Sensor deployment for pipeline leakage detection via optimal boundary control strategies
Chao Xu Yimeng Dong Zhigang Ren Huachen Jiang Xin Yu
Journal of Industrial & Management Optimization 2015, 11(1): 199-216 doi: 10.3934/jimo.2015.11.199
We consider a multi-agent control problem using PDE techniques for a novel sensing problem arising in the leakage detection and localization of offshore pipelines. A continuous protocol is proposed using parabolic PDEs and then a boundary control law is designed using the maximum principle. Both analytical and numerical solutions of the optimality conditions are studied.
keywords: Offshore pipeline network leakage detection optimal control. continuation approach multi-agent system partial differential equations Lagrangian sensor
Generalized and weighted Strichartz estimates
Jin-Cheng Jiang Chengbo Wang Xin Yu
Communications on Pure & Applied Analysis 2012, 11(5): 1723-1752 doi: 10.3934/cpaa.2012.11.1723
In this paper, we explore the relations between different kinds of Strichartz estimates and give new estimates in Euclidean space $\mathbb{R}^n$. In particular, we prove the generalized and weighted Strichartz estimates for a large class of dispersive operators including the Schrödinger and wave equation. As a sample application of these new estimates, we are able to prove the Strauss conjecture with low regularity for dimension $2$ and $3$.
keywords: Generalized Strichartz estimates weighted Strichartz estimates Strauss conjecture semilinear wave equations. angular regularity
The $C$-regularized semigroup method for partial differential equations with delays
Xin Yu Guojie Zheng Chao Xu
Discrete & Continuous Dynamical Systems - A 2016, 36(9): 5163-5181 doi: 10.3934/dcds.2016024
This paper is devoted to study the abstract functional differential equation (FDE) of the following form $$\dot{u}(t)=Au(t)+\Phi u_t,$$ where $A$ generates a $C$-regularized semigroup, which is the generalization of $C_0$-semigroup and can be applied to deal with many important differential operators that the $C_0$-semigroup can not be used to. We first show that the $C$-well-posedness of a FDE is equivalent to the $\mathscr{C}$-well-posedness of an abstract Cauchy problem in a product Banach space, where the operator $\mathscr{C}$ is related with the operator $C$ and will be defined in the following text. Then, by making use of a perturbation result of $C$-regularized semigroup, a sufficient condition is provided for the $C$-well-posedness of FDEs. Moreover, an illustrative application to partial differential equation (PDE) with delay is given in the last section.
keywords: perturbation functional differential equation $C$-regularized semigroup $C$-well-posedness partial differential equation.
On some Liouville type theorems for the compressible Navier-Stokes equations
Dong Li Xinwei Yu
Discrete & Continuous Dynamical Systems - A 2014, 34(11): 4719-4733 doi: 10.3934/dcds.2014.34.4719
We prove several Liouville type results for stationary solutions of the $d$-dimensional compressible Navier-Stokes equations. In particular, we show that when the dimension $d ≥ 4$, the natural requirements $\rho \in L^{\infty} ( \mathbb{R}^d )$, $v \in \dot{H}^1 (\mathbb{R}^d)$ suffice to guarantee that the solution is trivial. For dimensions $d=2,3$, we assume the extra condition $v \in L^{\frac{3d}{d-1}}(\mathbb R^d)$. This improves a recent result of Chae [1].
keywords: Compressible Navier-Stokes Liouville.
Weighted-average alternating minimization method for magnetic resonance image reconstruction based on compressive sensing
Yonggui Zhu Yuying Shi Bin Zhang Xinyan Yu
Inverse Problems & Imaging 2014, 8(3): 925-937 doi: 10.3934/ipi.2014.8.925
The problem of compressive-sensing (CS) L2-L1-TV reconstruction of magnetic resonance (MR) scans from undersampled $k$-space data has been addressed in numerous studies. However, the regularization parameters in models of CS L2-L1-TV reconstruction are rarely studied. Once the regularization parameters are given, the solution for an MR reconstruction model is fixed and is less effective in the case of strong noise. To overcome this shortcoming, we present a new alternating formulation to replace the standard L2-L1-TV reconstruction model. A weighted-average alternating minimization method is proposed based on this new formulation and a convergence analysis of the method is carried out. The advantages of and the motivation for the proposed alternating formulation are explained. Experimental results demonstrate that the proposed formulation yields better reconstruction results in the case of strong noise and can improve image reconstruction via flexible parameter selection.
keywords: magnetic resonance image reconstruction. weighted average alternating minimization method Compressive sensing
On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion
Xinwei Yu Zhichun Zhai
Communications on Pure & Applied Analysis 2012, 11(5): 1809-1823 doi: 10.3934/cpaa.2012.11.1809
In this article we study local and global well-posedness of the Lagrangian Averaged Euler equations. We show local well-posedness in Triebel-Lizorkin spaces and further prove a Beale-Kato-Majda type necessary and sufficient condition for global existence involving the stream function. We also establish new sufficient conditions for global existence in terms of mixed Lebesgue norms of the generalized Clebsch variables.
keywords: Global existence Euler equations Triebel-Lizorkin spaces. level sets vortex patch

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