On Initial Boundary Value Problem for Parabolic Differential Operator with Non-coercive Boundary Conditions

We consider initial boundary value problem for uniformly 2-parabolic differential operator of second order in cylinder domain in Rn with non-coercive boundary conditions. In this case there is a loss of smoothness of the solution in Sobolev type spaces compared with the coercive situation. Using by Faedo-Galerkin method we prove that problem has unique solution in special Bochner space

Error analysis and uncertainty quantification for the heterogeneous transport equation in slab geometry

We present an analysis of multilevel Monte Carlo (MLMC) techniques for the forward problem of uncertainty quantification for the radiative transport equation, when the coefficients (cross-sections) are heterogenous random fields. To do this we first give a new error analysis for the combined spatial and angular discretisation in the deterministic case, with error estimates that are explicit in the coefficients (and allow for very low regularity and jumps). This detailed error analysis is done for the one-dimensional space–one-dimensional angle slab-geometry case with classical diamond differencing. Under reasonable assumptions on the statistics of the coefficients, we then prove an error estimate for the random problem in a suitable Bochner space. Because the problem is not self-adjoint, stability can only be proved under a path-dependent mesh resolution condition. This means that, while the Bochner space error estimate is of order $\mathcal{O}(h^{\eta })$ for some $\eta $ where $h$ is a (deterministically chosen) mesh diameter, smaller mesh sizes might be needed for some realisations. We also show that the expected cost for computing a typical quantity of interest remains of the same order as for a single sample. This leads to rigorous complexity estimates for Monte Carlo (MC) and MLMC: for particular linear solvers, the multilevel version gives up to two orders of magnitude improvement over MC. We provide numerical results supporting the theory.

On Howland time-independent formulation of CP-divisible quantum evolutions

We extend Howland time-independent formalism to the case of completely positive and trace preserving dynamics of finite-dimensional open quantum systems governed by periodic, time-dependent Lindbladian in Weak Coupling Limit, expanding our result from previous papers. We propose the Bochner space of periodic, square integrable matrix-valued functions, as well as its tensor product representation, as the generalized space of states within the time-independent formalism. We examine some densely defined operators on this space, together with their Fourier-like expansions and address some problems related to their convergence by employing general results on Banach space-valued Fourier series, such as the generalized Carleson–Hunt theorem. We formulate Markovian dynamics in the generalized space of states by constructing appropriate time-independent Lindbladian in standard (Lindblad–Gorini–Kossakowski–Sudarshan) form, as well as one-parameter semigroup of bounded evolution maps. We show their similarity with Markovian generators and dynamical maps defined on matrix space, i.e. the generator still possesses a standard form (extended by closed perturbation) and the resulting semigroup is also completely positive, trace preserving and a contraction.

Integration in Orlicz-Bochner Spaces

Let (Ω,Σ,μ) be a complete σ-finite measure space, φ be a Young function, and X and Y be Banach spaces. Let Lφ(X) denote the Orlicz-Bochner space, and Tφ∧ denote the finest Lebesgue topology on Lφ(X). We study the problem of integral representation of (Tφ∧,·Y)-continuous linear operators T:Lφ(X)→Y with respect to the representing operator-valued measures. The relationships between (Tφ∧,·Y)-continuous linear operators T:Lφ(X)→Y and the topological properties of their representing operator measures are established.

k-Smooth Points in Some Banach Spaces

We characterize thek-smooth points in some Banach spaces. We will deal with injective tensor product, the Bochner spaceL∞(μ,X)of (equivalence classes of)μ-essentially bounded measurableX-valued functions, and direct sums of Banach spaces.

Isometric Stability Property of Certain Banach Spaces

Let E be one of the spaces C(K) and L1, F be an arbitrary Banach space, p > 1, and (X, σ) be a space with a finite measure. We prove that E is isometric to a subspace of the Lebesgue-Bochner space LP(X; F) only if E is isometric to a subspace of F. Moreover, every isometry T from E into Lp(X; F) has the form Te(x) = h(x)U(x)e, e ∊ E, where h: X —> R is a measurable function and, for every x ∊ X, U(x) is an isometry from E to F