arXiv subjects

Chanjin You

Publications and source records attributed to Chanjin You.

6 recordsLinked to original sources

arXiv2026

Modified Scattering for the Time-Dependent Kohn--Sham Equation

We study the long-time behavior of the (critical) Kohn--Sham equation in two and three dimensions, i.e.,\[ \mathrm{i} \partial_t {\gamma} = \Big[-\frac{1}{2}\Delta + \lambda \, |\cdot|^{-1} \ast \rho_{{\gamma}} + \mu \, \rho_{{\gamma}}^{1/d}, {\gamma} \Big] \quad \text{for} \quad d=2,3. \] By introducing a suitable ''square root'' of the density matrix and exploiting the pseudo-conformal transform, we establish global well-posedness for small initial data in an appropriate weighted Schatten norm. We also prove the optimal time decay of the particle density and establish modified scattering for small and localized solutions. In particular, our results provide a resolution to the open problems proposed by Pusateri and Sigal (2021) for the critical and subcritical regimes, rigorously proving their conjectures regarding modified scattering in the critical case and linear scattering in the subcritical cases. Our results place these scattering phenomena in the operator-valued setting of density matrices, thereby extending the classical scalar theory to a broader framework.

math.AP↗

arXiv2026

Asymptotic Stability of Hartree--Fock Homogenous Equilibria in $\mathbb{R}^d$

In this paper, we establish nonlinear Landau damping and asymptotic stability of a large class of translation-invariant steady solutions to the time-dependent Hartree--Fock equations in the presence of an {\em off-diagonal exchange operator}, which arises naturally in the meanfield theory of a large fermionic system, in the whole space $\mathbb{R}^d$, $d\ge 3$. Despite being a sub-order operator, the inclusion of the exchange term disturbs the classical Schr\"odinger dispersion and causes a complex linear response from the background electrons to the space density whose dispersion relation is no longer a Fourier multiplier as in the classical Vlasov and Hartree theory. In addition, the group velocity of each elementary waves involves a mixture of all other Fourier modes, leading to delicate {\em momentum-dependent echo resonances}. To overcome the issues, we develop a nonlinear iterative scheme that relies on a detailed resolvent analysis, makes use of a transport type dispersion in Fourier spaces, and propagates phase mixing and Landau damping in weighted $L^\infty_{k,p}$ norms.

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arXiv2025

Large Time Behavior of the Klein-Gordon-Schr\"{o}dinger system

We establish the global existence and scattering for small and localized solutions of the Klein-Gordon-Schr\"{o}dinger system in three dimensions. The system consists of coupled semilinear Schr\"{o}dinger and Klein-Gordon equations with quadratic nonlinearities. This model is motivated by the study of plasma oscillations arising from the Hartree equation near a translation-invariant equilibrium with the Coulomb potential. Our proof relies on the space-time resonance method. The main difficulty comes from the two dimensional space-time resonant set and the absence of null form structure.

math.AP↗

arXiv2024

Modified scattering for long-range Hartree equations of infinite rank near vacuum

We establish the asymptotic behavior and decay of solutions near vacuum to the Hartree equation with the Coulomb interaction potential in three dimensions. Our approach is direct, which consists of independently deriving the sharp dispersive decay estimates for the density function and establishing the boundedness of energy norms. The global in time well-posedness is done without introducing the phase correction, while the phase modification is explicit in terms of the density function.

math.AP↗

arXiv2024

Phase mixing estimates for the nonlinear Hartree equation of infinite rank

In this paper, we prove the phase mixing estimates for the density and its derivatives associated with the nonlinear Hartree equation around certain translation-invariant equilibria. Given a defocusing short-range interaction potential, we provide a precise criterion for the Penrose--Lindhard stability based on the marginal of the equilibrium. For linearly stable equilibria, pointwise decay estimates of the Green function associated with the linearized operator in Fourier space are established. The proof of phase mixing estimates is obtained through a nonlinear iterative scheme. An alternative proof of scattering is also provided.

math.AP↗

arXiv2023

Plasmons for the Hartree equations with Coulomb interaction

In this work, we establish the existence and decay of {\em plasmons}, the quantum of Langmuir's oscillatory waves found in plasma physics, for the linearized Hartree equations describing an interacting gas of infinitely many fermions near general translation-invariant steady states, including compactly supported Fermi gases at zero temperature, in the whole space $\RR^d$ for $d\ge 2$. Notably, these plasmons exist precisely due to the long-range pair interaction between the particles. Next, we provide a survival threshold of spatial frequencies, below which the plasmons purely oscillate and disperse like a Klein-Gordon's wave, while at the threshold they are damped by {\em Landau damping}, the classical decaying mechanism due to their resonant interaction with the background fermions. The explicit rate of Landau damping is provided for general radial homogenous equilibria. Above the threshold, the density of the excited fermions is well approximated by that of the free gas dynamics and thus decays rapidly fast for each Fourier mode via {\em phase mixing}. Finally, pointwise bounds on the Green function and dispersive estimates on the density are established.

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