Global solution of the 3‐D incompressible Navier‐Stokes equations in the Besov spaces B˙r1,r2,r3σ,1

In this paper, we construct a more general Besov spaces ${\dot{B}}_{r_1,r_2,r_3}^{\sigma ,q}$ and consider the global well‐posedness of incompressible Navier‐Stokes equations with small data in ${\dot{B}}_{r_1,r_2,r_3}^{\sigma ,\infty }$ for $\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}?\sigma =1,1?r_i<\infty$. In particular, we show that for any $\frac{2}{\gamma }+\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}?s=1,1<\gamma <\infty$ and $r_i?p_i<\infty$, the solution with initial data in ${\dot{B}}_{r_1,r_2,r_3}^{\sigma ,\infty }$ belong to , which, as far as we know, has not been discussed in other papers. Moreover, the smoothing effect of the solution to Navier‐Stokes equations is proved, which may have its own interest.     

Unique weak solutions of the d‐dimensional micropolar equation with fractional dissipation

This article examines the existence and uniqueness of weak solutions to the d‐dimensional micropolar equations (d=2 or d=3) with general fractional dissipation (?Δ)^αu and (?Δ)^βw. The micropolar equations with standard Laplacian dissipation model fluids with microstructure. The generalization to include fractional dissipation allows simultaneous study of a family of equations and is relevant in some physical circumstances. We establish that, when $\alpha \ge \frac{1}{2}$ and $\beta \ge \frac{1}{2}$, any initial data (u₀,w₀) in the critical Besov space $u_0\in B_{2,1}^{1+\frac{d}{2}?2\alpha }({?}^d)$ and $w_0\in B_{2,1}^{1+\frac{d}{2}?2\beta }({?}^d)$ yields a unique weak solution. For α ≥ 1 and β=0, any initial data $u_0\in B_{2,1}^{1+\frac{d}{2}?2\alpha }({?}^d)$ and $w_0\in B_{2,1}^{\frac{d}{2}}({?}^d)$ also leads to a unique weak solution as well. The regularity indices in these Besov spaces appear to be optimal and can not be lowered in order to achieve the uniqueness. Especially, the 2D micropolar equations with the standard Laplacian dissipation, namely, α=β=1, have a unique weak solution for $(u_0,w_0)\in B_{2,1}^0$. The proof involves the construction of successive approximation sequences and extensive a priori estimates in Besov space settings.     

Large‐time behavior of solutions to a 1D liquid crystal system

In this paper, we prove the large‐time behavior, as time tends to infinity, of solutions in $H^i\times H_0^i\times H^{i+1}(i=1,2)$ and $H^4\times H_0^4\times H^4$ for a system modeling the nematic liquid crystal flow, which consists of a subsystem of the compressible Navier‐Stokes equations coupling with a subsystem including a heat flow equation for harmonic maps.     

Blow‐up for wave equation with the scale‐invariant damping and combined nonlinearities

In this article, we study the blow‐up of the damped wave equation in the scale‐invariant case and in the presence of two nonlinearities. More precisely, we consider the following equation: $$u_{tt}?\mathrm{\Delta }u+\frac{\mu }{1+t}{u}_t=|u_t{|}^p+|u{|}^q,\text{in}{?}^N\times [0,\infty ),$$ with small initial data. For $\mu <\frac{N(q?1)}{2}$ and μ ∈ (0, μ_?) , where μ_? > 0 is depending on the nonlinearties' powers and the space dimension (μ_? satisfies $(q?1)\left((N+2{\mu }_{?}?1)p?2\right)=4$), we prove that the wave equation, in this case, behaves like the one without dissipation (μ = 0 ). Our result completes the previous studies in the case where the dissipation is given by $\frac{\mu }{{(1+t)}^{\beta }}{u}_t;\beta >1$, where, contrary to what we obtain in the present work, the effect of the damping is not significant in the dynamics. Interestingly, in our case, the influence of the damping term $\frac{\mu }{1+t}{u}_t$ is important.     

