Authors:Kazuo Yamazaki Pages: 1 - 40 Abstract: The theory of micropolar fluids emphasizes the micro-structure of fluids by coupling the Navier–Stokes equations with micro-rotational velocity, and is widely viewed to be well fit, better than the Navier–Stokes equations, to describe fluids consisting of bar-like elements such as liquid crystals made up of dumbbell molecules or animal blood. Following the work of Weinan et al. (Commun Math Phys 224:83–106, 2001), we prove the existence of a unique stationary measure for the stochastic micropolar fluid system with periodic boundary condition, forced by only the determining modes of the noise and therefore a type of finite-dimensionality of micropolar fluid flow. The novelty of the manuscript is a series of energy estimates that is reminiscent from analysis in the deterministic case. PubDate: 2019-02-01 DOI: 10.1007/s00245-017-9419-z Issue No:Vol. 79, No. 1 (2019)

Authors:Fatima Zahra Mokkedem; Xianlong Fu Pages: 41 - 67 Abstract: In this paper we study the standard optimal control and time optimal control problems for a class of semilinear evolution systems with infinite delay. We first establish the results of existence and uniqueness of mild solution and the compactness of the solution operator for the control system. Then, based on these results, we investigate the optimal control problem with integral cost function and the time optimal control problem respectively. Under some conditions we show the existence of optimal controls for the both cases of bounded and unbounded admissible control sets. We also obtain the existence of time optimal control to a target set. In addition, a convergence theorem of time optimal controls to a point target set is proved. Finally, an example is given to show the application of the main results. PubDate: 2019-02-01 DOI: 10.1007/s00245-017-9420-6 Issue No:Vol. 79, No. 1 (2019)

Authors:Annalisa Massaccesi; Edouard Oudet; Bozhidar Velichkov Pages: 69 - 86 Abstract: In this paper we propose a variational approach to the Steiner tree problem, which is based on calibrations in a suitable algebraic environment for polyhedral chains which represent our candidates. This approach turns out to be very efficient from numerical point of view and allows to establish whether a given Steiner tree is optimal. Several examples are provided. PubDate: 2019-02-01 DOI: 10.1007/s00245-017-9421-5 Issue No:Vol. 79, No. 1 (2019)

Authors:Jing Liu; Song Yao Pages: 87 - 129 Abstract: Given \(p \in (1,2]\) , the wellposedness of backward stochastic differential equations with jumps (BSDEJs) in \(\mathbb {L}^p\) sense gives rise to a so-called g-expectation with \(\mathbb {L}^p\) domain under the jump filtration (the one generated by a Brownian motion and a Poisson random measure). In this paper, we extend such a g-expectation to a nonlinear expectation \(\mathcal{E}\) with \(\mathbb {L}^p\) domain that is consistent with the jump filtration. We study the basic (martingale) properties of the jump-filtration consistent nonlinear expectation \(\mathcal{E}\) and show that under certain domination condition, the nonlinear expectation \(\mathcal{E}\) can be represented by some g-expectation. PubDate: 2019-02-01 DOI: 10.1007/s00245-017-9422-4 Issue No:Vol. 79, No. 1 (2019)

Authors:Qingying Hu; Hongwei Zhang; Gongwei Liu Pages: 131 - 144 Abstract: We consider the initial boundary value problem for a class of logarithmic wave equations with linear damping. By constructing a potential well and using the logarithmic Sobolev inequality, we prove that, if the solution lies in the unstable set or the initial energy is negative, the solution will grow as an exponential function in the \(H^1_0(\Omega )\) norm as time goes to infinity. If the solution lies in a smaller set compared with the stable set, we can estimate the decay rate of the energy. These results are extensions of earlier results. PubDate: 2019-02-01 DOI: 10.1007/s00245-017-9423-3 Issue No:Vol. 79, No. 1 (2019)

Authors:Stefan Ankirchner; Maike Klein; Thomas Kruse Pages: 145 - 177 Abstract: We consider the problem of optimally stopping a continuous-time process with a stopping time satisfying a given expectation cost constraint. We show, by introducing a new state variable, that one can transform the problem into an unconstrained control problem and hence obtain a dynamic programming principle. We characterize the value function in terms of the dynamic programming equation, which turns out to be an elliptic, fully non-linear partial differential equation of second order. We prove a classical verification theorem and illustrate its applicability with several examples. PubDate: 2019-02-01 DOI: 10.1007/s00245-017-9424-2 Issue No:Vol. 79, No. 1 (2019)

