arXiv subjects

Sung-Jin Oh

Publications and source records attributed to Sung-Jin Oh.

53 records · Page 3Linked to original sources

arXiv2015

Finite energy global well-posedness of the Yang-Mills equations on $\mathbb{R}^{1+3}$: An approach using the Yang-Mills heat flow

In this work, along with the companion work Oh (2012), we propose a novel approach to the problem of gauge choice for the \emph{Yang-Mills equations} on the Minkowski space $\mathbb{R}^{1+3}$. A crucial ingredient is the associated \emph{Yang-Mills heat flow}. As this approach does not possess the drawbacks of the previous approaches (as in Klainerman-Machedon (1995) and Tao (2003)), it is expected to be more robust and easily adaptable to other settings. Building on the results proved in the companion article Oh (2012), we prove, as one of the first applications of our approach, finite energy global well-posedness of the Yang-Mills equations on $\mathbb{R}^{1+3}$. This is a classical result first proved by S. Klainerman and M. Machedon (1995) using local Coulomb gauges. As opposed to their method, the present approach avoids the use of Uhlenbeck's lemma (1982), and hence does \emph{not} involve localization in space-time.

math.AP↗

arXiv2015

Quantitative decay rates for dispersive solutions to the Einstein-scalar field system in spherical symmetry

In this paper, we study the future causally geodesically complete solutions of the spherically symmetric Einstein-scalar field system. Under the a priori assumption that the scalar field $ϕ$ scatters locally in the scale-invariant bounded-variation (BV) norm, we prove that $ϕ$ and its derivatives decay polynomially. Moreover, we show that the decay rates are sharp. In particular, we obtain sharp quantitative decay for the class of global solutions with small BV norms constructed by Christodoulou. As a consequence of our results, for every future causally geodesically complete solution with sufficiently regular initial data, we show the dichotomy that either the sharp power law tail holds or that the spacetime blows up at infinity in the sense that some scale invariant spacetime norms blow up.

gr-qc↗

arXiv2015

Equivariant Wave Maps on the Hyperbolic Plane with Large Energy

In this paper we continue the analysis of equivariant wave maps from 2-dimensional hyperbolic space into surfaces of revolution that was initiated in [13, 14]. When the target is the hyperbolic plane we proved in [13] the existence and asymptotic stability of a 1-parameter family of finite energy harmonic maps indexed by how far each map wraps around the target. Here we conjecture that each of these harmonic maps is globally asymptotically stable, meaning that the evolution of any arbitrarily large finite energy perturbation of a harmonic map asymptotically resolves into the harmonic map itself plus free radiation. Since such initial data exhaust the energy space, this is the soliton resolution conjecture for this equation. The main result is a verification of this conjecture for a nonperturbative subset of the harmonic maps

math.AP↗

arXiv2015

On Nonperiodic Euler Flows with Hölder Regularity

In [Isett,13], the first author proposed a strengthening of Onsager's conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding $1/3$. This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space $L_t^\infty B_{3,\infty}^{1/3}$ due to low regularity of the energy profile. In this paper, we establish two theorems that may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than $1/5$. Our first result shows that any non-negative function with compact support and Hölder regularity $1/2$ can be prescribed as the energy profile of an Euler flow in the class $C^{1/5-ε}_{t,x}$. The exponent $1/2$ is sharp in view of a regularity result of [Isett,13]. The proof employs an improved greedy algorithm scheme that builds upon that in [Buckmaster-De Lellis-Székelyhidi, 13]. Our second result shows that any given smooth Euler flow can be perturbed in $C^{1/5-ε}_{t,x}$ on any pre-compact subset of ${\mathbb R}\times {\mathbb R}^3$ to violate energy conservation. In particular, there exist nonzero $C^{1/5-ε}_{t,x}$ solutions to Euler with compact space-time support, generalizing previous work of the first author [Isett,12] to the nonperiodic setting.

math.AP↗

arXiv2015

Local well-posedness of the (4+1)-dimensional Maxwell-Klein-Gordon equation at energy regularity

This paper is the first part of a trilogy dedicated to a proof of global well-posedness and scattering of the (4+1)-dimensional mass-less Maxwell-Klein-Gordon equation (MKG) for any finite energy initial data. The main result of the present paper is a large energy local well-posedness theorem for MKG in the global Coulomb gauge, where the lifespan is bounded from below by the energy concentration scale of the data. Hence the proof of global well-posedness is reduced to establishing non-concentration of energy. To deal with non-local features of MKG we develop initial data excision and gluing techniques at critical regularity, which might be of independent interest.

math.AP↗

arXiv2015

Energy dispersed solutions for the (4+1)-dimensional Maxwell-Klein-Gordon equation

This article is devoted to the mass-less energy critical Maxwell-Klein-Gordon system in 4+1 dimensions. In earlier work of the second author, joint with Krieger and Sterbenz, we have proved that this problem has global well-posedness and scattering in the Coulomb gauge for small initial data. This article is the second of a sequence of three papers of the authors, whose goal is to show that the same result holds for data with arbitrarily large energy. Our aim here is to show that large data solutions persist for as long as one has small energy dispersion; hence failure of global well-posedness must be accompanied with a non-trivial energy dispersion.

