arXiv subjects

Sung-Jin Oh

Publications and source records attributed to Sung-Jin Oh.

At least 37 records · Page 2Linked to original sources

arXiv2021

Global nonlinear stability of large dispersive solutions to the Einstein equations

We extend the monumental result of Christodoulou-Klainerman on the global nonlinear stability of the Minkowski spacetime to the global nonlinear stability of a class of large dispersive spacetimes. More precisely, we show that any regular future causally geodesically complete, asymptotically flat solution to the Einstein-scalar field system which approaches the Minkowski spacetime sufficiently fast for large times is future globally nonlinearly stable. Combining our main theorem with results of Luk-Oh, Luk-Oh-Yang and Kilgore, we prove that a class of large data spherically symmetric dispersive solutions to the Einstein-scalar field system are globally nonlinearly stable with respect to small non-spherically symmetric perturbations. This in particular gives the first construction of an open set of large asymptotically flat initial data for which the solutions to the Einstein-scalar field system are future causally geodesically complete.

gr-qc↗

arXiv2021

Gradient blow-up for dispersive and dissipative perturbations of the Burgers equation

We consider a class of dispersive and dissipative perturbations of the inviscid Burgers equation, which includes the fractional KdV equation of order $α$, and the fractal Burgers equation of order $β$, where $α, β\in [0,1)$, and the Whitham equation. For all $α, β\in [0,1)$, we construct solutions whose gradient blows up at a point, and whose amplitude stays bounded, which therefore display a "shock-like" singularity. We moreover provide an asymptotic description of the blow-up. To our knowledge, this constitutes the first proof of gradient blow-up for the fKdV equation in the range $α\in [2/3, 1)$, as well as the first description of explicit blow-up dynamics for the fractal Burgers equation in the range $β\in [2/3, 1)$. Our construction is based on modulation theory, where the well-known smooth self-similar solutions to the inviscid Burgers equation are used as profiles. A somewhat amusing point is that the profiles that are less stable under initial data perturbations (in that the number of unstable directions is larger) are more stable under perturbations of the equation (in that higher order dispersive and/or dissipative terms are allowed) due to their slower rates of concentration. Another innovation of this article, which may be of independent interest, is the development of a streamlined weighted $L^{2}$-based approach (in lieu of the characteristic method) for establishing the sharp spatial behavior of the solution in self-similar variables, which leads to the sharp Hölder regularity of the solution up to the blow-up time.

math.AP↗

arXiv2021

The Yang--Mills heat flow and the caloric gauge

This is the first part of the four-paper sequence, which establishes the Threshold Conjecture and the Soliton Bubbling vs.~Scattering Dichotomy for the energy critical hyperbolic Yang--Mills equation in the (4 + 1)-dimensional Minkowski space-time. The primary subject of this paper, however, is another PDE, namely the energy critical Yang--Mills heat flow on the 4-dimensional Euclidean space. Our first goal is to establish sharp criteria for global existence and asymptotic convergence to a flat connection for this system in $\dot{H}^{1}$, including the Dichotomy Theorem (i.e., either the above properties hold or a harmonic Yang--Mills connection bubbles off) and the Threshold Theorem (i.e., if the initial energy is less than twice that of the ground state, then the above properties hold). Our second goal is to use the Yang--Mills heat flow in order to define the caloric gauge, which will play a major role in the analysis of the hyperbolic Yang--Mills equation in the subsequent papers.

math.AP↗

arXiv2021

The threshold conjecture for the energy critical hyperbolic Yang--Mills equation

This article represents the fourth and final part of a four-paper sequence whose aim is to prove the Threshold Conjecture as well as the more general Dichotomy Theorem for the energy critical $4+1$ dimensional hyperbolic Yang--Mills equation. The Threshold Theorem asserts that topologically trivial solutions with energy below twice the ground state energy are global and scatter. The Dichotomy Theorem applies to solutions in arbitrary topological class with large energy, and provides two exclusive alternatives: Either the solution is global and scatters, or it bubbles off a soliton in either finite time or infinite time. Using the caloric gauge developed in the first paper, the continuation/scattering criteria established in the second paper, and the large data analysis in an arbitrary topological class at optimal regularity in the third paper, here we perform a blow-up analysis which shows that the failure of global well-posedness and scattering implies either the existence of a soliton with at most the same energy bubbling off, or the existence existence of a nontrivial self-similar solution. The proof is completed by showing that the latter solutions do not exist.