Authors:Nacira Agram; Bernt Øksendal Pages: 181 - 204 Abstract: By a memory mean-field process we mean the solution \(X(\cdot )\) of a stochastic mean-field equation involving not just the current state X(t) and its law \(\mathcal {L}(X(t))\) at time t, but also the state values X(s) and its law \(\mathcal {L}(X(s))\) at some previous times \(s<t.\) Our purpose is to study stochastic control problems of memory mean-field processes. We consider the space \(\mathcal {M}\) of measures on \(\mathbb {R}\) with the norm \( \cdot _{\mathcal {M}}\) introduced by Agram and Øksendal (Model uncertainty stochastic mean-field control. arXiv:1611.01385v5, [2]), and prove the existence and uniqueness of solutions of memory mean-field stochastic functional differential equations. We prove two stochastic maximum principles, one sufficient (a verification theorem) and one necessary, both under partial information. The corresponding equations for the adjoint variables are a pair of (time-advanced backward stochastic differential equations (absdes), one of them with values in the space of bounded linear functionals on path segment spaces. As an application of our methods, we solve a memory mean–variance problem as well as a linear–quadratic problem of a memory process. PubDate: 2019-02-01 DOI: 10.1007/s00245-017-9425-1 Issue No:Vol. 79, No. 1 (2019)

Authors:Nacira Agram; Bernt Øksendal Pages: 205 - 206 Abstract: The original version of this article unfortunately contained a few mistakes in Theorems and notation. The corrected information is given below. PubDate: 2019-02-01 DOI: 10.1007/s00245-018-9483-z Issue No:Vol. 79, No. 1 (2019)

Authors:Truong Minh Tuyen Pages: 207 - 227 Abstract: In this paper, we study the split common null point problem. Then, using the hybrid projection method and the metric resolvent of monotone operators, we prove a strong convergence theorem for an iterative method for finding a solution of this problem in Banach spaces. PubDate: 2019-02-01 DOI: 10.1007/s00245-017-9427-z Issue No:Vol. 79, No. 1 (2019)

Authors:Isaías Pereira de Jesus Pages: 229 - 229 Abstract: The original version of this article unfortunately contained a mistake in the equation. PubDate: 2019-02-01 DOI: 10.1007/s00245-018-9522-9 Issue No:Vol. 79, No. 1 (2019)

Authors:Eduard Feireisl; Madalina Petcu Abstract: We consider a model of a two phase flow proposed by Anderson et al. taking into account possible thermal fluctuations. The mathematical model consists of the compressible Navier–Stokes system coupled with the Cahn–Hilliard equation, where the latter is driven by a multiplicative temporal white noise accounting for thermal fluctuations. We show existence of dissipative martingale solutions satisfying the associated total energy balance. PubDate: 2019-02-13 DOI: 10.1007/s00245-019-09557-2

Authors:Da Xu Abstract: In this paper we study the observability properties of time discrete approximation schemes for some integro-differential equations. The equation is discretized in time by the back-ward Euler method in combination with convolution quadrature. We prove uniform observability results for time discretization schemes in which the high frequency components have been filtered. In this way, the well-known exact observability estimates of the integro-differential systems can be reproduced as the limit, as the time step \( {\varDelta }t\rightarrow 0 \) . The discrete observability estimates are established by means of a time-discrete version of the classical harmonic analysis approach. PubDate: 2019-02-01 DOI: 10.1007/s00245-019-09556-3

Authors:Jürgen Sprekels; Hao Wu Abstract: In this paper, we study an optimal control problem for a two-dimensional Cahn–Hilliard–Darcy system with mass sources that arises in the modeling of tumor growth. The aim is to monitor the tumor fraction in a finite time interval in such a way that both the tumor fraction, measured in terms of a tracking type cost functional, is kept under control and minimal harm is inflicted to the patient by administering the control, which could either be a drug or nutrition. We first prove that the optimal control problem admits a solution. Then we show that the control-to-state operator is Fréchet differentiable between suitable Banach spaces and derive the first-order necessary optimality conditions in terms of the adjoint variables and the usual variational inequality. PubDate: 2019-01-24 DOI: 10.1007/s00245-019-09555-4

