In this paper, we introduce the progress of the Euler equation and Onsager conjecture. We also introduce the Euler’s life, the researches about the incompressible Euler equation, and the Onsager conjecture.

In this paper, we study a fractional differential equation $$^cD^α _{0+} u(t) + f(t, u(t)) = 0, \ t ∈ (0, +∞)$$ satisfying the boundary conditions: $$u′(0)=0,\ \lim\limits_{t→+∞} \ ^cD^{α−1}_{0+} u(t) = g(u),$$ where $1<α\leq 2,$ $^cD^α_{0+}$ is the standard Caputo fractional derivative of order $α.$ The main tools used in the paper is a contraction principle in the Banach space and the fixed point theorem due to D. O’Regan. Under a compactness criterion, the existence of solutions is established.

This paper considers the unsteady boundary layer flow over a moving flat plate embedded in a porous medium with fractional Oldroyd-B viscoelastic fluid. The governing equations with mixed time-space fractional derivatives are solved numerically by using the finite difference method combined with an L1-algorithm. The effect of various physical parameters on the velocity and average skin friction are discussed and graphically illustrated in detail. Results show that the porosity $ϵ$ and fractional derivative $α$ enhance the flow of Oldroyd-B viscoelastic fluid within porous medium, but fractional derivative $β$ weakens the flow. Moreover, it is found that the average skin friction coefficient rises with the increase of fractional derivative $β.$

Based on the idea of local polynomial double-smoother, we propose an estimator of a conditional cumulative distribution function with dependent and left-truncated data. It is assumed that the observations form a stationary $α$-mixing sequence. Asymptotic normality of the estimator is established. The finite sample behavior of the estimator is investigated via simulations.

In this paper, we investigated an impulsive predator-prey model with mutual interference and Crowley-Martin response function. By the comparison theorem and the analysis technique of [12,14], sufficient conditions for the permanence of the impulsive model are obtained, which generalizes one of main results of [4].

Let $S(R^2)$ be the class of all infinitely differential functions which, as well as their derivatives, are rapidly decreasing on $R^2.$ Here we define a kind of semi-norms which is equivalent to the usual family of semi-norms on the Schwartz space $S(R^2).$

An autonomous stage-structured competitive systems with toxic effect is investigated in this paper. Sufficient conditions which guarantee the global attractivity of the system and the extinction of the partial species are obtained, respectively. Our results supplement and complement one of the main results of Liu and Li [Global stability analysis of a nonautonomous stage-structured competitive system with toxic effect and double maturation delays, Abstract and Applied Analysis, Volume 2014, Article ID 689573, 15 pages].

It is known that somebody’s behavior (decision) in a stochastic social network may be influenced by that of his (or her) friends. In this paper, we consider two stochastic social network game models (a) and (b) which can be defined respectively by two different utility functions. Some sufficient conditions for the existence of Nash equilibrium (NE) of the two network game models are obtained by analyzing the different effort relation between a player and his (or her) neighbors.

A graph $G$ is $k$-triangular if each of its edge is contained in at least $k$ triangles. It is conjectured that every 4-edge-connected triangular graph admits a nowhere-zero 3-flow. A triangle-path in a graph G is a sequence of distinct triangles $T_1T_2 · · · T_k$ in $G$ such that for $1 ≤ i ≤ k − 1,$ $|E(T_i) ∩ E(T_{i+1})| = 1$ and $E(T_i) ∩ E(T_j ) = ∅$ if $j > i + 1.$ Two edges $e,$ $e′ ∈ E(G)$ are triangularly connected if there is a triangle-path $T_1, T_2, · · · , T_k$ in $G$ such that $e ∈ E(T_1)$ and $e ′ ∈ E(T_k).$ Two edges $e,$ $e′ ∈ E(G)$ are equivalent if they are the same, parallel or triangularly connected. It is easy to see that this is an equivalent relation. Each equivalent class is called a triangularly connected component. In this paper, we prove that every 4-edge-connected triangular graph $G$ is $\mathbb{Z}_3$-connected, unless it has a triangularly connected component which is not $\mathbb{Z}_3$-connected but admits a nowhere-zero 3-flow.