For the 2D Euler equation in vorticity formulation, we construct localized smooth solutions whose critical Sobolev norms become large in a short period of time, and solutions which initially belong to L∞ ∩ H but escapes H immediately for t > 0. Our main observation is that a localized chunk of vorticity bounded in L∞ ∩H with odd-odd symmetry is able to generate a hyperbolic flow with large velo...

We apply the complex interpolation method to prove that, given two spacesB p0,a;s0 , B (n) p1,b;s1 of n-tuples of operators in the p-Schatten class of a Hilbert space H , endowed with weighted norms associated to positive and invertible operators a and b of B(H) then, the curve of interpolation (B p0,a;s0 , B (n) p1,b;s1 )[t] of the pair is given by the space of n-tuples of operators in the pt-...

This paper deals with Lipschitz triangular norms (t-norms). A partial answer to an open problem of Alsina, Frank and Schweizer is given with regard to strict t-norms with smooth additive generators. A new notion of local Lipschitz property for arbitrary t-norms is introduced. Some remarkable examples of non-Lipschitz continuous ones are provided.

A new example of a non-zero quasi-nilpotent operator T with reflexive commutant is presented. Norms ‖T n‖ converge to zero arbitrarily fast. Let H be a complex separable Hilbert space and let B(H) denote the algebra of all continuous linear operator on H. If T ∈ B(H) then {T}′ = {A ∈ B(H) : AT = TA} is called the commutant of T . By a subspace we always mean a closed linear subspace. If A ⊂ B(H...

Diagonals of continuous t-norms are studied. The characterization of all functions being diagonals of continuous t-norms is given. To a given diagonal, the class of all continuous t-norms with this diagonal is characterized.

This third and last part of a series of position papers on triangular norms (for Parts I and II see [25, 26]) presents the representation of continuous Archimedean t-norms by means of additive generators, and the representation of continuous t-norms by means of ordinal sums with Archimedean summands, both with full proofs. Finally some consequences of these representation theorems in the contex...