Interior local null controllability of one‐dimensional compressible flow near a constant steady state

We consider the one‐dimensional compressible Navier–Stokes equations with periodic boundary conditions, with initial conditions in a small neighborhood of a state of uniform density and uniform nonzero velocity. We prove that, with a control given only by a body force localized in a subinterval, we can steer the system to uniform density and velocity. Copyright © 2016 John Wiley & Sons, Ltd.     

A singular multi-dimensional piston problem in compressible flow

This paper concerns the multi-dimensional piston problem, which is a special initial boundary value problem of multi-dimensional unsteady potential flow equation. The problem is defined in a domain bounded by two conical surfaces, one of them is shock, whose location is also to be determined. By introducing self-similar coordinates, the problem can be reduced to a free boundary value problem of an elliptic equation. The existence of the problem is proved by using partial hodograph transformation and nonlinear alternating iteration. The result also shows the stability of the structure of shock front in symmetric case under small perturbation.     

Null controllability for the parabolic equation with a complex principal part

The paper is devoted to a study of the null controllability for the semilinear parabolic equation with a complex principal part. For this purpose, we establish a key weighted identity for partial differential operators (with real functions α and β), by which we develop a universal approach, based on global Carleman estimate, to deduce not only the desired explicit observability estimate for the linearized complex Ginzburg-Landau equation, but also all the known controllability/observability results for the parabolic, hyperbolic, Schrödinger and plate equations that are derived via Carleman estimates.     

Null controllability for the heat equation with singular inverse-square potentials

We prove the null controllability of the heat equation perturbed by a singular inverse-square potential arising in quantum mechanics and combustion theory. This is done within the range of subcritical coefficients of the singular potential, provided the control acts on an annular set around the singularity. Our proof uses a splitting argument on the domain, decomposition in spherical harmonics, new Carleman inequalities and refined Hardy inequalities.     

一维绝热流方程组的精确边界能控性

利用一阶拟线性双曲组混合初边值问题的精确能控性理论,通过对边界速度或压强的控制,实现了一维绝热流方程组的精确边界能控性.     

Exact boundary controllability of unsteady flows in a network of open canals

By means of the general results on the exact boundary controllability for quasilinear hyperbolic systems, the author establishes the exact boundary controllability of unsteady flows in both a single open canal and a network of open canals with star configuration respectively. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Well-posedness of the free boundary problem in compressible elastodynamics

We study the free boundary problem for the flow of a compressible isentropic inviscid elastic fluid. At the free boundary moving with the velocity of the fluid particles the columns of the deformation gradient are tangent to the boundary and the pressure vanishes outside the flow domain. We prove the local-in-time existence of a unique smooth solution of the free boundary problem provided that among three columns of the deformation gradient there are two which are non-collinear vectors at each point of the initial free boundary. If this non-collinearity condition fails, the local-in-time existence is proved under the classical Rayleigh–Taylor sign condition satisfied at the first moment. By constructing an Hadamard-type ill-posedness example for the frozen coefficients linearized problem we show that the simultaneous failure of the non-collinearity condition and the Rayleigh–Taylor sign condition leads to Rayleigh–Taylor instability.     

Implicit fully mesh-less method for compressible viscous flow calculations

A dual-time implicit mesh-less scheme is developed for solution of governing viscous flow equations. The computational efficiency of the method is enhanced by adopting accelerating techniques such as local time stepping and residual smoothing. The Taylor series least square method is used for approximation of derivatives at each node which leads to a central difference spatial discretization. The capabilities of the method are demonstrated by flow computations about standard cases at subsonic and transonic laminar flow conditions. Results are presented which indicate good agreements with the alternative numerical and experimental data. The computational time is considerably reduced when using the proposed mesh-less method compared with the explicit mesh-less and finite-volume counterparts using the same distribution of points.     

高阶拟线性双曲型方程的精确边界能控性

By means of the existence and uniqueness of semi-global C^1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues ,the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.     

Erratum: flow control in gas networks: exact controllability to a given demand

We correct a technical error in the paper of Gugat, Herty, Schleper, Math. Methods Appl. Sci. 34 (2011), where a framework for controllability of quasi–linear hyperbolic systems has been studied. The application to the case of gas networks is specified in more detail in the current work. Copyright © 2014 John Wiley & Sons, Ltd.     

Null controllability for a semilinear parabolic equation with gradient quadratic growth

This article is concerned with the null controllability of a semilinear parabolic equation with the nonlinear term involving the gradient quadratic term. The technique in this paper is a combination of Cole–Hopf transformation and some methods from [A.Y. Khapalov, Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: A qualitative approach, SIAM J. Control Optim. 41 (2003) 1886–1900].     

Exact boundary controllability for a kind of second‐order quasilinear hyperbolic systems and its applications

In this paper, by means of a constructive method based on the existence and uniqueness of the semi‐global C² solution, we establish the local exact boundary controllability for a kind of second‐order quasilinear hyperbolic systems. As an application, we obtain the one‐sided local exact boundary controllability for the first‐order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative eigenvalues are decoupled. Copyright © 2009 John Wiley & Sons, Ltd.     

Exact boundary controllability of nodal profile for quasilinear hyperbolic systems in a tree‐like network

In this paper, the exact boundary controllability of nodal profile is established for quasilinear hyperbolic systems with general nonlinear boundary and interface conditions in a tree‐like network with general topology. The basic principles for giving nodal profiles and for choosing boundary controls are presented, respectively. Copyright © 2011 John Wiley & Sons, Ltd.     

Exact boundary controllability of nodal profile for 1-D quasilinear wave equations

Based on the theory of semi-global C ² solution for 1-D quasilinear wave equations, the local exact boundary controllability of nodal profile for 1-D quasilinear wave equations is obtained by a constructive method, and the corresponding global exact boundary controllability of nodal profile is also obtained under certain additional hypotheses.     

Exact controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals

In this paper we establish the exact boundary controllability for quasilinear hyperbolic systems with interface conditions. As an application, we get the exact boundary controllability of unsteady flows in a string‐like network of open canals. Copyright © 2004 John Wiley & Sons, Ltd.     

On temporal decay estimates for the compressible nematic liquid crystal flow in

We use a general energy method recently developed by [Guo Y, Wang Y. Decay of dissipative equations and negative sobolev spaces. Commun. Partial Differ. Equ. 2012;37:2165–2208.] to prove the global existence and temporal decay rates of solutions to the three-dimensional compressible nematic liquid crystal flow in the whole space. In particular, the negative Sobolev norms of solutions are shown to be preserved along time evolution, and then the optimal decay rates of the higher order spatial derivatives of solutions are obtained by energy estimates and the interpolation inequalities.