Global dynamics of a 3 × 6 exponential system of difference equations

We study the global dynamics of following 3×6 exponential system of difference equations: $$x_{n+1}=\frac{{\alpha }_1+{\beta }_1{e}^{?x_n}}{{\gamma }_1+y_{n?1}},y_{n+1}=\frac{{\alpha }_2+{\beta }_2{e}^{?y_n}}{{\gamma }_2+z_{n?1}},z_{n+1}=\frac{{\alpha }_3+{\beta }_3{e}^{?z_n}}{{\gamma }_3+x_{n?1}},n=0,1,\text{?},$$ where parameters α_i,β_i,γ_i (i=1,2,3) and initial conditions x_i,y_i,z_i (i=0,?1) are nonnegative real numbers. The proposed work is considerably extended and improve some existing results in the literature. Finally, theoretical results are verified by numerical simulations.     

A note on the finite fractal dimension of the global attractors for dissipative nonlinear Schrödinger‐type equations

In this article, we present, throughout two basic models of damped nonlinear Schrödinger (NLS)–type equations, a new idea to bound from above the fractal dimension of the global attractors for NLS‐type equations. This could answer the following open issue: consider, for instance, the classical one‐dimensional cubic nonlinear Schrödinger equation $$u_t+i{u}_{xx}+i|u{|}^{2}u+\gamma u=f,f\in {\mathrm{??}}^2(?).$$ “How can we bound the fractal dimension of the associate global attractor without the need to assume that the external forcing term f has some decay at infinity (that is belonging to some weighted Lebesgue space)?”     

Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions of 2N‐point type

This paper is devoted to the study of a nonlinear wave equation with initial conditions and nonlocal boundary conditions of 2N‐point type, which connect the values of an unknown function u(x,t) at x = 1, x = 0, x = η_i(t) , and x = θ_i(t), where $0<{\eta }_1(t)<{\eta }_2(t)<\dots <{\eta }_{{}_{N?1}}(t)<1,$ $0<{\theta }_1(t)<{\theta }_2(t)<\dots <{\theta }_{{}_{N?1}}(t)<1,$ for all t ≥ 0. First, we prove local existence of a unique weak solution by using density arguments and applying the Banach's contraction principle. Next, under the suitable conditions, we show that the problem considered has a unique global solution u(t) with energy decaying exponentially as t → +∞ . Finally, we present numerical results.     

Norm of a multilinear integral operator from product of weighted-type spaces to weighted-type space

We investigate the following multilinear integral operator $$T_K^m(\overrightarrow{f})(x)={\int }_0^{\infty }\dots {\int }_0^{\infty }K(x,t_1,\dots ,t_m){\displaystyle \prod _{j=1}^m}{f}_j(t_j)d{t}_1\dots d{t}_m,$$ where $m\in ?$ and $K:{?}_{+}^{m+1}\to {?}_{+}$ is a continuous kernel function satisfying the condition $$K(x,g_1(x)s_1,\dots ,g_m(x)s_m)=h(x)K(1,s_1,\dots ,s_m),$$ for some functions $g_j,j=\stackrel{￣}{1,m}$, which are continuous, increasing, $g_j({?}_{+})={?}_{+},j=\stackrel{￣}{1,m}$, and a function $h:{?}_{+}\to {?}_{+}$, from a product of weighted-type spaces to weighted-type spaces of real functions. We calculate the norm of the operator, extending and complementing some results in the literature. We also give an explanation for a relation between integrals of an L^p integrable function and its radialization on ${?}^n$.     

