Applied Mathematics and Optimization最新文献

In this work, we consider a system of fast and slow time-scale stochastic partial differential equations with reflection, where the slow component is the one-dimensional stochastic Burgers equation, the fast component is the stochastic reaction-diffusion equation, and both the fast and slow components have two reflecting walls. The well-posedness of this system is established. Our approach is based on the penalized method by giving the delicate estimation of the penalized terms, which do not resort to splitting the reflected system into stochastic system without reflection and deterministic system with reflection. Then by means of penalized method and combining the classical Khasminskii’s time discretization, we prove the averaging principle for a class of reflected stochastic partial differential equations. In particular, due to the existence and uniqueness of invariant measure for fast component with frozen slow component, the ergodicity for frozen equations are given for different initial function spaces, which plays an important role.

This work focuses on the optimal control design for suppressing the singularity formation in chemotaxis governed by the parabolic-elliptic Patlak–Keller–Segel (PKS) system via flow advection. The main idea of this work lies in utilizing flow advection for enhancing diffusion as to control the nonlinear behavior of the system. The objective is to determine an optimal strategy for adjusting flow strength so that the possible finite time blow-up of the solution can be suppressed. Rigorous proof of the existence of an optimal solution and derivation of first-order optimality conditions for solving such a solution are presented. Spline collocation methods are employed for solving the optimality conditions. Numerical experiments based on 2D cellular flows in a rectangular domain are conducted to demonstrate our ideas and designs.

We study the relaxation problem for a two-fluid Euler-Poisson system. We prove the global-in-time convergence of the system for smooth solutions near the constant equilibrium states. The limit system is the two-fluid drift-diffusion system as the relaxation time tends to zero. In the proof, we establish uniform energy estimates of smooth solutions for all the parameters and the time. These estimates allow us to pass to the limit in the system to obtain the limit system. Moreover, the global convergence rate of the solutions is obtained by stream function techniques.

Inspired by some experimental (numerical) works on fractional diffusion PDEs, we develop a rigorous framework to prove that solutions to the fractional Navier–Stokes equations, which involve the fractional Laplacian operator ((-Delta )^{frac{alpha }{2}}) with (alpha <2), converge to a solution of the classical case, with (-Delta ), when (alpha ) goes to 2. Precisely, in the setting of mild solutions, we prove uniform convergence in the (L^{infty }_{t,x})-space and derive a precise convergence rate, revealing some phenomenological effects. As a bi-product, we prove strong convergence in the (L^{p}_{t}L^{q}_{x})-space. Finally, our results are also generalized to the coupled setting of the Magnetic-hydrodynamic system.

In this paper, we prove the following assertion for an absorbing Markov decision process (MDP) with the given initial distribution, which is also assumed to be semi-continuous: the continuity of the projection mapping from the space of strategic measures to the space of occupation measures, both endowed with their weak topologies, is equivalent to the MDP model being uniformly absorbing. An example demonstrates, among other interesting scenarios, that for an absorbing (but not uniformly absorbing) semi-continuous MDP with the given initial distribution, the space of occupation measures can fail to be compact in the weak topology.

We study the asymptotic behavior of the nonlinear MGT plate equation in the unbounded domain. By using semigroup theory, we first establish the well-posedness result for the Cauchy problem related to the linear MGT plate equation. By using the energy method in the Fourier space, we then prove the optimal decay estimate results for the non-critical case, in which the optimality is analyzed by considering the asymptotic expansion of the eigenvalues. By using the contraction mapping, we also show the local existence for the Cauchy problem of the nonlinear plate in appropriate function spaces, based on which we prove a global existence result for small data by using a priori energy estimates. Finally, based on the decay estimation of linear problems, the decay results of nonlinear problems are obtained.

In this article, we study an electrically conductive Rosensweig model for ferrofluids, whose Bloch–Torrey regularization was studied by Hamdache and Hamroun (Appl Math Optim 81(2):479–509, 2020). We mainly prove the global existence of weak solutions to the non-regularized model under a certain smallness condition on the electric conductivity. Hence, our result not only solves a problem that was left open by Hamdache and Hamroun, but it can also serve as a confirmation that ferrofluids are naturally poor conductors of electric current. The proof, which is interesting in itself, is quite involved and relies on the Helmohltz–Leray decomposition of the magnetic fields and the use of renormalized solutions for the magnetization. We also give a rigorous and detailed description of the convergence of the global weak solutions towards the quasi-equilibrium in the relaxation time limit regime (tau rightarrow 0).

In this work, we study the problem of existence and uniqueness of solutions of the stochastic Cahn-Hilliard-Navier-Stokes system with gradient-type noise. We show that such kind of noise is related to the problem of modelling turbulence. We apply a rescaling argument to transform the stochastic system into a random deterministic one. We split the latter into two parts: the Navier-Stokes part and the Cahn-Hilliard part, respectively. The rescale operators possess good properties which allow to show that the rescaled Navier-Stokes equations have a unique solution, by appealing to (delta -)monotone operators theory. While, well-posedness of the Cahn-Hilliard part is proved via a fixed point argument. Then, again a fixed point argument is used to prove global in time existence of a unique solution to the initial system. All the results are under the requirement that the initial data is in a certain small neighbourhood of the origin.

Controlled discrete time Markov processes are studied first with long run general discounting functional. It is shown that optimal strategies for average reward per unit time problem are also optimal for average generally discounting functional. Then long run risk sensitive reward functional with general discounting is considered. When risk factor is positive then optimal value of such reward functional is dominated by the reward functional corresponding to the long run risk sensitive control. In the case of negative risk factor we get an asymptotical result, which says that optimal average reward per unit time control is nearly optimal for long run risk sensitive reward functional with general discounting, assuming that risk factor is close to 0. For this purpose we show in Appendix upper estimates for large deviations of weighted empirical measures, which are of independent interest.