Nonlinear Volterra delay evolution inclusions subjected to nonlocal initial conditions

This paper deals with a nonlinear Volterra delay evolution inclusion subjected to a nonlocal implicit initial condition. The evolution inclusion involves an $m$-dissipative operator (possibly multivalued and/or nonlinear) and a noncompact interval. We first consider the evolution inclusion subjected to a local initial condition and prove an existence result for bounded $C^0$-solutions. Then, using a fixed point theorem for upper semicontinuous multifunctions with contractible values, we obtain a global solvability result for the original problem. Finally, we present an example to illustrate the abstract result.

The rate of convergence on fractional power dissipative operator on some sobolev type spaces

In [3], Chen, Deng, Ding and Fan proved that the fractional power dissipative operator is bounded on Lebesgue spaces Lp(ℝn), Hardy spaces Hp(ℝn) and general mixed norm spaces, which implies almost everywhere convergence of such operator. In this paper, we study the rate of convergence on fractional power dissipative operator on some sobolev type spaces.

The rate of convergence on fractional power dissipative operator on compact manifolds

On a compact connected manifold M $\mathbb{M}$ , we concern the fractional power dissipative operator e − t L α $e^{-t\left\vert \mathcal{L}\right\vert ^{\alpha}}$ , and obtain the almost-everywhere convergence rate (as t → 0+) of e − t L α f $e^{-t\left\vert \mathcal{L}\right\vert ^{\alpha}}\left( f\right)$ when f is in some Sobolev type Hardy spaces. The main result is a non-trivial extension of a recent result on ℝ n by Cao and Wang in 2.

Extensions of dissipative operators with closable imaginary part

Given a dissipative operator \(A\) on a complex Hilbert space \(\mathcal{H}\) such that the quadratic form \(f \mapsto \text{Im}\langle f, Af \rangle\) is closable, we give a necessary and sufficient condition for an extension of \(A\) to still be dissipative. As applications, we describe all maximally accretive extensions of strictly positive symmetric operators and all maximally dissipative extensions of a highly singular first-order operator on the interval.

Inviscid limit for the damped generalized incompressible Navier-Stokes equations on $ \mathbb{T}^2 $

<p style='text-indent:20px;'>In this paper, for the damped generalized incompressible Navier-Stokes equations on <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{T}^{2} $\end{document}</tex-math></inline-formula> as the index <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> of the general dissipative operator <inline-formula><tex-math id="M4">\begin{document}$ (-\Delta)^{\alpha} $\end{document}</tex-math></inline-formula> belongs to <inline-formula><tex-math id="M5">\begin{document}$ (0,\frac{1}{2}] $\end{document}</tex-math></inline-formula>, we prove the absence of anomalous dissipation of the long time averages of entropy. We also give a note to show that, by using the <inline-formula><tex-math id="M6">\begin{document}$ L^{\infty} $\end{document}</tex-math></inline-formula> bounds given in Caffarelli et al. [<xref ref-type="bibr" rid="b4">4</xref>], the absence of anomalous dissipation of the long time averages of energy for the forced SQG equations established in Constantin et al. [<xref ref-type="bibr" rid="b12">12</xref>] still holds under a slightly weaker conditions <inline-formula><tex-math id="M7">\begin{document}$ \theta_{0}\in L^{1}(\mathbb{R}^{2})\cap L^{2}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ f \in L^{1}(\mathbb{R}^{2})\cap L^{p}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula> with some <inline-formula><tex-math id="M9">\begin{document}$ p&gt;2 $\end{document}</tex-math></inline-formula>.</p>

Dilatations of Linear Operators

The article is devoted to building various dilatations of linear operators. The explicit construction of a unitary dilation of a compression operator is considered. Then the J -unitary dilatation of a bounded operator is constructed by means of the operator knot concept of a bounded linear operator. Using the Pavlov method, we construct the self-adjoint dilatation of a bounded dissipative operator. We consider spectral and translational representations of the self-adjoint dilatation of a densely defined dissipative operator with nonempty set of regular points. Using the concept of an operator knot for a bounded operator and the Cayley transform, we introduce an operator knot for a linear operator. By means of this concept, we construct the J -self-adjoint dilatation of a densely defined operator with a regular point. We obtain conditions of isomorphism of extraneous dilations and their minimality.

Dissipative Dirac operator with general boundary conditions on time scales

UDC 517.9 In this paper, we consider the symmetric Dirac operator on bounded time scales. With general boundary conditions, we describe extensions (dissipative, accumulative, self-adjoint and the other) of such symmetric operators. We construct a self-adjoint dilation of dissipative operator. Hence, we determine the scattering matrix of dilation. Later, we construct a functional model of this operator and define its characteristic function. Finally, we prove that all root vectors of this operator are complete.