Annals of Applied Mathematics AAM Volume 37, Number 1

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Volume 37, Number 1

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Convergence of an Embedded Exponential-Type Low-Regularity Integrators for the KdV Equation Without Loss of Regularity

Yongsheng Li, Yifei Wu & Fangyan Yao

10.4208/aam.OA-2020-0001

Ann. Appl. Math., 37 (2021), pp. 1-21.

Preview Full PDF 4281 50159 Abstract

Abstract

In this paper, we study the convergence rate of an Embedded exponential-type low-regularity integrator (ELRI) for the Korteweg-de Vries equation. We develop some new harmonic analysis techniques to handle the "stability" issue. In particular, we use a new stability estimate which allows us to avoid the use of the fractional Leibniz inequality,

and replace it by suitable inequalities without loss of regularity. Based on these techniques, we prove that the ELRI scheme proposed in [41] provides $\frac12$-order convergence accuracy in $H^\gamma$ for any initial data belonging to $H^\gamma$ with $\gamma >\frac32$, which does not require any additional derivative assumptions.

A Simple Proof of Regularity for $C^{1,\alpha}$ Interface Transmission Problems

Hongjie Dong

10.4208/aam.OA-2020-0002

Ann. Appl. Math., 37 (2021), pp. 22-30.

Preview Full PDF 3565 35003 Abstract

Abstract

We give an alternative proof of a recent result in [1] by Caffarelli, Soria-Carro, and Stinga about the $C^{1,\alpha}$ regularity of weak solutions to transmission problems with $C^{1,\alpha}$ interfaces. Our proof does not use the mean value property or the maximum principle, and also works for more general elliptic systems with variable coefficients. This answers a question raised in [1]. Some extensions to $C^{1,\text{Dini}}$ interfaces and to domains with multiple sub-domains are also discussed.

Stability of the Semi-Implicit Method for the Cahn-Hilliard Equation with Logarithmic Potentials

Dong Li & Tao Tang

10.4208/aam.OA-2020-0003

Ann. Appl. Math., 37 (2021), pp. 31-60.

Preview Full PDF 4499 52442 Abstract

Abstract

We consider the two-dimensional Cahn-Hilliard equation with logarithmic potentials and periodic boundary conditions. We employ the standard semi-implicit numerical scheme, which treats the linear fourth-order dissipation term implicitly and the nonlinear term explicitly. Under natural constraints on the time step we prove strict phase separation and energy stability of the semi-implicit scheme. This appears to be the first rigorous result for the semi-implicit discretization of the Cahn-Hilliard equation with singular potentials.

Initial and Boundary Value Problem for a System of Balance Laws from Chemotaxis: Global Dynamics and Diffusivity Limit

Zefu Feng, Jiao Xu, Ling Xue & Kun Zhao

10.4208/aam.OA-2020-0004

Ann. Appl. Math., 37 (2021), pp. 61-110.

Preview Full PDF 3978 36017 Abstract

Abstract

In this paper, we study long-time dynamics and diffusion limit of large-data solutions to a system of balance laws arising from a chemotaxis model with logarithmic sensitivity and nonlinear production/degradation rate. Utilizing energy methods, we show that under time-dependent Dirichlet boundary conditions, long-time dynamics of solutions are driven by their boundary data, and there is no restriction on the magnitude of initial energy. Moreover, the zero chemical diffusivity limit is established under zero Dirichlet boundary conditions, which has not been observed in previous studies on related models.

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