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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.

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.

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 $0<α<\frac{1}{4}$.

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.

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