arXiv subjects

Sung-Jin Oh

Publications and source records attributed to Sung-Jin Oh.

At least 19 recordsLinked to original sources

arXiv2026

The good commutator approach to global asymptotics for the Schrödinger equation with variable coefficients

We present a robust physical-space approach to establish time decay and global asymptotics of solutions to variable-coefficient Schrödinger equations in $(3+1)$-dimensions. As an immediate nonlinear application, we obtain new small data global existence and asymptotics results for quasilinear Schrödinger equations with cubic, Hamiltonian nonlinearity, variable coefficients in their linear part, and possibly outside obstacles, even in the presence of trapped bicharacteristics (provided that they are suitably unstable). Our approach relies on three primary ingredients. First is the concept of a good commutator, which extends Klainerman's classical commuting vector field method, and develops upon earlier approaches of Cuccagna--Georgiev--Visciglia and Rodnianski--Tao in the variable-coefficient case, and of Ifrim--Tataru, Ifrim--Koch--Tataru in the nonlinear case. Second, we apply Ifrim and Tataru's testing-by-wave-packets method, which is key for obtaining global asymptotics and handling sharp decay assumptions on the coefficients. Finally, we introduce a systematic technique for analyzing the control provided by the good commutator, termed two-scale elliptic analysis. Together, these techniques significantly broaden the applicability of physical-space methods to a wider class of variable-coefficient nonlinear problems.

math.AP↗

arXiv2026

Stability of the Minkowski spacetime in Newman-Unti gauge

We prove small-data global stability of the Minkowski solution to Einstein's equations in a centre-normalised outgoing null-geodesic gauge. Our scheme involves first using the $r^p$-estimates of Dafermos-Rodnianski to control certain components of the Weyl tensor which satisfy a decoupled tensorial wave equation. Having established this control, all remaining geometric quantities are controlled by transport equations, taking initial conditions at a regular central axis. This method establishes global stability for initial data which decay only weakly to flat space and can establish additional asymptotic control when the data are assumed to have more structure.

gr-qc↗

arXiv2026

Blow-up dynamics for radial self-dual Chern-Simons-Schrödinger equation with prescribed asymptotic profile

We construct finite energy blow-up solutions for the radial self-dual Chern-Simons-Schrödinger equation with a continuum of blow-up rates. Our result stands in stark contrast to the rigidity of blow-up of $H^{3}$ solutions proved by the first author for equivariant index $m \geq 1$, where the soliton-radiation interaction is too weak to admit the present blow-up scenarios. It is optimal (up to an endpoint) in terms of the range of blow-up rates and the regularity of the asymptotic profiles in view of the authors' previous proof of $H^{1}$ soliton resolution for the self-dual Chern-Simons-Schrödinger equation in any equivariance class. Our approach is a backward construction combined with modulation analysis, starting from prescribed asymptotic profiles and deriving the corresponding blow-up rates from their strong interaction with the soliton. In particular, our work may be seen as an adaptation of the method of Jendrej-Lawrie-Rodriguez (developed for energy critical equivariant wave maps) to the Schrödinger case. However, the Schrödinger nature of the equation (in particular, the lack of finite speed of propagation) and the optimal range (up to the $H^{1}$-endpoint) of our blow-up construction give rise to new challenges. Notably, the construction of (approximate) radiation from the prescribed asymptotic profile is one of our key novelties and might be of independent interest.

math.AP↗

arXiv2025

Late-time tail for a scalar quasilinear wave equation satisfying the weak null condition

We consider a class of scalar quasilinear wave equations in three spatial dimensions satisfying the weak null condition. For solutions arising from small, localized, smooth data, we give an asymptotic formula describing the global asymptotics towards the future. We prove that the late-time asymptotics is given by a continuous superposition of decay rates, in stark contrast to equations satisfying a null condition. The asymptotic formula we obtain is given in terms of a solution to the linear wave equation. Combining this with analysis on the linear wave equation, we strengthen some rigidity results of the third author, showing in particular that any solution with a faster time decay than expected away from the wave zone must vanish identically.

math.AP↗

arXiv2025

Integral formulas for under/overdetermined differential operators via recovery on curves and the finite-dimensional cokernel condition I: General theory