On infinite‐dimensional Banach spaces and weak forms of the axiom of choice

We study theorems from Functional Analysis with regard to their relationship with various weak choice principles and prove several results about them: “Every infinite‐dimensional Banach space has a well‐orderable Hamel basis” is equivalent to $\mathsf{AC}$; “ $\mathbb{R}$ can be well‐ordered” implies “no infinite‐dimensional Banach space has a Hamel basis of cardinality $<2^{{\aleph }_0}$”, thus the latter statement is true in every Fraenkel‐Mostowski model of $\mathsf{ZFA}$; “No infinite‐dimensional Banach space has a Hamel basis of cardinality $<2^{{\aleph }_0}$” is not provable in $\mathsf{ZF}$; “No infinite‐dimensional Banach space has a well‐orderable Hamel basis of cardinality $<2^{{\aleph }_0}$” is provable in $\mathsf{ZF}$; ${\mathsf{AC}}_{\mathsf{fin}}^{{\aleph }_{\mathsf{0}}}$ (the Axiom of Choice for denumerable families of non‐empty finite sets) is equivalent to “no infinite‐dimensional Banach space has a Hamel basis which can be written as a denumerable union of finite sets”; Mazur's Lemma (“If X is an infinite‐dimensional Banach space, Y is a finite‐dimensional vector subspace of X , and $\epsilon >0$, then there is a unit vector $x\in X$ such that $||y||\le (1+\epsilon )||y+\alpha x||$ for all $y\in Y$ and all scalars α”) is provable in $\mathsf{ZF}$; “A real normed vector space X is finite‐dimensional if and only if its closed unit ball $B_X=\{x\in X:||x||\le 1\}$ is compact” is provable in $\mathsf{ZF}$; $\mathsf{DC}$ (Principle of Dependent Choices) + “ $\mathbb{R}$ can be well‐ordered” does not imply the Hahn‐Banach Theorem ( $\mathsf{HB}$) in $\mathsf{ZF}$; $\mathsf{HB}$ and “no infinite‐dimensional Banach space has a Hamel basis of cardinality $<2^{{\aleph }_0}$” are independent from each other in $\mathsf{ZF}$; “No infinite‐dimensional Banach space can be written as a denumerable union of finite‐dimensional subspaces” lies in strength between ${\mathsf{AC}}^{{\aleph }_0}$ (the Axiom of Countable Choice) and ${\mathsf{AC}}_{\mathsf{fin}}^{{\aleph }_{\mathsf{0}}}$; $\mathsf{DC}$ implies “No infinite‐dimensional Banach space can be written as a denumerable union of closed proper subspaces” which in turn implies ${\mathsf{AC}}^{{\aleph }_0}$; “Every infinite‐dimensional Banach space has a denumerable linearly independent subset” is a theorem of $\mathsf{ZF}+{\mathsf{AC}}^{{\aleph }_0}$, but not a theorem of $\mathsf{ZF}$; and “Every infinite‐dimensional Banach space has a linearly independent subset of cardinality $\ge 2^{{\aleph }_0}$” implies “every Dedekind‐finite set is finite”.     

Two solutions for a class of singular Kirchhoff‐type problems with Hardy–Sobolev critical exponent II

In this article, we devote ourselves to investigate the following singular Kirchhoff‐type equation: $$\left\{\begin{array}{l}?\left(a+b{\int }_{\mathrm{\Omega }}|?u{|}^{2}dx\right)\mathrm{\Delta }u=\frac{u^{5?2s}}{|x{|}^s}+\frac{\lambda }{|x{|}^{\beta }{u}^{\gamma }},x\in \mathrm{\Omega },\\ u>0,x\in \mathrm{\Omega },\\ u=0,x\in ?\mathrm{\Omega },\end{array}\right.$$ where $\mathrm{\Omega }?{?}^3$ is a bounded domain with smooth boundary ?Ω,0∈Ω,a≥0,b,λ>0,0<γ,s<1, and $0\le \beta <\frac{5+\gamma }{2}.$ By using the variational and perturbation methods, we obtain the existence of two positive solutions, which generalizes and improves the recent results in the literature.     

Legendre G‐array pairs and the theoretical unification of several G‐array families

We investigate how Legendre $G$‐array pairs are related to several different perfect binary $G$‐array families. In particular we study the relations between Legendre $G$‐array pairs, Sidelnikov‐Lempel‐Cohn‐Eastman ${\mathbb{Z}}_{q?1}$‐arrays, Yamada‐Pott $G$‐array pairs, Ding‐Helleseth‐Martinsen ${\mathbb{Z}}_2\times {\mathbb{Z}}_p^m$‐arrays, Yamada ${\mathbb{Z}}_{\left(q?1\right)∕2}$‐arrays, Szekeres ${\mathbb{Z}}_p^m$‐array pairs, Paley ${\mathbb{Z}}_p^m$‐array pairs, and Baumert ${\mathbb{Z}}_{p_1}^{m_1}\times {\mathbb{Z}}_{p_2}^{m_2}$‐array pairs. Our work also solves one of the two open problems posed by Ding. Moreover, we provide several computer search‐based existence and nonexistence results regarding Legendre ${\mathbb{Z}}_n$‐array pairs. Finally, by using cyclotomic cosets, we provide a previously unknown Legendre ${\mathbb{Z}}_{57}$‐array pair.     

