Bochner Spaces Research Articles

Mean asymptotic behavior for stochastic Kuramoto–Sivashinshy equation in Bochner spaces

This paper is concerned with the mean asymptotic behavior of the Kuramoto–Sivashinshy equation with stochastic perturbation. We define the mean random dynamical systems for the stochastic Kuramoto–Sivashinshy equation in Bochner spaces. Then we obtain the so-called weak pullback mean random attractor for the stochastic Kuramoto–Sivashinshy equation with odd initial conditions.

Tracking and blind deconvolution of blood alcohol concentration from transdermal alcohol biosensor data: A population model-based LQG approach in Hilbert space

LQG control in Hilbert space, a novel approach for random abstract parabolic systems, and new transdermal alcohol biosensor technology are combined to yield tracking controllers that can be used to automate inpatient management of alcohol withdrawal syndrome and human subject intravenous alcohol infusion studies, and to blindly deconvolve blood or breath alcohol concentration from biosensor measured transdermal alcohol level. The approach taken is based on a full-body alcohol population model in the form of a random, nonlinear, hybrid system of ordinary and partial differential equations and its abstract formulation in a Gelfand triple of Bochner spaces. The efficacy of the approach is demonstrated through simulation studies based on laboratory collected drinking data.

A population model-based linear-quadratic Gaussian compensator for the control of intravenously infused alcohol studies and withdrawal symptom prophylaxis using transdermal sensing.

An output feedback LQG compensator (combined controller and state estimator) for the regulation of intravenous-infused alcohol studies and treatment using a noninvasive transdermal alcohol biosensor is developed. The design is based on a population model involving an abstract semi-linear parabolic hybrid reaction-diffusion system involving coupled partial and ordinary differential equations with random parameters known only up to their distributions. The scheme developed is based on a weak formulation of the model equations in an appropriately constructed Gelfand triple of Bochner spaces wherein the unknown random parameters are treated as additional spatial variables. Implementation relies on a Galerkin-based approximation and convergence theory and an abstract formulation involving linear semigroups of operators. The model is fit and validated using laboratory collected human subject data and the method of moments. The results of numerical simulations of controlled intravenous alcohol infusion are presented and discussed.

Inverse image of precompact sets and regular solutions to the Navier–Stokes equations

We consider the initial value problem for the Navier–Stokes equations over ${\mathbb R}^3 \times [0,T]$ with time $T>0$ in the spatially periodic setting. We prove that it induces open injective mappings ${\mathcal A}_s\colon B^{s}_1 \to B^{s-1}_2$ where $B^{s}_1$, $B^{s-1}_2$ are elements from scales of specially constructed function spaces of Bochner–Sobolev type parametrized with the smoothness index $s \in \mathbb N$. Finally, we prove that a map ${\mathcal A}_s$ is surjective if and only if the inverse image ${\mathcal A}_s ^{-1}(K)$ of any precompact set $K$ from the range of the map ${\mathcal A}_s$ is bounded in the Bochner space $L^{\mathfrak s} ([0,T], L^{{\mathfrak r}} ({\mathbb T}^3))$ with the Ladyzhenskaya–Prodi–Serrin numbers ${\mathfrak s}$, ${\mathfrak r}$.

Dynamics of stochastic Ginzburg–Landau equations driven by nonlinear noise

This paper is concerned with the well-posedness as well as long-term dynamics of stochastic Ginzburg–Landau equations driven by nonlinear noise. We will apply a specific method to solve stochastic Ginzburg–Landau equations, known as the variational approach. We prove the existence and uniqueness of the solutions by assuming that the coefficients satisfy certain monotonicity assumptions. The mean random dynamical system generated by the solution operators is proved to possess a unique weak pullback mean random attractor in a Bochner space. At the same time, the existence of invariant measures for the stochastic Ginzburg–Landau equations is also established.

