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Kwan Woo

Publications and source records attributed to Kwan Woo.

9 recordsLinked to original sources
arXiv2026

Failure of zero extension in parabolic Sobolev spaces

We show that spatial zero extension across the boundary may fail in parabolic Sobolev spaces $\mathring{\mathcal{H}}^1_p((0,T) \times \Omega)$, which can also be characterized as $$ L_p(0,T;\mathring{W}^1_p(\Omega))\cap W^1_p(0,T; W^{-1}_{p}(\Omega)). $$ More precisely, for any $p\in [1, \infty)$, we construct a function $u\in \mathring{\mathcal{H}}^1_p((0,T)\times \mathbb{R}^d_+)$ whose zero extension does not belong to $\mathcal{H}^1_p((0,T)\times \mathbb{R}^d)$. The obstruction occurs even for a flat boundary and is caused by a self-similar boundary layer concentrated at the initial-boundary corner, which produces a boundary supported normal flux defect after zero extension. We also discuss the suitability of various Sobolev-type spaces as solution spaces for parabolic equations in divergence form.

math.AP↗
arXiv2025

Existence and stability of Sadovskii vortices: from patch to smooth vortices

We establish a scaling-invariant variational framework for steadily translating dipoles of the two-dimensional incompressible Euler equations. Specifically, we consider the maximization of the kinetic energy subject to constraints on the impulse and the Lp-norm (1 4/3. By removing the mass constraint, we obtain a unified scaling-invariant variational principle valid for all 1<p\leq\infty. As a consequence of the variational structure, we establish a Lyapunov-type stability result, demonstrating that the axis-touching geometry persists under small perturbations. Finally, we derive a quantitative bound on the horizontal center of mass of perturbed solutions, showing that they propagate at nearly the same speed as the underlying Sadovskii vortex.

math.AP↗
arXiv2024

Emergence of peaked singularities in the Euler-Poisson system

We consider the one-dimensional Euler-Poisson system equipped with the Boltzmann relation and provide the exact asymptotic behavior of the peaked solitary wave solutions near the peak. This enables us to study the cold ion limit of the peaked solitary waves with the sharp range of H\"older exponents. Furthermore, we provide numerical evidence for $C^1$ blow-up solutions to the pressureless Euler-Poisson system, whose blow-up profiles are asymptotically similar to its peaked solitary waves and exhibit a different form of blow-up compared to the Burgers-type (shock-like) blow-up.

math.AP↗
arXiv2024

Fabes-Stroock approach to higher integrability of Green's functions and ABP estimates with $L_d$ drift

We explore the higher integrability of Green's functions associated with the second-order elliptic equation $a^{ij}D_{ij}u + b^i D_iu = f$ in a bounded domain $\Omega \subset \mathbb{R}^d$, and establish an enhanced version of Aleksandrov's maximum principle. In particular, we consider the drift term $b=(b^1, \ldots, b^d)$ in $L_d$ and the source term $f \in L_p$ for some $p < d$. This provides an alternative and analytic proof of a result by N. V. Krylov (\textit{Ann. Probab.}, 2021) concerning $L_d$ drifts. The key step involves deriving a Gehring-type inequality for Green's functions by using the Fabes-Stroock approach (\textit{Duke Math. J.}, 1984).

math.AP↗
arXiv2023

Characterizations of weighted Besov and Triebel-Lizorkin spaces with variable smoothness

In this paper, we study different types of weighted Besov and Triebel-Lizorkin spaces with variable smoothness. The function spaces can be defined by means of the Littlewood-Paley theory in the field of Fourier analysis, while there are other norms arising in the theory of partial differential equations such as Sobolev-Slobodeckij spaces. It is known that two norms are equivalent when one considers constant regularity function spaces without weights. We show that the equivalence still holds for variable smoothness and weights, which is accomplished by making use of shifted maximal functions, Peetre's maximal functions, and the reverse H\"older inequality. Moreover, we obtain a weighted regularity estimate for time-fractional evolution equations and a generalized Sobolev embedding theorem without weights.

math.CA↗
arXiv2023

On the trace theorem to Volterra-type equations with local or non-local derivatives

This paper considers traces at the initial time for solutions of evolution equations with local or non-local derivatives in vector-valued $L_p$ spaces with $A_p$ weight. To achieve this, we begin by introducing a generalized real interpolation method. Within the framework of generalized interpolation theory, we make use of stochastic process theory and two-weight Hardy's inequality to derive our trace and extension theorems. Our results encompass findings applicable to time-fractional equations with broad temporal weight functions.

math.AP↗
arXiv2023

Sobolev spaces and trace theorems for time-fractional evolution equations

We establish trace and extension theorems for evolutionary equations with the Caputo fractional derivatives in (weighted) $L_p$ spaces. To achieve this, we identify weighted Sobolev and Besov spaces with mixed norms that accommodate solution spaces and their initial values well-suited for equations involving time-fractional derivatives. Our analysis encompasses both time-fractional sub-diffusion and super-diffusion equations. We also provide observations on the initial behavior of solutions to time-fractional equations.

math.AP↗
arXiv2022

Parabolic equations with unbounded lower-order coefficients in Sobolev spaces with mixed norms

We prove the Lp,q-solvability of parabolic equations in divergence form with full lower-order terms. The coefficients and non-homogeneous terms belong to mixed Lebesgue spaces with the lowest integrability conditions. In particular, the coefficients for the lower-order terms are not necessarily bounded. We study both the Dirichlet and conormal derivative boundary value problems on irregular domains. We also prove embedding results for parabolic Sobolev spaces, the proof of which is of independent interest.

math.AP↗