math.AP↗

arXiv2015

Global well-posedness and scattering of the (4+1)-dimensional Maxwell-Klein-Gordon equation

This article constitutes the final and main part of a three-paper sequence, whose goal is to prove global well-posedness and scattering of the energy critical Maxwell-Klein-Gordon equation (MKG) on $\mathbb{R}^{1+4}$ for arbitrary finite energy initial data. Using the successively stronger continuation/scattering criteria established in the previous two papers, we carry out a blow-up analysis and deduce that the failure of global well-posedness and scattering implies the existence of a nontrivial stationary or self-similar solution to MKG. Then, by establishing that such solutions do not exist, we complete the proof.

math.AP↗

arXiv2015

A refined threshold theorem for (1+2)-dimensional wave maps into surfaces

The recently established threshold theorem for energy critical wave maps states that wave maps with energy less than that of the ground state (i.e., a minimal energy nontrivial harmonic map) are globally regular and scatter on (1+2)-dimensional Minkowski space. In this note we give a refinement of this theorem when the target is a closed orientable surface by taking into account an additional invariant of the problem, namely the topological degree. We show that the sharp energy threshold for global regularity and scattering is in fact twice the energy of the ground state for wave maps with degree zero, whereas wave maps with nonzero degree necessarily have at least the energy of the ground state. We also give a discussion on the formulation of a refined threshold conjecture for the energy critical $SU(2)$ Yang-Mills equation on (1+4)-dimensional Minkowski space.

math.AP↗

arXiv2015

Gap Eigenvalues and Asymptotic Dynamics of Geometric Wave Equations on Hyperbolic Space

In this paper we study $k$-equivariant wave maps from the hyperbolic plane into the $2$-sphere as well as the energy critical equivariant $SU(2)$ Yang-Mills problem on $4$-dimensional hyperbolic space. The latter problem bears many similarities to a $2$-equivariant wave map into a surface of revolution. As in the case of $1$-equivariant wave maps considered in~\cite{LOS1}, both problems admit a family of stationary solutions indexed by a parameter that determines how far the image of the map wraps around the target manifold. Here we show that if the image of a stationary solution is contained in a geodesically convex subset of the target, then it is asymptotically stable in the energy space. However, for a stationary solution that covers a large enough portion of the target, we prove that the Schrödinger operator obtained by linearizing about such a harmonic map admits a simple positive eigenvalue in the spectral gap. As there is no a priori nonlinear obstruction to asymptotic stability, this gives evidence for the existence of metastable states (i.e., solutions with anomalously slow decay rates) in these simple geometric models.

math.AP↗

arXiv2015

Proof of linear instability of the Reissner-Nordström Cauchy horizon under scalar perturbations

It has long been suggested that solutions to linear scalar wave equation $$\Box_gϕ=0$$ on a fixed subextremal Reissner-Nordström spacetime with non-vanishing charge are generically singular at the Cauchy horizon. We prove that generic smooth and compactly supported initial data on a Cauchy hypersurface indeed give rise to solutions with infinite nondegenerate energy near the Cauchy horizon in the interior of the black hole. In particular, the solution generically does not belong to $W^{1,2}_{loc}$. This instability is related to the celebrated blue shift effect in the interior of the black hole. The problem is motivated by the strong cosmic censorship conjecture and it is expected that for the full nonlinear Einstein-Maxwell system, this instability leads to a singular Cauchy horizon for generic small perturbations of Reissner-Nordström spacetime. Moreover, in addition to the instability result, we also show as a consequence of the proof that Price's law decay is generically sharp along the event horizon.

gr-qc↗

arXiv2014

Stability of stationary equivariant wave maps from the hyperbolic plane

In this paper we initiate the study of equivariant wave maps from 2d hyperbolic space into rotationally symmetric surfaces. This problem exhibits markedly different phenomena than its Euclidean counterpart due to the exponential volume growth of concentric geodesic spheres on the domain. In particular, when the target is the 2-sphere, we find a family of equivariant harmonic maps indexed by a parameter that measures how far the image of each harmonic map wraps around the sphere. These maps have energies taking all values between zero and the energy of the unique co-rotational Euclidean harmonic map, Q, from the Euclidean plane to the 2-sphere, given by stereographic projection. We prove that the harmonic maps are asymptotically stable for values of the parameter smaller than a threshold that is large enough to allow for maps that wrap more than halfway around the sphere. Indeed, we prove Strichartz estimates for the operator obtained by linearizing around such a harmonic map. However, for harmonic maps with energies approaching the Euclidean energy of Q, asymptotic stability via a perturbative argument based on Strichartz estimates is precluded by the existence of gap eigenvalues in the spectrum of the linearized operator. When the target is 2d hyperbolic space, we find a continuous family of asymptotically stable equivariant harmonic maps with arbitrarily small and arbitrarily large energies. This stands in sharp contrast to the corresponding problem on Euclidean space, where all finite energy solutions scatter to zero as time tends to infinity.