math.AP↗

arXiv2021

The hyperbolic Yang--Mills equation for connections in an arbitrary topological class

This is the third part of a four-paper sequence, which establishes the Threshold Conjecture and the Soliton-Bubbling vs.~Scattering Dichotomy for the energy critical hyperbolic Yang--Mills equation in the $(4+1)$-dimensional Minkowski space-time. This paper provides basic tools for considering the dynamics of the hyperbolic Yang--Mills equation in an arbitrary topological class at an optimal regularity. We generalize the standard notion of a topological class of connections on $\mathbb R^{d}$, defined via a pullback to the one-point compactification $\mathbb S^{d} = \mathbb R^{d} \cup \{\infty\}$, to rough connections with curvature in the critical space $L^{\frac{d}{2}}(\mathbb R^{d})$. Moreover, we provide excision and extension techniques for the Yang--Mills constraint (or Gauss) equation, which allow us to efficiently localize Yang--Mills initial data sets. Combined with the results in the previous paper \cite{OTYM2}, we obtain local well-posedness of the hyperbolic Yang--Mills equation on $\mathbb R^{1+d}$ $(d \geq 4)$ in an arbitrary topological class at optimal regularity in the temporal gauge (where finite speed of propagation holds). In addition, in the energy subcritical case $d = 3$, our techniques provide an alternative proof of the classical finite energy global well-posedness theorem of Klainerman--Machedon \cite{KlMa2}, while also removing the smallness assumption in the temporal-gauge local well-posedness theorem of Tao \cite{TaoYM}. Although this paper is a part of a larger sequence, the materials presented in this paper may be of independent and general interest. For this reason, we have organized the paper so that it may be read separately from the sequence.

math.AP↗

arXiv2021

The hyperbolic Yang--Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions

This is the second part in a four-paper sequence, which establishes the Threshold Conjecture and the Soliton Bubbling vs.~Scattering Dichotomy for the hyperbolic Yang--Mills equation in the $(4+1)$-dimensional space-time. This paper provides the key gauge-dependent analysis of the hyperbolic Yang--Mills equation. We consider topologically trivial solutions in the caloric gauge, which was defined in the first paper arXiv:1709.08599 using the Yang--Mills heat flow. In this gauge, we establish a strong form of local well-posedness, where the time of existence is bounded from below by the energy concentration scale. Moreover, we show that regularity and dispersive behavior of the solution persists as long as energy dispersion is small. We also observe that fixed-time regularity (but not dispersive) properties in the caloric gauge may be transferred to the temporal gauge without any loss, proving as a consequence small data global well-posedness in the temporal gauge. The results in this paper are used in the subsequent papers arXiv:1709.08604, arXiv:1709.08606 to prove the sharp Threshold Theorem in caloric gauge in the trivial topological class, and the dichotomy theorem in arbitrary topological classes.

math.AP↗

arXiv2021

On the Cauchy problem for the Hall and electron magnetohydrodynamic equations without resistivity I: illposedness near degenerate stationary solutions

In this article, we prove various illposedness results for the Cauchy problem for the incompressible Hall- and electron-magnetohydrodynamic (MHD) equations without resistivity. These PDEs are fluid descriptions of plasmas, where the effect of collisions is neglected (no resistivity), while the motion of the electrons relative to the ions (Hall current term) is taken into account. The Hall current term endows the magnetic field equation with a quasilinear dispersive character, which is key to our mechanism for illposedness. Perhaps the most striking conclusion of this article is that the Cauchy problems for the Hall-MHD (either viscous or inviscid) and the electron-MHD equations, under one translational symmetry, are ill-posed near the trivial solution in any sufficiently high regularity Sobolev space and even in Gevrey spaces. This result holds despite obvious wellposedness of the linearized equations near the trivial solution, as well as conservation of the nonlinear energy, by which the $L^{2}$ norm (energy) of the solution stays constant in time. The core illposedness mechanism is degeneration of certain high frequency wave packet solutions to the linearization around a class of linearly degenerate stationary solutions of these equations, which are essentially dispersive equations with degenerate principal symbols. The method developed in this work is sharp and robust, in that we also prove nonlinear $H^{s}$-illposedness in the presence of fractional dissipation of any order less than 1, matching known wellposedness results. The results in this article are complemented by a companion work, where we provide geometric conditions on the initial magnetic field that ensure wellposedness(!) of the Cauchy problem. In particular, in stark contrast to the results here, it is shown in the companion work that the nonlinear Cauchy problems are well-posed near any nonzero constant magnetic field.