Weak pullback mean random attractors for the stochastic convective Brinkman–Forchheimer equations and locally monotone stochastic partial differential equations

This work is concerned about the asymptotic behavior of the solutions of the two- and three-dimensional stochastic convective Brinkman–Forchheimer (SCBF) equations [Formula: see text] [Formula: see text] driven by white noise with nonlinear diffusion terms (for some [Formula: see text]). We prove the existence and uniqueness of weak pullback mean random attractors for the 2D SCBF equations (for [Formula: see text]) as well as 3D SCBF equations (for [Formula: see text], any [Formula: see text] and for [Formula: see text], [Formula: see text]) in Bochner spaces, when the diffusion terms are Lipschitz nonlinear functions. Furthermore, we establish the existence of weak pullback mean random attractors for a class of locally monotone stochastic partial differential equations.

Weak pullback attractors for stochastic Ginzburg-Landau equations in Bochner spaces

<p style='text-indent:20px;'>In this paper we discuss the weak pullback mean random attractors for stochastic Ginzburg-Landau equations defined in Bochner spaces. We prove the existence and uniqueness of weak pullback mean random attractors for the stochastic Ginzburg-Landau equations with nonlinear diffusion terms. We also establish the existence and uniqueness of such attractors for the deterministic Ginzburg-Landau equations with random initial data. In this case, the periodicity of the weak pullback mean random attractors is also proved whenever the external forcing terms are periodic in time.</p>

Generalizations of non-linear best simultaneous approximation in Köthe function spaces

This paper is concerned with the problem of nonlinear best simultaneous approximations in K?the Bochner function spaces with respect to Minkowski? norms in Euclidean spaces. Characterization results of the generalized best simultaneous approximation are established. These results are considered a generalization of the results concerning the Lebesgue Bochner spaces and the Orlicz Bochner spaces.

Well-defined forward operators in dynamic diffractive tensor tomography using viscosity solutions of transport equations

We consider a general setting for dynamic tensor field tomography in an inhomogeneous refracting and absorbing medium as an inverse source problem for the associated transport equation. Following Fermat's principle, the Riemannian metric in the considered domain is generated by the refractive index of the medium. There is a wealth of results for the inverse problem of recovering a tensor field from its longitudinal ray transform in a static Euclidean setting, whereas there are only a few inversion formulas and algorithms existing for general Riemannian metrics and time-dependent tensor fields. It is a well-known fact that tensor field tomography is equivalent to an inverse source problem for a transport equation where the ray transform serves as given boundary data. We prove that this result extends to the dynamic case. Interpreting dynamic tensor tomography as an inverse source problem represents a holistic approach in this field. To guarantee that the forward mappings are well defined, it is necessary to prove existence and uniqueness for the underlying transport equations. Unfortunately, the bilinear forms of the associated weak formulations do not satisfy the coercivity condition. To this end we transfer to viscosity solutions and prove their unique existence in appropriate Sobolev (static case) and Sobolev–Bochner (dynamic case) spaces under a certain assumption that allows only small variations of the refractive index. Numerical evidence is given that the viscosity solution solves the original transport equation if the viscosity term turns to zero.

Stochastic integration with respect to canonical α-stable cylindrical Lévy processes

In this work, we introduce a theory of stochastic integration with respect to symmetric α-stable cylindrical Lévy processes. Since α-stable cylindrical Lévy processes do not enjoy a semi-martingale decomposition, our approach is based on a decoupling inequality for the tangent sequence of the Radonified increments. This approach enables us to characterise the largest space of predictable Hilbert-Schmidt operator-valued processes which are integrable with respect to an α-stable cylindrical Lévy process as the collection of all predictable processes with paths in the Bochner space Lα. We demonstrate the power and robustness of the developed theory by establishing a dominated convergence result allowing the interchange of the stochastic integral and limit.

