Stability of boundary layers for the inflow compressible Navier-Stokes equations
Jing Wang Lining Tong
In this paper, we consider the boundary layer stability of the one-dimensional isentropic compressible Navier-Stokes equations with an inflow boundary condition. We assume only one of the two characteristics to the corresponding Euler equations is negative up to some small time. We prove the existence of the boundary layers, then instead of using the skew symmetric matrix, we give a higher convergence rate of the approximate solution than the previous results by a standard energy method as long as the strength of the boundary layers is suitably small.
keywords: inflow boundary condition asymptotic analysis energy estimate. Compressible Navier-Stokes equations degenerate viscosity matrix noncharacteristic boundary layers
On the Rayleigh-Taylor instability for the compressible non-isentropic inviscid fluids with a free interface
Jing Wang Feng Xie
In this paper, we study the Rayleigh-Taylor instability phenomena for two compressible, immiscible, inviscid, ideal polytropic fluids. Such two kind of fluids always evolve together with a free interface due to the uniform gravitation. We construct the steady-state solutions for the denser fluid lying above the light one. With an assumption on the steady-state temperature function, we find some growing solutions to the related linearized problem, which in turn demonstrates the linearized problem is ill-posed in the sense of Hadamard. By such an ill-posedness result, we can finally prove the solutions to the original nonlinear problem does not have the property EE(k). Precisely, the $H^3$ solutions to the original nonlinear problem can not Lipschitz continuously depend on their initial data.
keywords: Rayleigh-Taylor instability ill-posedness. compressible inviscid fluids non-isentropic free boundary
Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension
Jiahang Che Li Chen Simone GÖttlich Anamika Pandey Jing Wang

This article is devoted to the discussion of the boundary layer which arises from the one-dimensional parabolic elliptic type Keller-Segel system to the corresponding aggregation system on the half space case. The characteristic boundary layer is shown to be stable for small diffusion coefficients by using asymptotic analysis and detailed error estimates between the Keller-Segel solution and the approximate solution. In the end, numerical simulations for this boundary layer problem are provided.

keywords: Chemotaxis boundary layer asymptotic expansion
The mixed-mode oscillations in Av-Ron-Parnas-Segel model
Bo Lu Shenquan Liu Xiaofang Jiang Jing Wang Xiaohui Wang

Mixed-mode oscillations (MMOs) as complex firing patterns with both relaxation oscillations and sub-threshold oscillations have been found in many neural models such as the stellate neuron model, HH model, and so on. Based on the work, we discuss mixed-mode oscillations in the Av-Ron-Parnas-Segel model which can govern the behavior of the neuron in the lobster cardiac ganglion. By using the geometric singular perturbation theory we first explain why the MMOs exist in the reduced Av-Ron-Parnas-Segel model. Then the mixed-mode oscillatory phenomenon and aperiodic mixed-mode behaviors in the model have been analyzed numerically. Finally, we illustrate the influence of certain parameters on the model.

keywords: Geometric singular perturbation theory mixed-mode oscillations InterSpike Intervals Av-Ron-Parnas-Segel model lobster cardiac ganglion
Global exact controllability and asympotic stabilization of the periodic two-component $\mu\rho$-Hunter-Saxton system
Jingqun Wang Lixin Tian Weiwei Guo
In this paper, we discuss two main problems. In the first section, we establish a new global distributed exact controllability of the periodic two-component $\mu\rho$-Hunter-Saxton system on the circle by means of a distributed control. And in the second section, we present corresponding result of the asymptotic stabilization problem about the periodic two-component $\mu\rho$-Hunter-Saxton system. By presenting concrete form of the feedback law, an equivalent system is got.
keywords: Periodic two-component $\mu\rho$-Hunter-Saxton system global distributed exact controllability asymptotic stabilization.
Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation
Jingrui Wang Keyan Wang

In this paper we prove the almost sure existence of global weak solution to the 3D incompressible Navier-Stokes Equation for a set of large data in $\dot{H}^{-α}(\mathbb{R}^{3})$ or $\dot{H}^{-α}(\mathbb{T}^{3})$ with $0<α≤ 1/2$. This is achieved by randomizing the initial data and showing that the energy of the solution modulus the linear part keeps finite for all $t≥0$. Moreover, the energy of the solutions is also finite for all $t>0$. This improves the recent result of Nahmod, Pavlović and Staffilani on (SIMA) in which $α$ is restricted to $0<α<\frac{1}{4}$.

keywords: Navier-Stokes equation random initial data
Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent
Gabriel Fuhrmann Jing Wang

We study order-preserving $\mathcal{C}^1$-circle diffeomorphisms driven by irrational rotations with a Diophantine rotation number. We show that there is a non-empty open set of one-parameter families of such diffeomorphisms where the ergodic measures of nearly all family members are one-rectifiable, that is, absolutely continuous with respect to the restriction of the one-dimensional Hausdorff measure to a countable union of Lipschitz graphs.

keywords: Quasiperiodically forced circle maps rectifiability strange non-chaotic attractor/repeller non-uniform hyperbolicity multiscale analysis

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