arXiv subjects

Shiwu Yang

Publications and source records attributed to Shiwu Yang.

27 records · Page 2Linked to original sources

arXiv2017

Dynamical black holes with prescribed masses in spherical symmetry

We review our recent work on a construction of spherically symmetric global solutions to the Einstein--scalar field system with large bounded variation norms and large Bondi masses. We show that similar ideas, together with Christodoulou's short pulse method, allow us to prove the following result: Given $M_i \geq M_f>0$ and $ε>0$, there exists a spherically symmetric (black hole) solution to the Einstein--scalar field system such that up to an error of size $ε$, the initial Bondi mass is $M_i$ and the final Bondi mass is $M_f$. Moreover, if one assumes a continuity property of the final Bondi mass (which in principle follows from known techniques in the literature), then for $M_i>M_f>0$, the above result holds without an $ε$-error.

gr-qc↗

arXiv2016

Solutions to the Einstein-scalar-field system in spherical symmetry with large bounded variation norms

It is well-known that small, regular, spherically symmetric characteristic initial data to the Einstein-scalar-field system which are decaying towards (future null) infinity give rise to solutions which are foward-in-time global (in the sense of future causal geodesic completeness). We construct a class of spherically symmetric solutions which are global but the initial norms are consistent with initial data not decaying towards infinity. This gives the following consequences: (1) We prove that there exist foward-in-time global solutions with arbitrarily large (and in fact infinite) initial bounded variation (BV) norms and initial Bondi masses. (2) While general solutions with non-decaying data do not approach Minkowski spacetime, we show using the results of Luk--Oh that if a sufficiently strong asymptotic flatness condition is imposed on the initial data, then the solutions we construct (with large BV norms) approach Minkowski spacetime with a sharp inverse polynomial rate. (3) Our construction can be easily extended so that data are posed at past null infinity and we obtain solutions with large BV norms which are causally geodesically complete both to the past and to the future. Finally, we discuss applications of our method to construct global solutions for other nonlinear wave equations with infinite critical norms.

gr-qc↗

arXiv2015

On global behavior of solutions of the Maxwell-Klein-Gordon equations

It is known that the Maxwell-Klein-Gordon equations in $\mathbb{R}^{3+1}$ admit global solutions with finite energy data. In this paper, we present a new approach to study the asymptotic behavior of these global solutions. We show the quantitative energy flux decay of the solutions with data merely bounded in some weighted energy space. We also establish an integrated local energy decay and a hierarchy of $r$-weighted energy decay. The results in particular hold in the presence of large total charge. This is the first result to give a complete and precise description of the global behavior of large nonlinear charged scalar fields.

math.AP↗

arXiv2015

Decay of solutions of Maxwell-Klein-Gordon equations with large Maxwell field

It has been shown in the author's companion paper that solutions of Maxwell-Klein-Gordon equations in $\mathbb{R}^{3+1}$ possess some form of global strong decay properties with data bounded in some weighted energy space. In this paper, we prove pointwise decay estimates for the solutions for the case when the initial data are merely small on the scalar field but can be arbitrarily large on the Maxwell field. This extends the previous result of Lindblad-Sterbenz \cite{LindbladMKG}, in which smallness was assumed both for the scalar field and the Maxwell field.

math.AP↗

arXiv2015

On the geodesic hypothesis in general relativity

In this paper, we give a rigorous derivation of Einstein's geodesic hypothesis in general relativity. We use scaling stable solitons for nonlinear wave equations to approximate the test particle. Given a vacuum spacetime $([0, T]\times\mathbb{R}^3, h)$, we consider the scalar field coupled Einstein equations. For all sufficiently small $ε$ and $δ\leq ε^q$, $q>1$, where $δ$, $ε$ are the amplitude and size of the particle, we show the existence of solution $([0, T]\times\mathbb{R}^3, g, ϕ^ε)$ to the coupled Einstein equations with the property that the energy of the particle $ϕ^ε$ is concentrated along a timelike geodesic. Moreover, the gravitational field produced by $ϕ^ε$ is negligibly small in $C^1$, that is, the spacetime metric $g$ is $C^1$ close to $h$. These results generalize those obtained by D. Stuart.

math.AP↗

arXiv2013

Global solutions of nonlinear wave equations with large energy

In this paper, we give a criterion on the Cauchy data for the semilinear wave equations satisfying the null condition in $\mathbb{R}^+\times\mathbb{R}^{3}$ such that the energy of the data can be arbitrarily large while the solution is still globally in time in the future.

math.AP↗

arXiv2013

On the quasilinear wave equations in time dependent inhomogeneous media

We consider the problem of small data global existence for quasilinear wave equations with null condition on a class of Lorentzian manifolds $(\mathbb{R}^{3+1}, g)$ with \textbf{time dependent} inhomogeneous metric. We show that sufficiently small data give rise to a unique global solution for metric which is merely $C^1$ close to the Minkowski metric inside some large cylinder $\{\left.(t, x)\right||x|\leq R\}$ and approaches the Minkowski metric weakly as $|x|\rightarrow \infty$. Based on this result, we give weak but sufficient conditions on a given large solution of quasilinear wave equations such that the solution is globally stable under perturbations of initial data.

math.AP↗

arXiv2012

Global stability of solutions to nonlinear wave equations

We consider the problem of global stability of solutions to a class of semilinear wave equations with null condition in Minkowski space. We give sufficient conditions on the given solution which guarantees stability. Our stability result can be reduced to a small data global existence result for a class of semilinear wave equations with linear terms $B^{μν}\patial_μΦ(t, x)\patial_νϕ$, $L^μ(t,x)\patial_μϕ$ and quadratic terms $h^{μν}(t, x)\patial_μϕ\patial_νϕ$ where the functions $Φ(t, x)$, $L^μ(t, x)$, $h^{μν}(t, x)$ decay rather weakly and the constants $B^{μν}$ satisfy the null condition. We show the small data global existence result by using the new approach developed by M. Dafermos and I. Rodnianski. In particular, we prove the global stability result under weaker assumptions than those imposed by S. Alinhac.

math.AP↗

arXiv2012

Global solutions of nonlinear wave equations in time dependent inhomogeneous media

We consider the problem of small data global existence for a class of semilinear wave equations with null condition on a Lorentzian background $(\mathbb{R}^{3+1}, g)$ with a \textbf{time dependent metric $g$} coinciding with Minkowski metric outside the cylinder $\{(t, x)| |x|\leq R\}$. We show that the small data global existence result can be reduced to two integrated local energy estimates and demonstrate these estimates in the particular case when $g$ is merely $C^1$ close to the Minkowski metric. One of the novel aspects of this work is that it applies to equations on backgrounds which do not settle to any particular stationary metric.

math.AP↗