Ill-posedness of Naiver-Stokes equations and critical Besov-Morrey spaces

The blow up phenomenon in the first step of the classical Picard's scheme was proved in this paper. For certain initial spaces, Bourgain-Pavlovi\'c and Yoneda proved the ill-posedness of the Navier-Stokes equations by showing the norm inflation in certain solution spaces. But Chemin and Gallagher said the space $\dot{B}^{-1,\infty}_{\infty}$ seems to be optimal for some solution spaces best chosen. In this paper, we consider more general initial spaces than Bourgain-Pavlovi\'c and Yoneda did and establish ill-posedness result independent of the choice of solution space. Our result is a complement of the previous ill-posedness results on Navier-Stokes equations.Comment: 18 page

The canonical solution operator to d-bar restricted to Bergman spaces and spaces of entire functions

In this paper we obtain a necessary and sufficient condition for the canonical solution operator to $\overline \partial$ restricted to radial symmetric Bergman spaces to be a Hilbert-Schmidt operator. We also discuss compactness of the solution operator in spaces of entire functions in one variable. In the sequel we consider several examples and also treat the case of weighted spaces of entire functions in several variables.Comment: 16 page

A New Minimisation Principle for Poisson Equation Leading to a Flexible Finite Element Approach

We introduce a new minimisation principle for Poisson equation using two variables: the solution and the gradient of the solution. This principle allows us to use any conforming finite element spaces for both variables, where the finite element spaces do not need to satisfy a so-called inf-sup condition. A numerical example demonstrates the superiority of the approach

Hamilton's Ricci Flow on Finsler Spaces

Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, existence of solutions to the so called Hamilton Ricci flow on Finsler spaces is studied and a short time solution is found. To this end the Finslerian Ricci-DeTurck flow on Finsler spaces is defined and existence of its solution in short time is proved. Next, this solution is pulled back to determine a short time solution to the Hamilton Ricci flow on underlying Finsler space.Comment: 19 pages. arXiv admin note: text overlap with arXiv:math/0612069 by other author

Multiplication operators on non-commutative spaces

Boundedness and compactness properties of multiplication operators on quantum (non-commutative) function spaces are investigated. For endomorphic multiplication operators these properties can be characterized in the setting of quantum symmetric spaces. For non-endomorphic multiplication operators these properties can be completely characterized in the setting of quantum $L^p$-spaces and a partial solution obtained in the more general setting of quantum Orlicz spaces

Thermodynamics of non-extremal Kaluza-Klein multi-black holes in five dimensions

Using a solution-generating method, we derive an exact solution of the Einstein's field equations in five dimensions describing multi-black hole configurations. More specifically, this solution describes systems of non-extremal static black holes with Kaluza-Klein asymptotics. As expected, we find that, in general, there are conical singularities in-between the Kaluza-Klein black holes that cannot be completely eliminated. Notwithstanding the presence of these conical singularities, such solutions still exhibit interesting thermodynamical properties. By choosing an appropriate set of thermodynamic variables we show that the entropy of these objects still obeys the Bekenstein-Hawking law for spaces with Kaluza-Klein asymptotics. This extends the previously known thermodynamic description of asymptotically flat spaces with conical singularities to general spaces with Kaluza-Klein asymptotics with conical singularities. Finally, we obtain a charged generalization of this multi-black hole solution in the general Einstein-Maxwell-Dilaton theory and show how to recover the extremal multi-black hole solution as a particular case.Comment: 16 pages, 1 figure; v.2 added a new section and new reference

A solution to the problem of Lp−L^p-maximal regularity

We give a negative solution to the problem of the $L^p$-maximal regularity on various classes of Banach spaces including $L^q$-spaces with $1<q \neq 2<+\infty$.Comment: 9 page

Note on integrability of certain homogeneous Hamiltonian systems in 2D constant curvature spaces

We formulate the necessary conditions for the integrability of a certain family of Hamiltonian systems defined in the constant curvature two-dimensional spaces. Proposed form of potential can be considered as a counterpart of a homogeneous potential in flat spaces. Thanks to this property Hamilton equations admit, in a general case, a particular solution. Using this solution we derive necessary integrability conditions investigating differential Galois group of variational equations

On the Solution Existence of Nonconvex Quadratic Programming Problems in Hilbert Spaces

In this paper, we consider the quadratic programming problems under finitely many convex quadratic constraints in Hilbert spaces. By using the Legendre property of quadratic forms or the compactness of operators in the presentations of quadratic forms, we establish some sufficient conditions for the solution existence of the considered problems. As special cases, we obtain some existence solution results for the quadratic programming problems under linear constraints in Hilbert spaces

On the generation of groups of bounded linear operators on Fr\'{e}chet spaces

In this paper we present a general method for generation of uniformly continuous groups on abstract Fr\'{e}chet spaces (without appealing to spectral theory) and apply it to a such space of distributions, namely ${\mathscr F}L^{2}_{loc}(\mathbb{R}^{n})$, so that the linear evolution problem \begin{equation*} \left\{\begin{array}{l} u_{t} = a(D)u, t \in \mathbb{R} \\ u(0) = u_0 \end{array} \right. \end{equation*}always has a unique solution in such a space, for every pseudodifferential operator $a(D)$ with constant coefficients. We also provide necessary and sufficient conditions so that the spaces $L^{2}$ and ${\mathscr E}'$ are left invariant by this group; and we conclude that the solution of the heat equation on ${\mathscr F}L^{2}_{loc}(\mathbb{R}^{n})$ for all $t \in \mathbb{R}$ extends the standard solution on Hilbert spaces for $t \geqslant 0$