Communications in Analysis and Mechanics

&lt;p&gt;The objective of this paper is to examine the vanishing viscosity limit of the nonhomogeneous incompressible NS systerm, subject to the no-slip boundary condition. By adopting Kato's approach of constructing an artificial boundary layer &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b1"&gt;1&lt;/xref&gt;]&lt;/sup&gt;, within a smooth and bounded domain designated as $ \Omega\subseteq\mathbb{R}^{2} $, we derive a sufficient condition for the convergence to occur uniformly in time within the energy space $ L^{2}(\Omega) $.&lt;/p&gt;

Shuai Xi

A Kato-type criterion for the inviscid limit of the nonhomogeneous NS equations with no-slip boundary condition

非齐次NS方程无滑移边界条件下惯性极限的Kato型准则

&lt;p&gt;In this paper, we are concerned with an elliptic problem with mixed Dirichlet and Neumann boundary conditions that involve a mixed operator (i.e., the combination of classical Laplace operator and fractional Laplace operator) and critical nonlinearity. Also, we focus on identifying the optimal constant in the mixed Sobolev inequality, which we show is never achieved. Furthermore, by using variational methods, we provide an existence and nonexistence theory for both linear and superlinear perturbation cases.&lt;/p&gt;

Lovelesh Sharma

Brezis Nirenberg type results for local non-local problems under mixed boundary conditions

混合边界条件下局部非局部问题的Brezis-Nirenberg型结果

&lt;p&gt;Let $ L = -\Delta+|x|^2 $ be the Hermite operator on $ \mathbb R^{d} $, where $ \Delta $ is the Laplacian on $ \mathbb R^{d} $. In this paper, we will consider the Hardy-Sobolev spaces of higher order associated with $ L $. We also give some new characterizations of the Hardy spaces associated with $ L $.&lt;/p&gt;

Jizheng Huang

Shuangshuang Ying

Hardy-Sobolev spaces of higher order associated to Hermite operator

与Hermite算子相关的高阶Hardy-Sobolev空间

&lt;p&gt;In this paper, we were concerned with the existence of the solution related to the discontinuous differential equation, which corresponded to the stratification phenomenon in the Antarctic Circumpolar Current (ACC). By considering the piecewise vorticity function, we demonstrated the existence of solution corresponding to the discontinuous differential equation using Green's function, fixed point theory, and topological degree theory. This primarily included cases with piecewise constant vorticity, piecewise linear vorticity, and piecewise nonlinear vorticity. Additionally, we provided some examples to verify our results.&lt;/p&gt;

Michal Fečkan

Shan Li

JinRong Wang

Discontinuous differential equation for modelling the Antarctic Circumpolar Current

南极绕极流的非连续微分方程建模

&lt;p&gt;The following chemotaxis system has been considered:&lt;/p&gt;&lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} v_{t} = \Delta v-\xi \nabla\cdot(v \nabla w_{1})+\chi \nabla\cdot(v \nabla w_{2})+\lambda v-\mu v^{\kappa},\ &amp;amp;\ \ x\in \Omega, \ t&amp;gt;0,\\[2.5mm] w_{1t} = \Delta w_{1}-w_{1}+w^{\kappa_{1}}, \ 0 = \Delta w-w+v^{\kappa_{2}}, \ &amp;amp;\ \ x\in \Omega, \ t&amp;gt;0,\\[2.5mm] 0 = \Delta w_{2}-w_{2}+v^{\kappa_{3}}, \ &amp;amp;\ \ x\in \Omega, \ t&amp;gt;0 , \end{array} \right. \end{equation*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p&gt;under the boundary conditions of $ \frac{\partial{v}}{\partial{\nu}} = \frac{\partial{w_{1}}}{\partial{\nu}} = \frac{\partial{w}}{\partial{\nu}} = \frac{\partial{w_{2}}}{\partial{\nu}} $ on $ \partial \Omega, $ where $ \Omega $ was a bounded smooth domain of $ \mathbb{R}^{n}(n\geq 1), \; \nu $ was the normal vector of $ \partial\Omega, $ and the parameters were $ \lambda, \mu, \xi, \chi, \kappa_{1}, \; \kappa_{2}, \kappa_{3} &amp;gt; 0, $ and $ \kappa &amp;gt; 1. $ In this paper, we showed that if either $ \kappa_{1}\kappa_{2} &amp;lt; \max\{\frac{2}{n}, \kappa_{3}, \kappa-1\} $ or $ \kappa_{1}\kappa_{2} = \max\{\frac{2}{n}, \kappa_{3}, \kappa-1\} $ with the coefficients and initial data satisfying appropriate conditions, then the system possessed a global classical solution. Furthermore, we also have studied the convergence of solutions to a special case of the above system with $ \kappa = \delta+1, \kappa_{1} = 1, \kappa_{2} = \kappa_{3} = \delta $ for $ \delta &amp;gt; 0. $ It has been proven that if $ \mu &amp;gt; 0 $ is large enough, then the corresponding classical solutions exponentially converged to $ ((\frac{\lambda}{\mu})^{\frac{1}{\delta}}, \frac{\lambda}{\mu}, \frac{\lambda}{\mu}, \frac{\lambda}{\mu}), $ where the convergence rate could be formally expressed by the parameters of the system.&lt;/p&gt;

Chang-Jian Wang

Jia-Yue Zhu

Analysis of global dynamics in an attraction-repulsion model with nonlinear indirect signal and logistic source

具有非线性间接信号和逻辑源的吸引-排斥模型全局动力学分析

Publication Information	Publisher: American Institute of Mathematical Sciences，Publishing cycle: ，Journal Type: journal，Open Access Journals: Yes
Basic data	Year of publication: 2023，Proportion of original research papers: ，Self Citation Rate:， Gold OA Rate:
Average review cycle	10 weeks

Communications in Analysis and Mechanics

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A Kato-type criterion for the inviscid limit of the nonhomogeneous NS equations with no-slip boundary condition

Brezis Nirenberg type results for local non-local problems under mixed boundary conditions

Hardy-Sobolev spaces of higher order associated to Hermite operator

Discontinuous differential equation for modelling the Antarctic Circumpolar Current

Analysis of global dynamics in an attraction-repulsion model with nonlinear indirect signal and logistic source

A Kato-type criterion for the inviscid limit of the nonhomogeneous NS equations with no-slip boundary condition

Brezis Nirenberg type results for local non-local problems under mixed boundary conditions

Hardy-Sobolev spaces of higher order associated to Hermite operator

Discontinuous differential equation for modelling the Antarctic Circumpolar Current

Analysis of global dynamics in an attraction-repulsion model with nonlinear indirect signal and logistic source

ISSN	2836-3310
h-index	3