As a continuation of [Li, J. and Wang, Y. N., Structural stability of steady subsonic Euler flows in 2D finitely long nozzles with variable end pressure, J. Differential Equations, 413, 2014, 70–109], in this paper, the authors study the structural stability of three dimensional axisymmetric steady subsonic Euler flows in finitely long curved nozzles. The reference flow is a general subsonic shear flow in a three dimensional regular cylindrical nozzle with general size of vorticity and without stagnation points. The problem is described by the well-known steady compressible Euler system. With a class of admissible physical conditions and prescribed pressure at the entrance and the exit of the nozzle respectively, they establish the structural stability of this kind of axisymmetric subsonic shear flow with no-zero swirl velocity. Due to the hyperbolic-elliptic coupled form of the Euler system in subsonic regions, the problem is reformulated via a twofold normalized process, including straightening the lateral boundary of the nozzle under the natural Cartesian coordinates and reformulating the problem under the cylindrical coordinates. Accordingly, the Euler system is decoupled into an elliptic mode and three hyperbolic modes with some artificial singular terms under the cylindrical coordinates. The elliptic mode is a mixed type boundary value problem of first order elliptic system for the pressure and the radial velocity angle. Meanwhile, the hyperbolic modes are transport type to control the total energy, the specific entropy and the swirl velocity, respectively. The estimates as well as well-posedness are executed in a Banach space with optimal regularity under the natural Cartesian coordinates in place of the cylindrical coordinates. The authors develop a systematic framework to deal with the artificial singularity and the non-zero swirl velocity in three dimensional axisymmetric case. Their strategy is helpful for other three dimensional problems under axisymmetry.

This paper is concerned with the ergodic stochastic optimal control problem with Markov Regime-Switching in a dissipative system. The proposed approach primarily relies on duality techniques. The control system is described by controlled dissipative stochastic differential equations and modulated by a continuous-time, finite-state Markov chain. The cost functional is ergodic, which is the expected long-run mean average type. The control domain is assumed to be convex, and the convex variation technique is used. Both necessary condition version and sufficient condition version of the stochastic maximum principle are established for optimal control. An example is discussed to illustrate the significance of our results.

In this paper, the author considers a general control problem about the system of thermoelasticity of type I. By introducing some unique continuation property of the corresponding adjoint system and a suitable observability inequality for an elastic equation, using compact decoupling technique and variational approach, the exact-approximate controllability of the abstract thermoelasticity of type I is obtained. Finally, the author applies her abstract result to the exact-approximate controllability of the linear system of thermoelasticity.

Apéry-type (inverse) binomial series have appeared prominently in the calculations of Feynman integrals in recent years. In their previous work, the authors showed that a few large classes of the non-alternating Apéry-type (inverse) central binomial series can be evaluated using colored multiple zeta values of level four (i.e., special values of multiple polylogarithms at the fourth roots of unity) by expressing them in terms of iterated integrals. In this sequel, the authors will prove that for several classes of the alternating versions they need to raise the level to eight. Their main idea is to adopt hyperbolic trigonometric 1-forms to replace the ordinary trigonometric ones used in the non-alternating setting.

The author studies a stochastic linear quadratic (SLQ for short) optimal control problem for systems governed by stochastic evolution equations, where the control operator in the drift term may be unbounded. Under the condition that the cost functional is uniformly convex, the well-posedness of the operator-valued Riccati equation is proved. Based on that, the optimal feedback control of the control problem is given.

In this paper, the authors study the almost everywhere pointwise convergence problem along a class of restricted curves in ℝ × ℝ given by {(y, t): y ∈ Γ(x, t)} for each t ∈ [0, 1], where Γ(x, t) = {γ(x, t, θ): θ ∈ Θ} for a given compact set Θ in ℝ of the fractional Schrödinger propagator and Boussinesq operator. They focus on the relationship between the upper Minkowski dimension of Θ and the optimal s for which

The authors study the compact intertwining relation for the composition operators and integral operators between mixed-norm spaces and Zygmund spaces.