We introduce a new versatile method for constructing solution operators (i.e., right-inverses up to a finite rank operator) for a wide class of underdetermined PDEs $P u = f$, which are regularizing of optimal order and, more interestingly, whose integral kernels have certain prescribed support properties. By duality, we simultaneously obtain integral representation formulas (i.e., left-inverses up to a finite rank operator) for overdetermined PDEs $P^{\ast} v = g$ with analogous properties, which lead to Poincaré- or Korn-type inequalities. Our method applies to operators such as the divergence, linearized scalar curvature, and linearized Einstein constraint operators (which are underdetermined), as well as the gradient, Hessian, trace-free part of the Hessian, Killing, and conformal Killing operators (which are overdetermined). The starting point for our construction is a condition - dubbed the recovery on curves condition (RC) - that leads to Green's functions for $P$ supported on prescribed curves. Then the desired integral solution operators (and, by duality, integral representation formulas) are obtained by taking smooth averages over a suitable family of curves. This procedure generalizes the previous constructions of Bogovskii, Oh-Tataru, and Reshetnyak. We furthermore identify a simple algebraic sufficient condition for (RC), namely, that the principal symbol $p(x, ξ)$ of $P$ is full-rank for all non-zero complex vectors $ξ$ (as opposed to real, as in ellipticity). When the principal symbol has constant coefficients, this is equivalent to (RC) and also to the condition that the formal cokernel of $P$ (without any boundary conditions) is finite-dimensional; for this reason, we call it the finite-dimensional cokernel condition (FC). We give a short proof that all operators above satisfy (FC), and thus (RC). Various applications will be considered in subsequent papers.

math.AP↗

arXiv2025

Illposedness via degenerate dispersion for generalized surface quasi-geostrophic equations with singular velocities

We prove strong nonlinear illposedness results for the generalized SQG equation $$\partial_t θ+ \nabla^\perp Γ[θ] \cdot \nabla θ= 0 $$ in any sufficiently regular Sobolev spaces, when $Γ$ is a singular in the sense that its symbol satisfies $|Γ(ξ)|\to\infty$ as $|ξ|\to\infty$ with some mild regularity assumptions. The key mechanism is degenerate dispersion, i.e., the rapid growth of frequencies of solutions around certain shear states, and the robustness of our method allows one to extend linear and nonlinear illposedness to fractionally dissipative systems, as long as the order of dissipation is lower than that of $Γ$. Our illposedness results are completely sharp in view of various existing wellposedness statements as well as those from our companion paper. Key to our proofs is a novel construction of degenerating wave packets for the class of linear equations $$\partial_t ϕ+ ip(t,X,D)ϕ= 0$$ where $p(t,X,D)$ is a pseudo-differential operator which is self-adjoint in $L^2$, degenerate, and dispersive. Degenerating wave packets are approximate solutions to the above linear equation with spatial and frequency support localized at $(X(t),Ξ(t))$, which are solutions to the bicharacteristic ODE system associated with $p(t,x,ξ)$. These wave packets explicitly show degeneration as $X(t)$ approaches a point where $p$ vanishes, which in particular allows us to prove illposedness in topologies finer than $L^2$. While the equation for the wave packet can be formally obtained from a Taylor expansion of the symbol near $ξ=Ξ(t)$, the difficult part is to rigorously control the error in sufficiently long timescales, which is obtained by sharp estimates for not only degenerating wave packets but also for oscillatory integrals which naturally appear in the error estimate.

math.AP↗

arXiv2025

Stability of the Catenoid for the Hyperbolic Vanishing Mean Curvature Equation Outside Symmetry

