arXiv subjects

Yunjoo Kim

Publications and source records attributed to Yunjoo Kim.

5 recordsLinked to original sources

arXiv2026

Stable $C^{7/9}$ cusp formation for the Novikov equation

We establish stable cusp formation for the Novikov equation, a cubically nonlinear Camassa--Holm-type equation. We identify an open set of smooth initial data for which the first gradient blow-up produces a cusp with sharp H\"older regularity $C^{7/9}$. This result shows that, in nonlocal wave-breaking problems, the sharp regularity of the cusp is not determined by the nonlocal or nonlinear structure alone. While the conserved $H^1$-type quantity excludes the $C^{1/3}$ cusp associated with Burgers-type gradient blow-up, the precise H\"older exponent is selected by the coupling between the nonlocal term and the algebraic structure of the nonlinearity. In the Novikov equation, this coupling yields the exponent $7/9$, rather than the $3/5$ exponent known for the Camassa--Holm and Hunter--Saxton equations. The main difficulty is that the naive high-frequency limit retains the cubic character of the equation and therefore does not exhibit a self-similar leading flow. We overcome this by introducing a Galilean-type change of variables around a nonzero background, which reveals a quadratic Hunter--Saxton-type leading equation. Its self-similar profiles determine the $C^{7/9}$ cusp, while the nonlocal and cubic remainders are controlled perturbatively in modulated similarity variables.

math.AP↗

arXiv2026

Asymptotic self-similar blow-up for the regularized Saint-Venant equations

We investigate singularity formation in the regularized Saint--Venant (rSV) equations, a conservative, non-dispersive shallow water system that is formally regarded as a Hamiltonian regularization of the isentropic Euler equations. While it is known that smooth solutions to the rSV system can develop gradient blow-up in finite time, the precise structure of such singularities has not been rigorously characterized. In this work, we establish stability of self-similar blow-up profiles of the Hunter--Saxton equation within the rSV framework, using a nonlinear bootstrap argument in dynamically rescaled coordinates. Our analysis captures the detailed space-time dynamics of solutions near the singularity, and proves their sharp $C^{3/5}$ H\"older regularity at the singular time. This regularity differs from the $C^{1/3}$ H\"older regularity of the cubic-root singularities found in the compressible Euler and inviscid Burgers equations. This contrast highlights the structural influence of the Hamiltonian regularization on singularity formation. To illuminate this effect, we also show that the same $C^{3/5}$ blow-up profile emerges in the regularized Burgers equation, a scalar analogue of the rSV system.

math.AP↗

arXiv2024

Sharp regularity of gradient blow-up solutions in the Camassa-Holm equation

We study the formation of singularities in the Camassa-Holm (CH) equation, providing a detailed description of the blow-up dynamics and identifying the precise Hölder regularity of the gradient blow-up solutions. To this end, we first construct self-similar blow-up profiles and examine their properties, including the asymptotic behavior at infinity, which determines the type of singularity. Using these profiles as a reference and employing modulation theory, we establish global pointwise estimates for the blow-up solutions in self-similar variables, thereby demonstrating the stability of the self-similar profiles we construct. Our results indicate that the solutions, evolving from smooth initial data within a fairly general open set, form $C^{3/5}$ cusps as the first singularity in finite time. These singularities are analogous to \emph{pre-shocks} emerging in the Burgers equation, exhibiting unbounded gradients while the solutions themselves remain bounded. However, the nature of the singularity differs from that of the Burgers equation, which is a cubic-root singularity, i.e., $C^{1/3}$. Our work provides the first proof that generic pre-shocks of the CH equation exhibit $C^{3/5}$ Hölder regularity. It is our construction of new self-similar profiles, incorporating the precise leading-order correction to the CH equation in the blow-up regime, that enables us to identify the sharp Hölder regularity of the singularities and to capture the detailed spatial and temporal dynamics of the blow-up. We also show that the generic singularities developed in the Hunter-Saxton (HS) equation are of the same type as those in the CH equation.

math.AP↗

arXiv2024

Delta-shock for the pressureless Euler-Poisson system

We study singularity formation for the pressureless Euler-Poisson system of cold ion dynamics. In contrast to the Euler-Poisson system with pressure, when its smooth solutions experience $C^1$ blow-up, the $L^\infty$ norm of the density becomes unbounded, which is often referred to as a delta-shock. We provide a constructive proof of singularity formation to obtain an exact blow-up profile and the detailed asymptotic behavior of the solutions near the blow-up point in both time and space. Our result indicates that at the blow-up time $t=T_\ast$, the density function is unbounded but is locally integrable with the profile of $ρ(x,T_\ast) \sim (x-x_*)^{-2/3}$ near the blow-up point $x=x_\ast$. This profile is not yet a Dirac measure. On the other hand, the velocity function has $C^{1/3}$ regularity at the blow-up point. Loosely following our analysis, we also obtain an exact blow-up profile for the pressureless Euler equations.

math.AP↗

arXiv2024

Structure of singularities for the Euler-Poisson system of ion dynamics

We study the formation of singularity for the isothermal Euler-Poisson system arising from plasma physics. Contrast to the previous studies yielding only limited information on the blow-up solutions, for instance, sufficient conditions for the blow-up and the temporal blow-up rate along the characteristic curve, we rather give a constructive proof of singularity formation from smooth initial data. More specifically, employing the stable blow-up profile of the Burgers equation in the self-similar variables, we establish the global stability estimate in the self-similar time, which yields the asymptotic behavior of blow-up solutions near the singularity point. Our analysis indicates that the smooth solution to the Euler-Poisson system can develop a cusp-type singularity; it exhibits $C^1$ blow-up in a finite time, while it belongs to $C^{1/3}$ at the blow-up time, provided that smooth initial data are sufficiently close to the blow-up profile in some weighted $C^4$-topology. We also present a similar result for the isentropic case, and discuss noteworthy differences in the analysis.

math.AP↗