arXiv subjects

Sehyun Ji

Publications and source records attributed to Sehyun Ji.

6 recordsLinked to original sources

arXiv2026

Finite time singularities in the Landau equation with very hard potentials

We consider the inhomogeneous Landau equation with $γ\in (\sqrt{3},2]$ and construct smooth, strictly positive initial data that develop a finite time singularity. The $C^α$-norm of the distribution function blows up for every $α>0$, whereas its $L^{\infty}$-norm remains uniformly bounded. In self-similar variables, the solution becomes asymptotically hydrodynamic - the distribution function converges to a local Maxwellian, while the hydrodynamic fields develop an asymptotically self-similar implosion whose profile coincides with a smooth imploding profile of the compressible Euler equations. To our knowledge, this provides the first example of a collisional kinetic model which is globally well-posed in the homogeneous setting, but admits finite time singularities for inhomogeneous data.

math.AP↗

arXiv2024

Dissipation estimates of the Fisher information for the Landau equation

We establish an a priori estimate for the dissipation of the Fisher information for the space-homogeneous Landau equation with very soft potentials. This work is motivated by the recent breakthrough by Guillen and Silvestre, which proves that the Fisher information is monotone decreasing. As a direct consequence, we show that the Fisher information becomes instantaneously bounded, even if it is not initially bounded. This leads to a proof of the global existence of smooth solutions for the space-homogeneous Landau equation with very soft potentials, given initial data $f_0 \in L^1_{2-γ} \cap L \log L$. This result includes the case of the Coulomb potential.

math.AP↗

arXiv2024

Bounds for the optimal constant of the Bakry-Émery $Γ_2$ criterion inequality on $ RP^{d-1}$

We prove upper and lower bounds on the optimal constant $Λ_d$ of the Bakry-Émery $Γ_2$ criterion for positive symmetric functions on the unit sphere $S^{d-1}$, which also can be identified as positive functions on the real projective space $RP^{d-1}$. The Bakry-Émery $Γ_2$ criterion inequality was crucially used to prove the monotonicty of the Fisher information for the Landau equation by Guillen and Silvestre recently. Therefore, a better bound on the optimal constant $Λ_d$ expands the range of interaction potentials that exhibits the monotonicity of the Fisher information. In particular, we compute that $Λ_3$ is between $5.5$ and $5.739$.

math.AP↗

arXiv2024

Entropy dissipation estimates for the Landau equation with Coulomb potentials

We prove a lower bound for the entropy dissipation of the Landau equation with Coulomb potentials by a weighted Lebesgue norm $L^3_{-5/3}$. In particular, we enhance the weight exponent from $-5$, which was established by Desvillettes, to $-5/3$. Moreover, we prove that the weighted Lebesgue norm $L^3_{-5/3}$ is optimal for both exponents.

math.AP↗

arXiv2024

Global Existence for an Isotropic Landau Model

Following the recent ideas of Guillen and Silvestre in $[9]$, we prove that the Fisher information is non-increasing along the flow of the isotropic Landau equation. We then use this fact to deduce global existence for the equation $\partial_t f = (-Δ)^{-1}f \cdot Δf + f^2$ under a relatively lax set of conditions on the initial data. In particular, we remove the restrictive radially decreasing assumption of previous works.

math.AP↗

arXiv2024

Local regularity for the space-homogeneous Landau equation with very soft potentials

This paper deals with the space-homogenous Landau equation with very soft potentials, including the Coulomb case. This nonlinear equation is of parabolic type with diffusion matrix given by the convolution product of the solution with the matrix $a_{ij} (z)=|z|^γ(|z|^2 δ_{ij} - z_iz_j)$ for $γ\in [-3,-2)$. We derive local truncated entropy estimates and use them to establish two facts. Firstly, we prove that the set of singular points (in time and velocity) for the weak solutions constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998), 273-307] has zero $\mathscr{P}^{m_\ast}$ parabolic Hausdorff measure with $m_\ast:= \frac72 |2+γ|$. Secondly, we prove that if such a weak solution is axisymmetric, then it is smooth away from the symmetry axis. In particular, radially symmetric weak solutions are smooth away from the origin.

math.AP↗