We study the problem of stability of the catenoid, which is an asymptotically flat rotationally symmetric minimal surface in Euclidean space, viewed as a stationary solution to the hyperbolic vanishing mean curvature equation in Minkowski space. The latter is a quasilinear wave equation that constitutes the hyperbolic counterpart of the minimal surface equation in Euclidean space. Our main result is the nonlinear asymptotic stability, modulo suitable translation and boost (i.e., modulation), of the $n$-dimensional catenoid with respect to a codimension one set of initial data perturbations without any symmetry assumptions, for $n \geq 5$. The modulation and the codimension one restriction on the data are necessary and optimal in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid. In a broader context, this paper fits in the long tradition of studies of soliton stability problems. From this viewpoint, our aim here is to tackle some new issues that arise due to the quasilinear nature of the underlying hyperbolic equation. Ideas introduced in this paper include a new profile construction and modulation analysis to track the evolution of the translation and boost parameters of the stationary solution, a new scheme for proving integrated local energy decay for the perturbation in the quasilinear and modulation-theoretic context, and an adaptation of the vectorfield method in the presence of dynamic translations and boosts of the stationary solution.

math.AP↗

arXiv2025

Well-posedness for Ohkitani model and long-time existence for surface quasi-geostrophic equations

We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani, $$\partial_t θ- \nabla^\perp \log(10+(-Δ)^{\frac12})θ\cdot \nabla θ= 0 ,$$ and establish local existence and uniqueness of smooth solutions in the scale of Sobolev spaces with exponent decreasing with time. Such a decrease of the Sobolev exponent is necessary, as we have shown in the companion paper that the problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence of the Sobolev exponent can be removed when there is a dissipation term strictly stronger than log. These results improve wellposedness statements by Chae, Constantin, Córdoba, Gancedo, and Wu in \cite{CCCGW}. This well-posedness result can be applied to describe the long-time dynamics of the $δ$-SQG equations, defined by $$\partial_t θ+ \nabla^\perp (10+(-Δ)^{\frac12})^{-δ}θ\cdot \nabla θ= 0,$$ for all sufficiently small $δ>0$ depending on the size of the initial data. For the same range of $δ$, we establish global well-posedness of smooth solutions to the logarithmically dissipative counterpart: $$\partial_t θ+ \nabla^\perp (10+(-Δ)^{\frac12})^{-δ}θ\cdot \nabla θ+ \log(10+(-Δ)^{\frac12})θ= 0.$$

math.AP↗

arXiv2024

Beale--Kato--Majda-type continuation criteria for Hall- and electron-magnetohydrodynamics

We show that regular solutions to electron-MHD with resistivity can be continued as long as the time integral of the supremum of the current gradient remains finite. This dimensionless continuation criterion is analogous to the celebrated result of Beale--Kato--Majda for the incompressible Euler and Navier--Stokes equations. A similar continuation criterion, formulated in terms of the time integral of the supremum of the vorticity, velocity gradient and current gradient, is established for the Hall-MHD with resistivity as well.

math.AP↗

arXiv2024

Stability of the 3-dimensional catenoid for the hyperbolic vanishing mean curvature equation

We prove that the $3$-dimensional catenoid is asymptotically stable as a solution to the hyperbolic vanishing mean curvature equation in Minkowski space, modulo suitable translation and boost (i.e., modulation) and with respect to a codimension one set of initial data perturbations. The modulation and the codimension one restriction on the initial data are necessary (and optimal) in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid. The $3$-dimensional problem is more challenging than the higher (specifically, $5$ and higher) dimensional case addressed in the previous work of the authors with J.~Lührmann, due to slower temporal decay of waves and slower spatial decay of the catenoid. To overcome these issues, we introduce several innovations, such as a proof of Morawetz- (or local-energy-decay-) estimates for the linearized operator with slowly decaying kernel elements based on the Darboux transform, a new method to obtain Price's-law-type bounds for waves on a moving catenoid, as well as a refined profile construction designed to capture a crucial cancellation in the wave-catenoid interaction. In conjunction with our previous work on the higher dimensional case, this paper outlines a systematic approach for studying other soliton stability problems for $(3+1)$-dimensional quasilinear wave equations.

math.AP↗

arXiv2024

On illposedness of the Hall and electron magnetohydrodynamic equations without resistivity on the whole space

It has been shown in our previous work that the incompressible and irresistive Hall- and electron-magnetohydrodynamic (MHD) equations are illposed on flat domains $M = \mathbb{R}^k \times \mathbb{T}^{3-k}$ for $0 \le k \le 2$. The data and solutions therein were assumed to be independent of one coordinate, which not only significantly simplifies the systems but also allows for a large class of steady states. In this work, we remove the assumption of independence and conclude strong illposedness for compactly supported data in $\mathbb{R}^3$. This is achieved by constructing degenerating wave packets for linearized systems around time-dependent axisymmetric magnetic fields. A few main additional ingredients are: a more systematic application of the generalized energy estimate, use of the Bogovskiǐ operator, and a priori estimates for axisymmetric solutions to the Hall- and electron-MHD systems.

