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 ${?}^n$.     

The local Borg–Marchenko uniqueness theorem of matrix-valued Dirac-type operators for coefficients locally smooth at the right endpoint

We provide an expression of the Weyl–Titchmarsh matrix associated with a self-adjoint matrix-valued Dirac-type operator defined on [0, b) for . As a concrete application of it, we establish asymptotics of the difference $(M_{+}-{\stackrel{\sim }{M}}_{+})(z)$ of two Weyl–Titchmarsh matrices $M_{+},{\stackrel{\sim }{M}}_{+}$ corresponding to two Dirac-type operators $L=Jd/dx-B(x)$, $\stackrel{\sim }{L}=Jd/dx-\stackrel{\sim }{B}(x)$ with $B=\stackrel{\sim }{B}$ a.e. on [0, a], in $L_2{(0,b)}^{2m}$ with , when the $B(x),\stackrel{\sim }{B}(x)$ are sufficiently smooth in a right neighborhood of the point a and their right derivatives at a coincide up to a certain order. In addition to this, we also provide new proofs of the local Borg–Marchenko theorem and asymptotic high-energy expansion of Weyl–Titchmarsh matrices associated with Dirac-type operators.     

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.     

Complete latticeability in vector-valued sequence spaces

Using a technique due to Jiménez–Rodríguez, first we prove the complete latticeability of the set of disjoint non-norm null weakly null sequences and of the set of disjoint non-norm null regular-polynomially null sequences in Banach lattices. Then we apply the mother vector technique to prove the complete latticeability of ${\lambda }_{\pi }(E)\setminus {\lambda }_s(E)$, which implies the complete latticeability of $({\ell }_p{\widehat{\otimes }}_{|\pi |}E)\setminus ({\ell }_p{\widehat{\otimes }}_{\pi }E)$, where E is a Banach lattice and .     

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.     

Extensions of valuations to rational function fields over completions

Given a valued field $(K,v)$ and its completion $(\widehat{K},v)$, we study the set of all possible extensions of v to $\widehat{K}(X)$. We show that any such extension is closely connected with the underlying subextension $(K(X)|K,v)$. The connections between these extensions are studied via minimal pairs, key polynomials, pseudo-Cauchy sequences, and implicit constant fields. As a consequence, we obtain strong ramification theoretic properties of $(\widehat{K},v)$. We also give necessary and sufficient conditions for $(K(X),v)$ to be dense in $(\widehat{K}(X),v)$.     

Regularity via one vorticity component for the 3D axisymmetric MHD equations

In this paper, we investigate the regularity criteria of axisymmetric weak solutions to the three-dimensional (3D) incompressible magnetohydrodynamics (MHD) equations with nonzero swirl component. By making use of techniques of the Littlewood–Paley decomposition, we show that weak solutions to the 3D axisymmetric MHD equations become regular if the swirl component of vorticity satisfies that $w_{\theta }{e}_{\theta }\in L^1(0,T;{\dot{B}}_{\infty ,\infty }^0)$, which partially gives a positive answer to the marginal case for the regularity of MHD equations.     

Non-uniqueness for a critical heat equation in two dimensions with singular data

Nonlinear heat equations in two dimensions with singular initial data are studied. In recent works nonlinearities with exponential growth of Trudinger-Moser type have been shown to manifest critical behavior: well-posedness in the subcritical case and non-existence for certain supercritical data. In this article we propose a specific model nonlinearity with Trudinger-Moser growth for which we obtain surprisingly complete results: a) for initial data strictly below a certain singular threshold function $\stackrel{?}{u}$ the problem is well-posed, b) for initial data above this threshold function $\stackrel{?}{u}$, there exists no solution, c) for the singular initial datum $\stackrel{?}{u}$ there is non-uniqueness. The function $\stackrel{?}{u}$ is a weak stationary singular solution of the problem, and we show that there exists also a regularizing classical solution with the same initial datum $\stackrel{?}{u}$.     

More about divisibility in βN

We continue the research of an extension $\stackrel{～}{\mid }$ of the divisibility relation to the Stone-?ech compactification $\beta \mathbb{N}$. First we prove that ultrafilters we call prime actually possess the algebraic property of primality. Several questions concerning the connection between divisibilities in $\beta \mathbb{N}$ and nonstandard extensions of $\mathbb{N}$ are answered, providing a few more equivalent conditions for divisibility in $\beta \mathbb{N}$. Results on uncountable chains in $(\beta \mathbb{N},\stackrel{～}{\mid })$ are proved and used in a construction of a well-ordered chain of maximal cardinality. Probably the most interesting result is the existence of a chain of type $(\mathbb{R},<)$ in $(\beta \mathbb{N},\stackrel{～}{\mid })$. Finally, we consider ultrafilters without divisors in $\mathbb{N}$ and among them find the greatest class.     

Even (s̄,t̄)-core partitions and self-associate characters of S̃n

We extend a method of Olsson and Bessenrodt to determine the number of even partitions that are simultaneously $\stackrel{?}{s}$-core and $\stackrel{?}{t}$-core. When $p$ and $q$ are distinct primes, this also determines the number of self-associate characters of ${\stackrel{?}{S}}_n$ that are simultaneously defect 0 for $p$ and $q$.     

On Yau's problem of evolving one curve to another: Convex case

A revised Yau's Curvature Difference Flow is considered to deform one convex curve $X_0$ to another one $\stackrel{?}{X}$. It is proved that this flow exists globally on time interval $[0,+\infty )$ and the evolving curve, preserving its convexity and bounded area A, converges to a fixed limiting curve $X_{\infty }$ (congruent to $\sqrt{A/\stackrel{?}{A}}\stackrel{?}{X}$) as time tends to infinity, where $\stackrel{?}{A}$ is the area bounded by the target curve $\stackrel{?}{X}$.     

A time-periodic reaction–diffusion–advection equation with a free boundary and sign-changing coefficients

We consider a reaction–diffusion–advection equation of the form: $u_t=u_{xx}-\beta \left(t\right)u_x+f(t,u)$ for $x\in [0,h\left(t\right))$, where $\beta \left(t\right)$ is a $T$-periodic function, $f(t,u)$ is a $T$-periodic Fisher–KPP type of nonlinearity with $a\left(t\right)≔f_u(t,0)$ changing sign, $h\left(t\right)$ is a free boundary satisfying the Stefan condition. We study the long time behavior of solutions and find that there are two critical numbers $\bar{c}$ and $B\left(\tilde{\beta }\right)$ with $B\left(\tilde{\beta }\right)>\bar{c}>0$, $\bar{\beta }≔\frac{1}{T}{\int }_0^T\beta \left(t\right)dt$ and $\tilde{\beta }\left(t\right)≔\beta \left(t\right)-\bar{\beta }$, such that a vanishing–spreading dichotomy result holds when $\left|\bar{\beta }\right|<\bar{c}$; a vanishing–transition–virtual spreading trichotomy result holds when $\bar{\beta }\in [\bar{c},B\left(\tilde{\beta }\right))$; all solutions vanish when $\bar{\beta }⩾B\left(\tilde{\beta }\right)$ or $\bar{\beta }⩽-\bar{c}$.