Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent
Gabriel Fuhrmann Jing Wang
Discrete & Continuous Dynamical Systems - A 2017, 37(11): 5747-5761 doi: 10.3934/dcds.2017249

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
Stability of boundary layers for the inflow compressible Navier-Stokes equations
Jing Wang Lining Tong
Discrete & Continuous Dynamical Systems - B 2012, 17(7): 2595-2613 doi: 10.3934/dcdsb.2012.17.2595
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
Discrete & Continuous Dynamical Systems - B 2016, 21(8): 2767-2784 doi: 10.3934/dcdsb.2016072
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
The mixed-mode oscillations in Av-Ron-Parnas-Segel model
Bo Lu Shenquan Liu Xiaofang Jiang Jing Wang Xiaohui Wang
Discrete & Continuous Dynamical Systems - S 2017, 10(3): 487-504 doi: 10.3934/dcdss.2017024

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
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
Communications on Pure & Applied Analysis 2017, 16(3): 1013-1036 doi: 10.3934/cpaa.2017049

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
Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers
Jing Wang Lining Tong
Communications on Pure & Applied Analysis 2019, 18(2): 887-910 doi: 10.3934/cpaa.2019043

In this paper, we study the limiting behavior of solutions to a 1D two-point boundary value problem for viscous conservation laws with genuinely-nonlinear fluxes as $\varepsilon$ goes to zero. We here discuss different types of non-characteristic boundary layers occurring on both sides. We first construct formally the three-term approximate solutions by using the method of matched asymptotic expansions. Next, by energy method we prove that the boundary layers are nonlinearly stable and thus it is proved the boundary layer effects are just localized near both boundaries. Consequently, the viscous solutions converge to the smooth inviscid solution uniformly away from the boundaries. The rate of convergence in viscosity is optimal.

keywords: Two-point boundary value problem multiple boundary layers weak boundary layer strong boundary layer matched asymptotic expansions stability analysis

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