arXiv subjects

Willie Wai Yeung Wong

Publications and source records attributed to Willie Wai Yeung Wong.

8 recordsLinked to original sources

arXiv2020

Geometric analysis of 1+1 dimensional quasilinear wave equations

We prove global well-posedness of the initial value problem for a class of variational quasilinear wave equations, in one spatial dimension, with initial data that is not-necessarily small. Key to our argument is a form of quasilinear null condition (a "nilpotent structure") that persists for our class of equations even in the large data setting. This in particular allows us to prove global well-posedness for $C^2$ initial data of moderate decrease, provided the data is sufficiently close to that which generates a simple traveling wave. We take here a geometric approach inspired by works in mathematical relativity and recent works on shock formation for fluid systems. First we recast the equations of motion in terms of a dynamical double-null coordinate system; we show that this formulation semilinearizes our system and decouples the wave variables from the null structure equations. After solving for the wave variables in the double-null coordinate system, we next analyze the null structure equations, using the wave variables as input, to show that the dynamical coordinates are $C^1$ regular and covers the entire space-time.

math.AP↗

arXiv2020

Dispersive estimates for Klein-Gordon equations via a physical space approach

Building on the hyperboloidal foliation approach of Lefloch and Ma, we extend Klainerman's physical-space approach to dispersive estimates to recover the frequency-restricted $L^1$--$L^\infty$ dispersive estimates for Klein-Gordon equations. The hyperboloidal foliation approach naturally only provide estimates within a fixed forward light-cone, and is based on an initial data norm that is not translation invariant. Both of these problems can be handled with frequency-dependent physical-space truncations. To handle the lack of scaling symmetry for the Klein-Gordon equation and complete the argument, we also need to keep track of the effectiveness of our estimates in the vanishing mass limit.

math.AP↗

arXiv2020

Global nearly-plane-symmetric solutions to the membrane equation

We prove that any simple planar travelling wave solution to the membrane equation in spatial dimension $d \geq 3$ with bounded spatial extent is globally nonlinearly stable under sufficiently small compactly-supported perturbations, where the smallness depends on the size of the support of the perturbation as well as on the initial travelling wave profile. The main novelty of the argument is the lack of higher-order peeling in our vector-field based method. In particular, the higher order energies (in fact, all energies at order $2$ or higher) are allowed to grow polynomially (but in a controlled way) in time. This is in contrast with classical global stability arguments where only the "top" order energies used in the bootstrap argument exhibit growth, and reflects the fact that the background travelling wave solution has "infinite energy" and the coefficients of the perturbation equation are not asymptotically Lorentz invariant. Nonetheless, we can prove that the perturbation converges to zero in $C^2$ by carefully analyzing the nonlinear interactions and exposing a certain "vestigial" null structure in the equations.

math.AP↗

arXiv2019

Small data global existence and decay for two dimensional wave maps

We prove the small-data global existence for the wave-map equation on $\mathbb{R}^{1,2}$ using a variant of the vector field method. The main innovations lie in the introduction of two new linear estimates. First is the control of the dispersive decay of the solution $ϕ$ itself (as opposed to its derivatives), via a logarithmic weighted Hardy inequality. This control has not been previously established using purely physical space methods in two spatial dimensions. Second is a point-wise decay estimate for a twisted derivative of $ϕ$ associated to the Morawetz $K$ multiplier, that cannot be reduced to point-wise decay estimates associated to the standard commutator vector fields. As the linear theory is largely similar between dimensions, and in view of the novelty of the second innovation even in higher dimensions, we include a discussion of the method for $\mathbb{R}^{1,d}$ in general. Both linear estimates are used crucially in our study: the control of the wave-map equation in the small data regime necessarily requires understanding the dispersive behavior of the bare solution $ϕ$ itself, by virtue of the equation it satisfies. The point-wise decay for the twisted derivative allows us to avoid certain top-order logarithmic energy growths; this is indispensable for extending our argument from the case of compactly supported initial data (to which our methods are most naturally adapted) to initial data that are strongly localized but not necessarily of compact support, via an iterative construction.