math.AP↗

arXiv2019

Local smoothing estimates for Schrödinger equations on hyperbolic space

We establish global-in-time frequency localized local smoothing estimates for Schrödinger equations on hyperbolic space $\mathbb{H}^d$. In the presence of symmetric first and zeroth order potentials, which are possibly time-dependent, possibly large, and have sufficiently fast polynomial decay, these estimates are proved up to a localized lower order error. Then in the time-independent case, we show that a spectral condition (namely, absence of threshold resonances) implies the full local smoothing estimates (without any error), after projecting to the continuous spectrum. In the process, as a means to localize in frequency, we develop a general Littlewood-Paley machinery on $\mathbb{H}^d$ based on the heat flow. Our results and techniques are motivated by applications to the problem of stability of solitary waves to nonlinear Schrödinger-type equations on $\mathbb{H}^{d}$. Specifically, some of the estimates established in this paper play a crucial role in the authors' proof of the nonlinear asymptotic stability of harmonic maps under the Schrödinger maps evolution on the hyperbolic plane; see [29]. As a testament of the robustness of approach, which is based on the positive commutator method and a heat flow based Littlewood-Paley theory, we also show that the main results are stable under small time-dependent perturbations, including polynomially decaying second order ones, and small lower order nonsymmetric perturbations.

math.AP↗

arXiv2019

Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane Under the Schrödinger Maps Evolution

We consider the Cauchy problem for the Schrödinger maps evolution when the domain is the hyperbolic plane. An interesting feature of this problem compared to the more widely studied case on the Euclidean plane is the existence of a rich new family of finite energy harmonic maps. These are stationary solutions, and thus play an important role in the dynamics of Schrödinger maps. The main result of this article is the asymptotic stability of (some of) such harmonic maps under the Schrödinger maps evolution. More precisely, we prove the nonlinear asymptotic stability of a finite energy equivariant harmonic map $Q$ under the Schrödinger maps evolution with respect to non-equivariant perturbations, provided $Q$ obeys a suitable linearized stability condition. This condition is known to hold for all equivariant harmonic maps with values in the hyperbolic plane and for a subset of those maps taking values in the sphere. One of the main technical ingredients in the paper is a global-in-time local smoothing and Strichartz estimate for the operator obtained by linearization around a harmonic map, proved in the companion paper [36].

math.AP↗

arXiv2019

Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data I. The interior of the black hole region

This is the first and main paper of a two-part series, in which we prove the $C^{2}$-formulation of the strong cosmic censorship conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. For this model, it is known through the works of Dafermos and Dafermos-Rodnianski that the maximal globally hyperbolic future development of any admissible two-ended asymptotically flat Cauchy initial data set possesses a non-empty Cauchy horizon, across which the spacetime is $C^{0}$-future-extendible (in particular, the $C^{0}$-formulation of the strong cosmic censorship conjecture is false). Nevertheless, the main conclusion of the present series of papers is that for a generic (in the sense of being open and dense relative to appropriate topologies) class of such data, the spacetime is future-inextendible with a Lorentzian metric of higher regularity (specifically, $C^{2}$). In this paper, we prove that the solution is $C^{2}$-future-inextendible under the condition that the scalar field obeys an $L^{2}$-averaged polynomial lower bound along each of the event horizons. This, in particular, improves upon a previous result of Dafermos, which required instead a pointwise lower bound. Key to the proof are appropriate stability and instability results in the interior of the black hole region, whose proofs are in turn based on ideas from the work of Dafermos-Luk on the stability of the Kerr Cauchy horizon (without symmetry) and from our previous paper on linear instability of Reissner-Nordström Cauchy horizon. In the second paper of the series (arXiv:1702.05716), which concerns analysis in the exterior of the black hole region, we show that the $L^2$-averaged polynomial lower bound needed for the instability result indeed holds for a generic class of admissible two-ended asymptotically flat Cauchy initial data.

gr-qc↗

arXiv2019

Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data II. The exterior of the black hole region

