler equations of the universe filled with a large class of perfect fluids (the equations of state are allowed to be certain nonlinear or linear types both). Several previous results as specific examples can be covered in the results of this article. We emphasize that the future stability of FLRW metric for polytropic fluids with positive cosmological constant has been a difficult problem and can not be directly generalized from the previous known results. Our result in this article has not only covered this difficult case for the polytropic fluids, but also unified more types of fluids in a same scheme of proofs.
We establish a global existence theory for the equation governing the evolution of a relativistic membrane in a (possibly curved) Lorentzian manifold, when the spacetime metric is a perturbation of the Minkowski metric. Relying on the Hyperboloidal Foliation Method introduced by LeFloch and Ma in 2014, we revisit a theorem established earlier by Lindblad (who treated membranes in the flat Minkowski spacetime) and we provide a simpler proof of existence, which is also valid in a curved spacetime and, most importantly, leads to the important property that the total energy of the membrane is globally bounded in time.
We analyze the global nonlinear stability of FRW (Friedmann-Robertson-Walker) spacetimes in presence of an irrotational perfect fluid. We assume that the fluid is governed by the so-called (generalized) Chaplygin equation of state relating the pressure to the mass-energy density. We express the Einstein equations in wave gauge as a systems of coupled nonlinear wave equations and by performing a suitable conformal transformation, we are able to analyze the global behavior of solutions in future timelike directions. We establish that the (3+1)-spacetime metric and the mass density and velocity vector describing the evolution of the fluid remain globally close to a reference FRW solution, under small initial data perturbations. Our analysis provides also the precise asymptotic behavior of the perturbed solutions in the future directions.
In this paper, we investigate the formation and propagation of singularities for the system for one-dimensional Chaplygin gas, which is described by a quasilinear hyperbolic system with linearly degenerate characteristic fields. The phenomena of concentration and the formation of ``$δ$-shock'' waves are identified and analyzed systematically for this system under suitably large initial data. In contrast to the Rankine-Hogoniot conditions for classical shock, the generalized Rankine-Hogoniot conditions for ``$δ$-shock'' waves are established. Finally, it is shown that the total mass and momentum related to the solution are independent of time.
In this paper we investigate the formation and propagation of singularities for the system for one-dimensional Chaplygin gas. In particular, under suitable assumptions we construct a physical solution with a new type of singularities called "Delta-like" solution for this kind of quasilinear hyperbolic system with linearly degenerate characteristics. By careful analysis, we study the behavior of the solution in a neighborhood of a blowup point. The formation of this new kind of singularities is due to the envelope of the different families of characteristics instead of the same family of characteristics in the traditional situation. This shows that the blowup phenomenon of solution for the system with linearly degenerate characteristics is quite different from the situation of shock formation for the system with genuinely nonlinear characteristics. Different initial data can lead to kinds of different Delta-like singularities: the Delta-like singularity with point-shape and Delta-like singularity with line-shape.