arXiv subjects

Rong Dong

Publications and source records attributed to Rong Dong.

4 recordsLinked to original sources

arXiv2024

Interior Second Order H\"{o}lder Regularity for Stokes systems

Global second order H\"{o}lder regularity for Stokes systems can be obtained by global Schauder estimates, which are actually a priori estimates and were established by Solonnikov [20] and [23] with appropriate compatible conditions. This paper will investigate the corresponding interior regularity which unfortunately may fail in general from Serrin's counterexample (cf. [19]). However, we discover interior $C^{2,\alpha}$ regularity for velocity and interior $C^{1,\alpha}$ regularity for pressure in spatial variables, and furthermore, for curl of velocity, we find its gradient belongs to $C^{\alpha, \frac{\alpha}2}$, that is, possesses H\"{o}lder continuity in both space and time directions. The interesting phenomenon here is that no continuity in time variable is assumed for both the coefficients and the righthand side terms. The estimates for velocity and its curl are achieved pointwisely and the results are sharp indicated by a counterexample.

math.AP↗

arXiv2023

Interior pointwise $C^{1,α}$ estimates for Stokes systems in divergence form

Interior pointwise $C^{1,α}$ estimates are established for Stokes systems in divergence form where no continuity in time variable is assumed for the coefficients and the given data. The estimates are attained by iteration and are presented by Campanato's characterization. The sharpness of the conclusions can be seen from Serrin's counterexample.

math.AP↗

arXiv2022

Interior $L_{p}$ regularity for Stokes systems

A new iteration method is represented to study the interior $L_{p}$ regularity for Stokes systems both in divergence form and in non-divergence form. By the iteration, we improve the integrability of derivatives of solutions for Stokes systems step by step; after infinitely many steps, $L_{p}$ regularity is achieved; and in each step, the maximal function method is used where solutions and their derivatives are involved simultaneously in each scale. The H\"{o}lder continuity of the coefficients in spatial variables is assumed to compensate the different scalings between the solutions and their derivatives.

math.AP↗

arXiv2011

Dark-bright gap solitons in coupled-mode one-dimensional saturable waveguide arrays

In the present work, we consider the dynamics of dark solitons as one mode of a defocusing photorefractive lattice coupled with bright solitons as a second mode of the lattice. Our investigation is motivated by an experiment which illustrates that such coupled states can exist with both components in the first gap of the linear band spectrum. This finding is further extended by the examination of different possibilities from a theoretical perspective, such as symbiotic ones where the bright component is supported by states of the dark component in the first or second gap, or non-symbiotic ones where the bright soliton is also a first-gap state coupled to a first or second gap state of the dark component. While the obtained states are generally unstable, these instabilities typically bear fairly small growth rates which enable their observation for experimentally relevant propagation distances.

nlin.PS↗