In this paper, we study the Cauchy problem for the 3D generalized Navier-Stokes-Boussinesq equations with fractional diffusion:
With the help of the smoothing effect of the fractional diffusion operator and a logarithmic estimate, we prove the global well-posedness for this system with α≥ 5/4. Moreover, the uniqueness and continuity of the solution with weaker initial data is based on Fourier localization technique. Our results extend ones on the 3D Navier-Stokes equations with fractional diffusion.
This work develops near-optimal controls for systems given by differential equations with wideband noise and random switching. The random switching is modeled by a continuous-time, time-inhomogeneous Markov chain. Under broad conditions, it is shown that there is an associated limit problem, which is a switching jump diffusion. Using near-optimal controls of the limit system, we then build controls for the original systems. It is shown that such constructed controls are nearly optimal.
Firstly, we use Nehari manifold and Mountain Pass Lemma to prove an existence result of positive solutions for a class of nonlocal elliptic system with Kirchhoff type. Then a multiplicity result is established by cohomological index of Fadell and Rabinowitz. We also consider the critical case and prove existence of positive least energy solution when the parameter β is sufficiently large.
The quasilinear hyperbolic equation with nonlinear damping is considered in this paper, a new asymptotic profile for the solution to the equation is obtained by suitably choosing the initial data of the corresponding parabolic equation, the convergence rates of the new profile are better than that obtained by Nishihara (1997, J. Differential Equations 137, 384–395) and H.-J. Zhao (2000, J. Differential Equations 167, 467–494).
An r-acyclic edge chromatic number of a graph G, denoted by a_{r}'(G), is the minimum number of colors used to produce an edge coloring of the graph such that adjacent edges receive different colors and every cycle C has at least min {|C|, r} colors. We prove that a_{r}'(G)≤(4r+ 1) Δ(G), when the girth of the graph G equals to max{50, Δ(G)} and 4 ≤ r ≤7. If we relax the restriction of the girth to max {220, Δ(G)}, the upper bound of a_{r}'(G) is not larger than (2r + 5) (G) with 4 ≤ r ≤ 10.
In this paper, bifurcation of small amplitude limit cycles from the degenerate equilibrium of a three-dimensional system is investigated. Firstly, the method to calculate the focal values at nilpotent critical point on center manifold is discussed. Then an example is studied, by computing the quasi-Lyapunov constants, the existence of at least 4 limit cycles on the center manifold is proved. In terms of degenerate singularity in high-dimensional systems, our work is new.
In this paper, we consider the following problem
where λ> 0 is a parameter, p = (N+2)/(N-2). We will prove that there exists a positive constant 0 < λ^{*} < +∞ such that (*) has a minimal positive solution for λ ∈ (0, λ^{*}), no solution for λ> λ^{*}, a unique solution for λ= λ^{*}. Furthermore, (*) possesses at least two positive solutions when λ ∈ (0, λ^{*}) and 3 ≤N ≤5. For N ≥6, under some monotonicity conditions of h we show that there exists a constant 0 < λ^{**} < λ^{*} such that problem (*) possesses a unique solution for λ ∈ (0, λ^{**}).
Frankl and Füredi in [1] conjectured that the r-graph with m edges formed by taking the first m sets in the colex ordering of N^{(r)} has the largest Lagrangian of all r-graphs with m edges. Denote this r-graph by C_{r,m} and the Lagrangian of a hypergraph by λ(G). In this paper, we first show that if ≤m < , G is a left-compressed 3-graph with m edges and on vertex set [t], the triple with minimum colex ordering in G^{c} is (t-2-i)(t-2)t, then λ(G) ≤ (C_{3,m}). As an implication, the conjecture of Frankl and Füredi is true for -6≤m ≤.
In this paper, a system of reaction-diffusion equations arising in a nutrient-phytoplankton popula-tions is investigated. The equations model a situation in which phytoplankton population is divided into two groups, namely susceptible phytoplankton and infected phytoplankton. A number of existence and non-existence results about the non-constant steady states of a reaction diffusion system are given. If the diffusion coefficient of the infected phytoplankton is treated as bifurcation parameter, non-constant positive steady-state solutions may bifurcate from the constant steady-state solution under some conditions.
This paper presents a method of constructing a mixed graph which can be used to analyze the causality for multivariate time series. We construct a partial correlation graph at first which is an undirected graph. For every undirected edge in the partial correlation graph, the measures of linear feedback between two time series can help us decide its direction, then we obtain the mixed graph. Using this method, we construct a mixed graph for futures sugar prices in Zhengzhou (ZF), spot sugar prices in Zhengzhou (ZS) and futures sugar prices in New York (NF). The result shows that there is a bi-directional causality between ZF and ZS, an unidirectional causality from NF to ZF, but no causality between NF and ZS.