Small combination of slices, dentability and stability results of small diameter properties in Banach spaces

In this work we study three different versions of small diameter properties of the unit ball in a Banach space and its dual. The related concepts for all closed bounded convex sets of a Banach space was initiated and developed in [9], [4], [12], [13] was extensively studied in the context of dentability, huskability, Radon Nikodym Property and Krein Milman Property in [14]. We introduce the Ball Huskable Property (BHP), namely, the unit ball has relatively weakly open subsets of arbitrarily small diameter. We compare this property to two related properties, BSCSP namely, the unit ball has convex combination of slices of arbitrarily small diameter and BDP namely, the closed unit ball has slices of arbitrarily small diameter. We show BDP implies BHP which in turn implies BSCSP and none of the implications can be reversed. We prove similar results for the w⁎-versions. We prove that all these properties are stable under lp sum for 1≤p≤∞,c0 sum and Lebesgue Bochner spaces. Finally, we explore the stability of these with properties in the light of three space property. We show that BHP is a three space property provided X/Y is finite dimensional and same is true for BSCSP when X has BSCSP and X/Y is strongly regular ([14]).

Weak pullback mean random attractors for stochastic evolution equations and applications

In this paper, we investigate the existence and uniqueness of weak pullback mean random attractors for abstract stochastic evolution equations with general diffusion terms in Bochner spaces. As applications, the existence and uniqueness of weak pullback mean random attractors for some stochastic models such as stochastic reaction–diffusion equations, the stochastic p-Laplace equation and stochastic porous media equations are established.

Dynamics and invariant measures of multi-stochastic sine-Gordon lattices with random viscosity and nonlinear noise

We investigate mean dynamics and invariant measures for a multi-stochastic discrete sine-Gordon equation driven by random viscosity, stochastic forces, and infinite-dimensional nonlinear noise. We first show the existence of a unique solution when the random viscosity is bounded and the nonlinear diffusion of noise is locally Lipschitz continuous, which leads to the existence of a mean random dynamical system. We then prove that such a mean random dynamical system possesses a unique weak pullback mean random attractor in the Bochner space. Finally, we show the existence of an invariant measure. Some difficulties arise from dealing with the term of random viscosity in all uniform estimates (including the tail-estimate) of solutions, which lead to the tightness of a family of distribution laws of solutions.

Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs

By combining a certain approximation property in the spatial domain, and weighted 𝓁2-summability of the Hermite polynomial expansion coefficients in the parametric domain obtained in Bachmayr et al. [ESAIM: M2AN 51 (2017) 341–363] and Bachmayr et al. [SIAM J. Numer. Anal. 55 (2017) 2151–2186], we investigate linear non-adaptive methods of fully discrete polynomial interpolation approximation as well as fully discrete weighted quadrature methods of integration for parametric and stochastic elliptic PDEs with lognormal inputs. We construct such methods and prove convergence rates of the approximations by them. The linear non-adaptive methods of fully discrete polynomial interpolation approximation are sparse-grid collocation methods which are certain sums taken over finite nested Smolyak-type indices sets of mixed tensor products of dyadic scale successive differences of spatial approximations of particular solvers, and of successive differences of their parametric Lagrange interpolating polynomials. The Smolyak-type sparse interpolation grids in the parametric domain are constructed from the roots of Hermite polynomials or their improved modifications. Moreover, they generate in a natural way fully discrete weighted quadrature formulas for integration of the solution to parametric and stochastic elliptic PDEs and its linear functionals, and the error of the corresponding integration can be estimated via the error in the Bochner space L1 (ℝ∞, V, γ) norm of the generating methods where γ is the Gaussian probability measure on ℝ∞ and V is the energy space. We also briefly consider similar problems for parametric and stochastic elliptic PDEs with affine inputs, and problems of non-fully discrete polynomial interpolation approximation and integration. In particular, the convergence rates of non-fully discrete polynomial interpolation approximation and integration obtained in this paper significantly improve the known ones.

Stability of diametral diameter two properties

We prove that the diametral diameter two properties are inherited by F-ideals (e.g., M-ideals). On the other hand, these properties are lifted from an M-ideal to the superspace under strong geometric assumptions. We also show that all of the diametral diameter two properties are stable under the formation of corresponding Köthe–Bochner spaces (e.g., \(L_p\)-Bochner spaces). Finally, we investigate when the projective tensor product of two Banach spaces has some diametral diameter two property.