A critical Kirchhoff‐type problem driven by a p (·)‐fractional Laplace operator with variable s (·) ‐order

The paper deals with the following Kirchhoff‐type problem $$\left\{\begin{array}{lll}& M\left({?}_{{?}^{2N}}\frac{1}{p(x,y)}\frac{|v(x)?v(y){|}^{p(x,y)}}{|x?y{|}^{N+p(x,y)s(x,y)}}dxdy\right){(?\mathrm{\Delta })}_{p(·)}^{s(·)}v(x)=\mu g(x,v)+|v{|}^{r(x)?2}v& \text{in}\mathrm{\Omega },\\ & v=0& \text{in}{?}^N\backslash \mathrm{\Omega },\end{array}\right.$$ where M models a Kirchhoff coefficient, ${(?\mathrm{\Delta })}_{p(·)}^{s(·)}$ is a variable s(·) ‐order p(·) ‐fractional Laplace operator, with $s(·):{?}^{2N}\to (0,1)$ and $p(·):{?}^{2N}\to (1,\infty )$. Here, $\mathrm{\Omega }?{?}^N$ is a bounded smooth domain with N > p(x, y)s(x, y) for any $(x,y)\in \overline{\mathrm{\Omega }}\times \overline{\mathrm{\Omega }}$, μ is a positive parameter, g is a continuous and subcritical function, while variable exponent r(x) could be close to the critical exponent $p_s^{?}(x)=N\overline{p}(x)/(N?\overline{s}(x)\overline{p}(x))$, given with $\overline{p}(x)=p(x,x)$ and $\overline{s}(x)=s(x,x)$ for $x\in \overline{\mathrm{\Omega }}$. We prove the existence and asymptotic behavior of at least one non‐trivial solution. For this, we exploit a suitable tricky step analysis of the critical mountain pass level, combined with a Brézis and Lieb‐type lemma for fractional Sobolev spaces with variable order and variable exponent.     

Decay rates for the energy of a singular nonlocal viscoelastic system

This work deals with decay rates for the energy of an initial boundary value problem with a nonlocal boundary condition for a system of nonlinear singular viscoelastic equations. We prove the decay rates for the energy of a singular one‐dimensional viscoelastic system with a nonlinear source term and nonlocal boundary condition of relaxation kernels described by the inequality $g_i^{\prime }\left(t\right)\le ?H\left(g_i\left(t\right)\right),\left(i=1,2\right)$ for all t ≥ 0, with H convex.     

A note on 3‐partite graphs without 4‐cycles

Let $C_4$ be a cycle of order 4. Write $ex\left(n,n,n,C_4\right)$ for the maximum number of edges in a balanced 3‐partite graph whose vertex set consists of three parts, each has $n$ vertices that have no subgraph isomorphic to $C_4$. In this paper, we show that $ex\left(n,n,n,C_4\right)\ge \frac{3}{2}n\left(p+1\right)$, where $n=\frac{p\left(p?1\right)}{2}$ and $p$ is a prime number. Note that $ex\left(n,n,n,C_4\right)\le \left(\frac{3\sqrt{2}}{2}+o\left(1\right)\right)n^{\frac{3}{2}}$ from Tait and Timmons's works. Since for every integer $m$, one can find a prime $p$ such that $m\le p\le \left(1+o\left(1\right)\right)m$, we obtain that $\underset{n\to \infty }{\lim }\frac{ex\left(n,n,n,C_4\right)}{\frac{3\sqrt{2}}{2}{n}^{\frac{3}{2}}}=1$.