We investigate the nonlinear Schrödinger equation iut+ u+|u|^{p-1}u = 0 with 1+ 4/N < p < 1+ 4/(N-2) (when N = 1, 2, 1 + 4/N< p < ∞) in energy space H^{1} and study the divergent property of infinite-variance and nonradial solutions. If M(u)^{(1-sc)/sc)}E(u) < M(Q)^{(1-sc)/sc)}E(Q) and ||u_{0}||^{(1-sc)/sc)}_{2}||∇u_{0}||_{2} > ||Q||^{(1-sc)/sc)}_{2}||∇Q||_{2}, then either u(t) blows up in finite forward time or u(t exists globally for positive time and there exists a time sequence t_{n} → +∞ such that ||∇u(t_{n})||_{2} → +∞. Here Q is the ground state solution of -(1-s_{c})Q+ΔQ+|Q|^{p-1}Q = 0. A similar result holds for negative time. This extend the result of the 3D cubic Schrödinger equation obtained by Holmer to the general mass-supercritical and energy-subcritical case.
Regime switching, which is described by a Markov chain, is introduced in a Markov copula model. We prove that the marginals (X,H^{i}), i = 1, 2, 3 of the Markov copula model (X,H) are still Markov processes and have martingale property. In this proposed model, a pricing formula of credit default swap (CDS) with bilateral counterparty risk is derived.
This paper is concerned with the large-time behavior of solutions to an initial-boundary-value problem for full compressible Navier-Stokes equations on the half line (0,∞), which is named impermeable wall problem. It is shown that the 3-rarefaction wave is stable under partially large initial perturbation if the adiabatic exponent γ is close to 1. Here partially large initial perturbation means that the perturbation of absolute temperature is small, while the perturbations of specific volume and velocity can be large. The proof is given by the elementary energy method.
In this paper, we focus on the vertex-fault-tolerant cycles embedding on enhanced hypercube, which is an attractive variant of hypercube and is obtained by adding some complementary edges from hypercube. Let F_{v} be the set of faulty vertices in the n-dimensional enhanced hypercube Q_{n,k} (1 ≤ k ≤ n-1). When |F_{v}| = 2, we showed that Q_{n,k}-F_{v} contains a fault-free cycle of every even length from 4 to 2^{n}-4 where n (n ≤ 3) and k have the same parity; and contains a fault-free cycle of every even length from 4 to 2^{n}-4, simultaneously, contains a cycle of every odd length from n-k + 2 to 2^{n}-3 where n (n ≥3) and k have the different parity. Furthermore, when |F_{v}| = f_{v} ≤ n-2, we proof that there exists the longest fault-free cycle, which is of even length 2^{n}-2f_{v} whether n (n ≥3) and k have the same parity or not; and there exists the longest fault-free cycle, which is of odd length 2^{n}-2f_{v}-1 in Q_{n,k}-F_{v} where n (n ≥3) and k have the different parity.
The purpose of this paper is to use a very recent three critical points theorem due to Bonanno and Marano to establish the existence of at least three solutions for the quasilinear second order differential equation on a compact interval [a, b] ⊂R
under appropriate hypotheses. We exhibit the existence of at least three (weak) solutions and, and the results are illustrated by examples.
Nearly orthogonal Latin squares are useful for conducting experiments eliminating heterogeneity in two directions and using different interventions each at each level. In this paper, some constructions of mutually nearly orthogonal Latin squares are provided. It is proved that there exist 3 MNOLS(2m) if and only if m≥ 3 and there exist 4 MNOLS(2m) if and only if m≥ 4 with some possible exceptions.
Recurrent events data and gap times between recurrent events are frequently encountered in many clinical and observational studies, and often more than one type of recurrent events is of interest. In this paper, we consider a proportional hazards model for multiple type recurrent gap times data to assess the effect of covariates on the censored event processes of interest. An estimating equation approach is used to obtain the estimators of regression coefficients and baseline cumulative hazard functions. We examine asymptotic properties of the proposed estimators. Finite sample properties of these estimators are demonstrated by simulations.
In this paper we study the existence of infinitely many periodic solutions for second-order Hamiltonian systems
where F(t, u) is even in u, and ∇F(t, u) is of sublinear growth at infinity and satisfies the Ahmad-Lazer-Paul condition.