ORCID Profile
0000-0002-3522-9580
Current Organisations
University of Wrocław
,
Macquarie University
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Publisher: Springer Science and Business Media LLC
Date: 09-09-2019
DOI: 10.1007/S00021-019-0456-Z
Abstract: We study 2d vortex sheets with unbounded support. First we show a version of the Biot–Savart law related to a class of objects including such vortex sheets. Next, we give a formula associating the kinetic energy of a very general class of flows with certain moments of their vorticities. It allows us to identify a class of vortex sheets of unbounded support being only $$\\sigma $$ σ -finite measures (in particular including measures $$\\omega $$ ω such that $$\\omega (\\mathbb {R}^2)=\\infty $$ ω ( R 2 ) = ∞ ), but with locally finite kinetic energy. One of such ex les are celebrated Kaden approximations. We study them in details. In particular our estimates allow us to show that the kinetic energy of Kaden approximations in the neighbourhood of an origin is dissipated, actually we show that the energy is pushed out of any ball centered in the origin of the Kaden spiral. The latter result can be interpreted as an artificial viscosity in the center of a spiral.
Publisher: Springer Science and Business Media LLC
Date: 02-2016
Publisher: Springer Science and Business Media LLC
Date: 05-11-2018
Publisher: Elsevier BV
Date: 12-2012
Publisher: Springer Science and Business Media LLC
Date: 14-03-2013
Publisher: International Press of Boston
Date: 10-2010
Publisher: Springer Science and Business Media LLC
Date: 28-03-2020
DOI: 10.1007/S13163-020-00354-Y
Abstract: We consider a nonnegative self-adjoint operator L on $$L^2(X)$$ L 2 ( X ) , where $$X\subseteq {{\mathbb {R}}}^d$$ X ⊆ R d . Under certain assumptions, we prove atomic characterizations of the Hardy space $$\begin{aligned} H^1(L) = \left\{ f\in L^1(X) \ : \ \left\| \sup _{t } \left| \exp (-tL)f \right| \right\| _{L^1(X)} \infty \right\} . \end{aligned}$$ H 1 ( L ) = f ∈ L 1 ( X ) : sup t 0 exp ( - t L ) f L 1 ( X ) ∞ . We state simple conditions, such that $$H^1(L)$$ H 1 ( L ) is characterized by atoms being either the classical atoms on $$X\subseteq {\mathbb {R}^d}$$ X ⊆ R d or local atoms of the form $$|Q|^{-1}\chi _Q$$ | Q | - 1 χ Q , where $$Q\subseteq X$$ Q ⊆ X is a cube (or cuboid). One of our main motivation is to study multidimensional operators related to orthogonal expansions. We prove that if two operators $$L_1, L_2$$ L 1 , L 2 satisfy the assumptions of our theorem, then the sum $$L_1 + L_2$$ L 1 + L 2 also does. As a consequence, we give atomic characterizations for multidimensional Bessel, Laguerre, and Schrödinger operators. As a by-product, under the same assumptions, we characterize $$H^1(L)$$ H 1 ( L ) also by the maximal operator related to the subordinate semigroup $$\exp (-tL^\nu )$$ exp ( - t L ν ) , where $$\nu \in (0,1)$$ ν ∈ ( 0 , 1 ) .
Publisher: Springer Science and Business Media LLC
Date: 19-02-2019
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2017
Publisher: Elsevier BV
Date: 02-2012
Publisher: American Mathematical Society (AMS)
Date: 13-07-2022
DOI: 10.1090/TRAN/8695
Abstract: We investigate the Hardy space H L 1 H^1_L associated with a self-adjoint operator L L defined in a general setting by Hofmann, Lu, Mitrea, Mitrea, and Yan [Mem. Amer. Math. Soc. 214 (2011), pp. vi+78]. We assume that there exists an L L -harmonic non-negative function h h such that the semigroup exp ( − t L ) \\exp (-tL) , after applying the Doob transform related to h h , satisfies the upper and lower Gaussian estimates. Under this assumption we describe an illuminating characterisation of the Hardy space H L 1 H^1_L in terms of a simple atomic decomposition associated with the L L -harmonic function h h . Our approach also yields a natural characterisation of the B M O BMO -type space corresponding to the operator L L and dual to H L 1 H^1_L in the same circumstances. The applications include surprisingly wide range of operators, such as: Laplace operators with Dirichlet boundary conditions on some domains in R n {\\mathbb {R}^n} , Schrödinger operators with certain potentials, and Bessel operators.
Publisher: Wiley
Date: 09-02-2018
Publisher: Springer Science and Business Media LLC
Date: 11-2009
Publisher: Springer Science and Business Media LLC
Date: 27-07-2011
No related grants have been discovered for Marcin Preisner.