math.AP↗

arXiv2024

Late time tail of waves on dynamic asymptotically flat spacetimes of odd space dimensions

We introduce a general method for understanding the late time tail for solutions to wave equations on asymptotically flat spacetimes with odd space dimensions. In particular, for a large class of equations, we prove that the precise late time tail is determined by the limits of higher radiation field at future null infinity. In the setting of stationary linear equations, we recover and generalize the Price law decay rates. In particular, in addition to reproving known results on $(3+1)$-dimensional black holes, this allows one to obtain the sharp decay rate for the wave equation on higher dimensional black hole spacetimes, which exhibits an anomalous rate due to subtle cancellations. More interesting, our method goes beyond the stationary linear case and applies to both equations on dynamical background and nonlinear equations. In this case, our results can be used to show that in general there is a correction to the Price law rates.

gr-qc↗

arXiv2024

Wellposedness of the electron MHD without resistivity for large perturbations of the uniform magnetic field

We prove the local wellposedness of the Cauchy problems for the electron magnetohydrodynamics equations (E-MHD) without resistivity for possibly large perturbations of nonzero uniform magnetic fields. While the local wellposedness problem for (E-MHD) has been extensively studied in the presence of resistivity (which provides dissipative effects), this seems to be the first such result without resistivity. (E-MHD) is a fluid description of plasma in small scales where the motion of electrons relative to ions is significant. Mathematically, it is a quasilinear dispersive equation with nondegenerate but nonelliptic second-order principal term. Our result significantly improves upon the straightforward adaptation of the classical work of Kenig--Ponce--Rolvung--Vega on the quasilinear ultrahyperbolic Schrödinger equations, as the regularity and decay assumptions on the initial data are greatly weakened to the level analogous to the recent work of Marzuola--Metcalfe--Tataru in the case of elliptic principal term. A key ingredient of our proof is a simple observation about the relationship between the size of a symbol and the operator norm of its quantization as a pseudodifferential operator when restricted to high frequencies. This allows us to localize the (non-classical) pseudodifferential renormalization operator considered by Kenig--Ponce--Rolvung--Vega, and produce instead a classical pseudodifferential renormalization operator. We furthermore incorporate the function space framework of Marzuola--Metcalfe--Tataru to the present case of nonelliptic principal term.

math.AP↗

arXiv2023

Illposedness for dispersive equations: Degenerate dispersion and Takeuchi--Mizohata condition

We provide a unified viewpoint on two illposedness mechanisms for dispersive equations in one spatial dimension, namely degenerate dispersion and (the failure of) the Takeuchi--Mizohata condition. Our approach is based on a robust energy- and duality-based method introduced in an earlier work of the authors in the setting of Hall-magnetohydynamics. Concretely, the main results in this paper concern strong illposedness of the Cauchy problem (e.g., non-existence and unboundedness of the solution map) in high-regularity Sobolev spaces for various quasilinear degenerate Schrödinger- and KdV-type equations, including the Hunter--Smothers equation, $K(m, n)$ models of Rosenau--Hyman, and the inviscid surface growth model. The mechanism behind these results may be understood in terms of combination of two effects: degenerate dispersion -- which is a property of the principal term in the presence of degenerating coefficients -- and the evolution of the amplitude governed by the Takeuchi--Mizohata condition -- which concerns the subprincipal term. We also demonstrate how the same techniques yield a more quantitative version of the classical $L^{2}$-illposedness result by Mizohata for linear variable-coefficient Schrödinger equations with failed Takeuchi--Mizohata condition.

math.AP↗

arXiv2023

Initial data gluing in the asymptotically flat regime via solution operators with prescribed support properties

