We consider the steady Navier-Stokes equations in a 2-dimensional unbounded multiply connected domain Ω under the general outflow condition. Let T be a 2-dimensional straight channel R × (−1, 1). We suppose that Ω ∩ {x1 < 0} is bounded and that Ω ∩ {x1 > −1} = T ∩ {x1 > −1}. Let V be a Poiseuille flow in T and µ the flux of V . We look for a solution which tends to V as x1 → ∞. Assuming that the domain and the boundary data are symmetric with respect to the x1-axis, and that the axis intersects every component of the boundary, we have shown the existence of solutions if the flux is small (Morimoto-Fujita [8]). Some improvement will be reported in this note. We also show certain regularity and asymptotic properties of the solutions.
Let $L(H)$ denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space $H$ into itself. Given $A\in L(H)$, we define the elementary operator $\Delta _A\colon L(H)\longrightarrow L(H)$ by $\Delta _A(X)=AXA-X$. In this paper we study the class of operators $A\in L(H)$ which have the following property: $ATA=T$ implies $AT^{\ast }A=T^{\ast }$ for all trace class operators $T\in C_1(H)$. Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of $\Delta _A$ is closed under taking adjoints. We give a characterization and some basic results concerning generalized quasi-adjoints operators.
In the paper a new proof of Lemma 11 in the above-mentioned paper is given. Its original proof was based on Theorem 3 which has been shown to be incorrect.
In 2000, Figallo and Sanza introduced n × m-valued Lukasiewicz-Moisil algebras which are both particular cases of matrix Lukasiewicz algebras and a generalization of n-valued Lukasiewicz-Moisil algebras. Here we initiate an investigation into the class tLMn×m of tense n × m-valued Lukasiewicz-Moisil algebras (or tense LMn×m-algebras), namely n×m-valued Lukasiewicz-Moisil algebras endowed with two unary operations called tense operators. These algebras constitute a generalization of tense Lukasiewicz-Moisil algebras (or tense LMn-algebras). Our most important result is a representation theorem for tense LMn×m-algebras. Also, as a corollary of this theorem, we obtain the representation theorem given by Georgescu and Diaconescu in 2007, for tense LMn-algebras.