Authors:Jérome Lemoine; Arnaud Münch; Pablo Pedregal Abstract: We analyze two \(H^{-1}\) -least-squares methods for the steady Navier–Stokes system of incompressible viscous fluids. Precisely, we show the convergence of minimizing sequences for the least-squares functional toward solutions. Numerical experiments support our analysis. PubDate: 2019-01-24 DOI: 10.1007/s00245-019-09554-5

Authors:Constantin Christof; Gerd Wachsmuth Abstract: This paper is concerned with the differential sensitivity analysis of variational inequalities in Banach spaces whose solution operators satisfy a generalized Lipschitz condition. We prove a sufficient criterion for the directional differentiability of the solution map that turns out to be also necessary for elliptic variational inequalities in Hilbert spaces (even in the presence of asymmetric bilinear forms, nonlinear operators and nonconvex functionals). Our method of proof is fully elementary. Moreover, our technique allows us to also study those cases where the variational inequality at hand is not uniquely solvable and where directional differentiability can only be obtained w.r.t. the weak or the weak-star topology of the underlying space. As tangible examples, we consider a variational inequality arising in elastoplasticity, the projection onto prox-regular sets, and a bang–bang optimal control problem. PubDate: 2019-01-10 DOI: 10.1007/s00245-018-09553-y

Authors:Yu Zhuo; Yuchao Dong; Jiangyan Pu Abstract: In this paper, we consider the stochastic recursive control problem under non-Lipschitz framework. More precisely, we assume that the generator of the backward stochastic differential equation that describes the cost functional is monotonic with respect to the first unknown variable and uniformly continuous in the second unknown variable. A dynamic programming principle is established by making use of a Girsanov transformation argument and the BSDE methods. The value function is then shown to be the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation via truncation methods, approximation techniques and the stability result of viscosity solutions. PubDate: 2019-01-02 DOI: 10.1007/s00245-018-9551-4

Authors:Yanjie Zhang; Huijie Qiao; Jinqiao Duan Abstract: This work is about a slow-fast data assimilation system under non-Gaussian noisy fluctuations. Firstly, we show the existence of a random invariant manifold for a stochastic dynamical system with non-Gaussian noise and two-time scales. Secondly, we obtain a low dimensional reduction of this system via a random invariant manifold. Thirdly, we prove that the low dimensional filter on the random invariant manifold approximates the original filter, in a probabilistic sense. PubDate: 2019-01-01 DOI: 10.1007/s00245-018-9552-3

Authors:Lijun Bo; Claudia Ceci Abstract: We discuss dynamic hedging of counterparty risk for a portfolio of credit derivatives by the local risk-minimization approach. We study the problem from the perspective of an investor who, trading with credit default swaps (CDS) referencing the counterparty, wants to protect herself/himself against the loss incurred at the default of the counterparty. We propose a credit risk intensity-based model consisting of interacting default intensities by taking into account direct contagion effects. The portfolio of defaultable claims is of generic type, including CDS portfolios, risky bond portfolios and first-to-default claims with payments allowed to depend on the default state of the reference firms and counterparty. Using the martingale representation of the conditional expectation of the counterparty risk price payment stream under the minimal martingale measure, we recover a closed-form representation for the locally risk minimizing strategy in terms of classical solutions to nonlinear recursive systems of Cauchy problems. We also discuss applications of our framework to the most prominent classes of credit derivatives. PubDate: 2019-01-01 DOI: 10.1007/s00245-018-9549-y

Authors:Roberto Triggiani Abstract: We consider a heat–plate interaction model where the 2-dimensional plate is subject to viscoelastic (strong) damping. Coupling occurs at the interface between the two media, where each component evolves through differential operators. In this paper, we apply “high” boundary interface conditions, which involve the two classical boundary operators of a physical plate: the bending moment operator \(B_1\) and the shear forces operator \(B_2\) . We prove three main results: analyticity of the corresponding contraction semigroup on the natural energy space; sharp location of the spectrum of its generator, which does not have compact resolvent, and has the point \(\lambda =-\,1\) in its continuous spectrum; exponential decay of the semigroup with sharp decay rate. Here analyticity cannot follow by perturbation. PubDate: 2019-01-01 DOI: 10.1007/s00245-018-9547-0