Square Functions for Commuting Families of Ritt Operators

In this paper, we investigate the role of square functions defined for a d-tuple of commuting Ritt operators \((T_1,\ldots ,T_d)\) acting on a general Banach space X. Firstly, we prove that if the d-tuple admits a \(H^\infty \) joint functional calculus, then it verifies various square function estimates. Then we study the converse when every \(T_k\) is a R-Ritt operator. Under this last hypothesis, and when X is a K-convex space, we show that square function estimates yield dilation of \((T_1,\ldots ,T_d)\) on some Bochner space \(L_p(\Omega ;X)\) into a d-tuple of isomorphisms with a \(C(\mathbb {T}^d)\) bounded calculus. Finally, we compare for a d-tuple of Ritt operators its \(H^\infty \) joint functional calculus with its dilation into a d-tuple of polynomially bounded isomorphisms.

Solving an inverse heat convection problem with an implicit forward operator by using a projected quasi-Newton method

We consider the quasilinear 1D inverse heat convection problem of determining the enthalpy-dependent heat fluxes from noisy internal enthalpy measurements. This problem arises in the accelerated cooling process of producing thermomechanically controlled processed heavy plates made of steel. In order to adjust the complex microstructure of the underlying material, the Leidenfrost behavior of the hot surfaces with respect to the application of the cooling fluid has to be studied. Since the heat fluxes depend on the enthalpy and hence on the solution of the underlying initial boundary value problem (IBVP), the parameter-to-solution operator, and thus the forward operator of the inverse problem, can only be defined implicitly. To guarantee well-defined operators, we study two approaches for showing existence and uniqueness of solutions of the IBVP. One approach deals with the theory of pseudomonotone operators and so-called strong solutions in Sobolev–Bochner spaces. The other theory uses classical solutions in Hölder spaces. Whereas the first approach yields a solution under milder assumptions, it fails to show the uniqueness result in contrast to the second approach. Furthermore, we propose a convenient parametrization approach for the nonlinear heat fluxes in order to decouple the parameter-to-solution relation and use an iterative solver based on a projected quasi-Newton (PQN) method together with box-constraints to solve the inverse problem. For numerical experiments, we derive the necessary gradient information of the objective functional and use the discrepancy principle as a stopping rule. Numerical tests show that the PQN method outperforms the Landweber method with respect to computing time and approximation accuracy.

Dual of Köthe–Bochner Spaces and Radon–Nikodým Property

The Köthe–Bochner spaces Lρ(X) are the vector valued version of the scalar Köthe spaces Lρ, which generalize the Lebesgue spaces Lp, the Orlicz spaces and many other functional spaces. We study the possible identification between the dual (Lρ(X))′ and the associated space Lρ′(X′) in connection with the Radon–Nikodým property of X′.

On Orthogonally Additive Operators in Köthe–Bochner Spaces

In this article we consider some classes of orthogonally additive operators in Kothe–Bochner spaces in the setting of the theory of lattice-normed spaces and dominated operators. The our first main result asserts that the C-compactness of a dominated orthogonally additive operator $$S:E(X)\rightarrow F(Y)$$ implies the C-compactness of its exact dominant $$|\!|S|\!|:E\rightarrow F$$ . Then we show that a dominated orthogonally additive operator $$S:E(X)\rightarrow F$$ from a Kothe–Bochner space to a Banach lattice F with an order continuous norm is narrow if and only if its exact dominant $$|\!|S|\!|:E\rightarrow F$$ is. Finally we prove that every laterally-to-norm continuous dominated orthogonally additive operator from E(X) to a sequence Banach lattice F is narrow.

Space-Time Finite Element Discretization of Parabolic Optimal Control Problems with Energy Regularization

In this paper, we analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{-1}(\Omega))$. By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary value problem. When eliminating the control, we end up with the reduced optimality system that is nothing but the variational formulation of the coupled forward-backward primal and adjoint equations. Using Babuška's theorem, we prove unique solvability in the continuous case. Furthermore, we establish the discrete inf-sup condition for any conforming space-time finite element discretization yielding quasi-optimal discretization error estimates. Various numerical examples confirm the theoretical findings. We emphasize that the energy regularization results in a more localized control with sharper contours for discontinuous target functions, which is demonstrated by a comparison with an $L^2$-regularization and with a sparse optimal control approach.