arXiv subjects

Ming-Yuan Chang

Publications and source records attributed to Ming-Yuan Chang.

3 recordsLinked to original sources

arXiv2026

A Hölder estimate for the trajectories of the Navier-Stokes equations

We study solutions to the Navier-Stokes equations in the class $L^\infty_t C^α_x$. Landau and Lifshitz [LL87] predicted that the Eulerian and Lagrangian temporal structure functions for turbulence exhibit $1/3$ and $1/2$ scaling laws, respectively. These laws were justified for the Euler equations in [Ise23,Ise25], assuming the spatial structure functions satisfies a $1/3$ scaling law. We demonstrate them in a viscous setting by proving that the $C^α_{t,x}$-norm of the solution and the $C^{1/(1-α)}$-norm of any fluid trajectory can be estimated by the $L^\infty_tC^α_x$-norm independently of the viscosity parameter $ν>0$, for times bounded away from zero by a positive power of $ν$.

math.AP↗

arXiv2026

Level sets of fractional Sobolev functions

We prove a coarea-type result for scalar functions $f$ in fractional Sobolev spaces $W^{s, p} (Ω)$ with $Ω\subset \mathbb R^n$, $0<s<1$, and $1\leq p < \infty$. Our theorem shows that a.e. level set has zero Hausdorff $\mathcal{H}^{n-s}$ measure, where the level set $f^{-1} (y)$ is defined as the set all points at which $y$ is between the $\liminf$ and the $\limsup$ (as $r\downarrow 0$) of the averages of $f$ over the balls $B_r (y)$. A quite general construction of random series of wavelets shows also that with probability $1$ (many) level sets have indeed dimension $n-s$.

math.AP↗

arXiv2023

$L^2$-estimates for the Dirac-Dolbeault operator and Bergman kernel asymptotics on some classes of non-compact complex manifolds

For high power $k$, the $L^2$-estimates for the Dirac-Dolbeault operator with coefficient $L^k\otimes E$ can be obtained from the Bochner-Kodaira-Nakano identity if $L$ has positive curvature. In this article, we generalize the classical method to obtain $L^2$-estimates for mixed curvature case, and give a bound to the extra error term. Modifying the $L^2$-estimates and existence theorems for $\bar{\partial}$-operator, we can get a local spectral gap of the Kodaira Laplacian $\Box$ and thus a full asymptotic expansion for Bergman kernel.

math.CV↗