math.AP↗

arXiv2014

Profile decompositions for wave equations on hyperbolic space with applications

The goal for this paper is twofold. Our first main objective is to develop Bahouri-Gerard type profile decompositions for waves on hyperbolic space. Recently, such profile decompositions have proved to be a versatile tool in the study of the asymptotic dynamics of solutions to nonlinear wave equations with large energy. With an eye towards further applications, we develop this theory in a fairly general framework, which includes the case of waves on hyperbolic space perturbed by a time-independent potential. Our second objective is to use the profile decomposition to address a specific nonlinear problem, namely the question of global well-posedness and scattering for the defocusing, energy critical, semi-linear wave equation on three-dimensional hyperbolic space, possibly perturbed by a repulsive time-independent potential. Using the concentration compactness/rigidity method introduced by Kenig and Merle, we prove that all finite energy initial data lead to a global evolution that scatters to linear waves in infinite time. This proof will serve as a blueprint for the arguments in a forthcoming work, where we study the asymptotic behavior of large energy equivariant wave maps on the hyperbolic plane.

math.AP↗

arXiv2014

A heat flow approach to Onsager's conjecture for the Euler equations on manifolds

We give a simple proof of Onsager's conjecture concerning energy conservation for weak solutions to the Euler equations on any compact Riemannian manifold, extending the results of Constantin-E-Titi and Cheskidov-Constantin-Friedlander-Shvydkoy in the flat case. When restricted to $\mathbb{T}^{d}$ or $\mathbb{R}^{d}$, our approach yields an alternative proof of the sharp result of the latter authors. Our method builds on a systematic use of a smoothing operator defined via a geometric heat flow, which was considered by Milgram-Rosenbloom as a means to establish the Hodge theorem. In particular, we present a simple and geometric way to prove the key nonlinear commutator estimate, whose proof previously relied on a delicate use of convolutions.

math.AP↗

arXiv2013

Decay and Scattering for the Chern-Simons-Schrödinger Equations

We consider the Chern-Simons-Schrödinger model in 1+2 dimensions, and prove scattering for small solutions of the Cauchy problem in the Coulomb gauge. This model is a gauge covariant Schrödinger equation, with a potential decaying like $r^{-1}$ at infinity. To overcome the difficulties due to this long range decay we start by performing $L^2$-based estimates covariantly. This gives favorable commutation identities so that only curvature terms, which decay faster than $r^{-1}$, appear in our weighted energy estimates. We then select the Coulomb gauge to reveal a genuinely cubic null structure, which allows us to show sharp decay by Fourier methods.

math.AP↗

arXiv2013

Finite energy global well-posedness of the Chern-Simons-Higgs equations in the Coulomb gauge

In a recent paper, Selberg-Tesfahun proved that the abelian Chern-Simons-Higgs system (CSH) is globally well-posed for finite energy initial data under the Lorenz gauge condition. It has been suspected by Huh, however, that such a result should hold in the Coulomb gauge as well. In this note, we give an affirmative answer to this question by first establishing low regularity local well-posededness of (CSH) in the Coulomb gauge for initial data set $(f, g) \in H^γ_{x} \times H^{γ-1}_{x}$ for any $γ> 3/4$. Then by conservation of energy, global well-posedness for (CSH) in the energy space $(f, g) \in H^{1}_{x} \times L^{2}_{x}$ follows rather immediately.

math.AP↗

arXiv2013

One approach for determining susceptibilities and order parameters in multi-band Hubbard model

We present an approach for determining susceptibilities and order parameters in multi-band Hubbard model within functional renormalization group method, and apply it to study various instabilities of the FeAs-based high temperature superconductor. First, we derive the formulae of susceptibilities and order parameters of superconducting parings, spin density waves and charge density waves with diverse wave vectors which could occur in multi-band Hubbard model. Second, we apply it to the FeAs-based high temperature superconductor and find an electronic-driven superconducting pairing instability within a five band model with pure repulsive interactions. Our study shows that for doping of our concern, there is competition between antiferromagnetic and superconducting instabilities. In addition, we show that for doping of 0.1, the susceptibility of extended s-wave pairing is dominant over others, while a staggered superconducting pairing with has the second largest value of susceptibility.

cond-mat.str-el↗

arXiv2012

Low regularity solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs equations in the Lorenz gauge

In this paper, we address the problem of local well-posedness of the Chern-Simons-Dirac (CSD) and the Chern-Simons-Higgs (CSH) equations in the Lorenz gauge for low regularity initial data. One of our main contributions is the uncovering of a null structure of (CSD). Combined with the standard machinery of $X^{s,b}$ spaces, we obtain local well-posedness of (CSD) for initial data $a_μ, ψ\in H^{1/4+ε}_{x}$. Moreover, it is observed that the same techniques applied to (CSH) lead to a quick proof of local well-posedness for initial data $a_μ \in H^{1/4+ε}_{x}$, $(ϕ, \partial_{t} ϕ) \in H^{3/4+ε}_{x} \times H^{-1/4+ε}_{x}$, which improves the previous result of Selberg-Tesfahun (2012).

math.AP↗