This is the second and last paper of a two-part series in which we prove the $C^2$-formulation of the strong cosmic censorship conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. In the first paper (arXiv:1702.05715), we showed that the maximal globally hyperbolic future development of an admissible asymptotically flat Cauchy initial data set is $C^2$-future-inextendible provided that an $L^2$-averaged (inverse) polynomial lower bound for the derivative of the scalar field holds along each horizon. In this paper, we show that this lower bound is indeed satisfied for solutions arising from a generic set of Cauchy initial data. Roughly speaking, the generic set is open with respect to a (weighted) $C^1$ topology and is dense with respect to a (weighted) $C^\infty$ topology. The proof of the theorem is based on extensions of the ideas in our previous work on the linear instability of Reissner-Nordström Cauchy horizon, as well as a new large data asymptotic stability result which gives good decay estimates for the difference of the radiation fields for small perturbations of an arbitrary solution.

gr-qc↗

arXiv2017

The Threshold Theorem for the $(4+1)$-dimensional Yang--Mills equation: an overview of the proof

This article is devoted to the energy critical hyperbolic Yang--Mills system in the $(4+1)$ dimensional Minkowski space, which is considered by the authors in a sequence of four papers. The final outcome of these papers is twofold: (i) the Threshold Theorem, which asserts that global well-posedness and scattering hold for all topologically trivial initial data with energy below twice the ground state energy, and (ii) the Dichotomy Theorem, which for larger data in arbitrary topological classes provides a choice of two outcomes, either a global, scattering solution or a soliton bubbling off. In the last case, the bubbling off phenomena can happen either (a) in finite time, triggering a finite time blow-up, or (b) in infinite time. Our goal here is to describe these results, and to provide an overview of the flow of ideas within their proofs.

math.AP↗

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

Global well-posedness of high dimensional Maxwell-Dirac for small critical data

In this paper, we prove global well-posedness of the massless Maxwell-Dirac equation in Coulomb gauge on $\mathbb{R}^{1+d}$ $(d \geq 4)$ for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of our proof are A) uncovering null structure of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru), which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon.

math.AP↗

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

Small data global existence and decay for relativistic Chern-Simons equations

We establish a general small data global existence and decay theorem for Chern-Simons theories with a general gauge group, coupled with a massive relativistic field of spin 0 or 1/2. Our result applies to a wide range of relativistic Chern-Simons theories considered in the literature, including the abelian/non-abelian self-dual Chern-Simons-Higgs equation and the Chern-Simons-Dirac equation. A key idea is to develop and employ a gauge invariant vector field method for relativistic Chern-Simons theories, which allows us to avoid the long range effect of charge.

math.AP↗

arXiv2015

The Cauchy problem for wave maps on hyperbolic space in dimensions $d \geq 4$

We establish global well-posedness and scattering for wave maps from $d$-dimensional hyperbolic space into Riemannian manifolds of bounded geometry for initial data that is small in the critical Sobolev space for $d \geq 4$. The main theorem is proved using the moving frame approach introduced by Shatah and Struwe. However, rather than imposing the Coulomb gauge we formulate the wave maps problem in Tao's caloric gauge, which is constructed using the harmonic map heat flow. In this setting the caloric gauge has the remarkable property that the main `gauged' dynamic equations reduce to a system of nonlinear scalar wave equations on $\mathbb{H}^{d}$ that are amenable to Strichartz estimates rather than tensorial wave equations (which arise in other gauges such as the Coulomb gauge) for which useful dispersive estimates are not known. This last point makes the heat flow approach crucial in the context of wave maps on curved domains.

math.AP↗

arXiv2015

Gauge choice for the Yang-Mills equations using the Yang-Mills heat flow and local well-posedness in $H^{1}$

In this work, we introduce a novel approach to the problem of gauge choice for the Yang-Mills equation on the Minkowski space $\mathbb{R}^{1+3}$, which uses the Yang-Mills heat flow in a crucial way. As this approach does not possess the drawbacks of the previous approaches, it is expected to be more robust and easily adaptable to other settings. As the first demonstration of the `structure' offered by this new approach, we will give an alternative proof of the local well-posedness of the Yang-Mills equations for initial data in $(\dot{H}^{1}_{x} \cap L^{3}_{x}) \times L^{2}_{x}$, which is a classical result of S. Klainerman and M. Machedon that had been proved using a different method (local Coulomb gauges). The new proof does not involve localization in space-time, which had been the key drawback of the previous method. Based on the results proved in this paper, a new proof of finite energy global well-posedness of the Yang-Mills equations, also using the Yang-Mills heat flow, is established in a companion article.

math.AP↗