math.AP↗

arXiv2019

Global versions of Gagliardo-Nirenberg-Sobolev inequality and applications to wave and Klein-Gordon equations

We prove global, or space-time weighted, versions of the Gagliardo-Nirenberg interpolation inequality, with $L^p$ ($p < \infty$) endpoint, adapted to a hyperboloidal foliation. The corresponding versions with $L^\infty$ endpoint was first introduced by Klainerman and is the basis of the classical vector field method, which is now one of the standard techniques for studying long-time behavior of nonlinear evolution equations. We were motivated in our pursuit by settings where the vector field method is applied to an energy hierarchy with growing higher order energies. In these settings the use of the $L^p$ endpoint versions of Sobolev inequalities can allow one to gain essentially one derivative in the estimates, which would then give a corresponding gain of decay rate. The paper closes with the analysis of one such model problem, where our new estimates provide an improvement.

math.AP↗

arXiv2017

A commuting-vector-field approach to some dispersive estimates

We prove the pointwise decay of solutions to three linear equations: (i) the transport equation in phase space generalizing the classical Vlasov equation, (ii) the linear Schrodinger equation, (iii) the Airy (linear KdV) equation. The usual proofs use explicit representation formulae, and either obtain $L^1$---$L^\infty$ decay through directly estimating the fundamental solution in physical space, or by studying oscillatory integrals coming from the representation in Fourier space. Our proof instead combines "vector field" commutators that capture the inherent symmetries of the relevant equations with conservation laws for mass and energy to get space-time weighted energy estimates. Combined with a simple version of Sobolev's inequality this gives pointwise decay as desired. In the case of the Vlasov and Schrodinger equations we can recover sharp pointwise decay; in the Schrodinger case we also show how to obtain local energy decay as well as Strichartz-type estimates. For the Airy equation we obtain a local energy decay that is almost sharp from the scaling point of view, but nonetheless misses the classical estimates by a gap. This work is inspired by the work of Klainerman on $L^2$---$L^\infty$ decay of wave equations, as well as the recent work of Fajman, Joudioux, and Smulevici on decay of mass distributions for the relativistic Vlasov equation.

math.AP↗

arXiv2017

Warped Product Space-times

Many classical results in relativity theory concerning spherically symmetric space-times have easy generalizations to warped product space-times, with a two-dimensional Lorentzian base and arbitrary dimensional Riemannian fibers. We first give a systematic presentation of the main geometric constructions, with emphasis on the Kodama vector field and the Hawking energy; the construction is signature independent. This leads to proofs of general Birkhoff-type theorems for warped product manifolds; our theorems in particular apply to situations where the warped product manifold is not necessarily Einstein, and thus can be applied to solutions with matter content in general relativity. Next we specialize to the Lorentzian case and study the propagation of null expansions under the assumption of the dominant energy condition. We prove several non-existence results relating to the Yamabe class of the fibers, in the spirit of the black-hole topology theorem of Hawking-Galloway-Schoen. Finally we discuss the effect of the warped product ansatz on matter models. In particular we construct several cosmological solutions to the Einstein-Euler equations whose spatial geometry is generally not isotropic.

gr-qc↗

arXiv2016

Global Existence for the Minimal Surface Equation on $\mathbb{R}^{1,1}$

In a 2004 paper, Lindblad demonstrated that the minimal surface equation on $\mathbb{R}l^{1,1}$ describing graphical time-like minimal surfaces embedded in $\mathbb{R}^{1,2}$ enjoy small data global existence for compactly supported initial data, using Christodoulou's conformal method. Here we give a different, geometric proof of the same fact, which exposes more clearly the inherent null structure of the equations, and which allows us to also close the argument using relatively few derivatives and mild decay assumptions at infinity.

math.AP↗