We give new proofs of general relativistic initial data gluing results on unit-scale annuli based on explicit solution operators for the linearized constraint equation around the flat case with prescribed support properties. These results retrieve and optimize - in terms of positivity, regularity, size and/or spatial decay requirements - a number of known theorems concerning asymptotically flat initial data, including Kerr exterior gluing by Corvino-Schoen and Chruściel-Delay, interior gluing (or "fill-in") by Bieri-Chruściel, and obstruction-free gluing by Czimek-Rodnianski. In particular, our proof of the strengthened obstruction-free gluing theorem relies on purely spacelike techniques, rather than null gluing as in the original approach.

math.AP↗

arXiv2023

Soliton resolution for equivariant self-dual Chern-Simons-Schrödinger equation in weighted Sobolev class

We consider the self-dual Chern-Simons-Schrödinger equation (CSS) under equivariant symmetry, which is a $L^{2}$-critical equation. It is known that (CSS) admits solitons and finite-time blow-up solutions. In this paper, we show soliton resolution for any solutions with equivariant data in the weighted Sobolev space $H^{1,1}$: every maximal solution decomposes into at most one modulated soliton and a radiation. A striking fact is that the nonscattering part must be a single modulated soliton. To our knowledge, this is the first result on soliton resolution in a class of nonlinear Schrödinger equations which are not known to be completely integrable. The key ingredient is the defocusing nature of the equation in the exterior of a soliton profile. This is a consequence of two distinctive features of (CSS): self-duality and non-local nonlinearity.

math.AP↗

arXiv2023

Blow-up dynamics for smooth finite energy radial data solutions to the self-dual Chern-Simons-Schrödinger equation

We consider the finite-time blow-up dynamics of solutions to the self-dual Chern-Simons-Schrödinger (CSS) equation (also referred to as the Jackiw-Pi model) near the radial soliton $Q$ with the least $L^{2}$-norm (ground state). While a formal application of pseudoconformal symmetry to $Q$ gives rise to an $L^{2}$-continuous curve of initial data sets whose solutions blow up in finite time, they all have infinite energy due to the slow spatial decay of $Q$. In this paper, we exhibit initial data sets that are smooth finite energy radial perturbations of $Q$, whose solutions blow up in finite time. Interestingly, their blow-up rate differs from the pseudoconformal rate by a power of logarithm. Applying pseudoconformal symmetry in reverse, this also yields a first example of an infinite-time blow-up solution, whose blow-up profile contracts at a logarithmic rate. Our analysis builds upon the ideas of previous works of the first two authors on (CSS) [21,22], as well as the celebrated works on energy-critical geometric equations by Merle, Raphaël, and Rodnianski [33,38]. A notable feature of this paper is a systematic use of nonlinear covariant conjugations by the covariant Cauchy-Riemann operators in all parts of the argument. This not only overcomes the nonlocality of the problem, which is the principal challenge for (CSS), but also simplifies the structure of nonlinearity arising in the proof.

math.AP↗

arXiv2022

A scattering theory approach to Cauchy horizon instability and applications to mass inflation

Motivated by the strong cosmic censorship conjecture, we study the linear scalar wave equation in the interior of subextremal strictly charged Reissner-Nordström black holes by analyzing a suitably-defined "scattering map" at $0$ frequency. The method can already be demonstrated in the case of spherically symmetric scalar waves on Reissner-Nordström: we show that assuming suitable ($L^2$-averaged) upper and lower bounds on the event horizon, one can prove ($L^2$-averaged) polynomial lower bound for the solution (1) on any radial null hypersurface transversally intersecting the Cauchy horizon, and (2) along the Cauchy horizon towards timelike infinity. Taken together with known results regarding solutions to the wave equation in the exterior, (1) above in particular provides yet another proof of the linear instability of the Reissner-Nordström Cauchy horizon. As an application of (2) above, we prove a conditional mass inflation result for a nonlinear system, namely, the Einstein-Maxwell-(real)-scalar field system in spherical symmetry. For this model, it is known that for a generic class of Cauchy data $\mathcal G$, the maximal globally hyperbolic future developments are $C^2$-future-inextendible. We prove that if a (conjectural) improved decay result holds in the exterior region, then for the maximal globally hyperbolic developments arising from initial data in $\mathcal G$, the Hawking mass blows up identically on the Cauchy horizon.

gr-qc↗