JIMO
Sensor deployment for pipeline leakage detection via optimal boundary control strategies
Chao Xu Yimeng Dong Zhigang Ren Huachen Jiang Xin Yu
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
DCDS
The $C$-regularized semigroup method for partial differential equations with delays
Xin Yu Guojie Zheng Chao Xu
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.
JIMO
Computational optimal control of 1D colloid transport by solute gradients in dead-end micro-channels
Tehuan Chen Chao Xu Zhigang Ren

Diffusiophoresis is a common phenomenon that occurs when colloids are placed in the non-uniform solute concentration. It generates solute gradients which force the colloids to transfer toward or away from the higher solute concentration side. In this paper, we consider the input sequence control of the colloid transport in a dead-end micro-channel with a boundary solute concentration being manipulated, which has a wide range of applications such as drug delivery, biology transport, oil recovery system and so on. We model this process by a coupled system, which involves the solute diffusion equation and the colloid transport model. Then an optimal control problem is formulated, in which the goal is to minimize colloid density distribution deviation between the computational one and the target at a pre-specified terminal time. To solve this partial differential equation (PDE) optimal control problem, we first apply the control parameterization method to discretize the boundary control and transfer it into an optimal parameter selection problem. Then, using the variational method, the gradient of the objective function with respect to the decision parameters can be derived, which depends on the solution of the coupled system and the costate system. Based on this, we propose an effective computational method and a gradient-based optimization algorithm to solve the optimal control problem numerically. Finally, we give the simulation results to demonstrate that the objective function based on the proposed method is less nearly two orders of magnitude than that of a constant value control strategy, which well illustrates the effectiveness of the proposed method.

keywords: Diffusiophoresis dead-end micro-channel colloid transport control parameterization variational method pde-constrained optimization computational optimal control
KRM
A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation
Yoshinori Morimoto Karel Pravda-Starov Chao-Jiang Xu
We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a close-to-equilibrium framework and show that the Cauchy problem for the fluctuation around the Maxwellian equilibrium distribution enjoys a Gelfand-Shilov regularizing effect in the class $S_{1/2}^{1/2}(\mathbb{R}^d)$, implying the ultra-analyticity and the production of exponential moments of the fluctuation, for any positive time.
keywords: ultra-analyticity smoothing effect. Gelfand-Shilov regularity Landau equation
KRM
Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation
Léo Glangetas Hao-Guang Li Chao-Jiang Xu
In this work, we study the Cauchy problem for the spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules. We prove that this Cauchy problem enjoys Gelfand-Shilov's regularizing effect, meaning that the smoothing properties are the same as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator. The power of the fractional exponent is exactly the same as the singular index of the non-cutoff collisional kernel of the Boltzmann equation. Therefore, we get the sharp regularity of solutions in the Gevrey class and also the sharp decay of solutions with an exponential weight. We also give a method to construct the solution of the Boltzmann equation by solving an infinite system of ordinary differential equations. The key tool is the spectral decomposition of linear and non-linear Boltzmann operators.
keywords: Boltzmann equation spectral decomposition Gelfand-Shilov class.
KRM
Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential
Radjesvarane Alexandre Yoshinori Morimoto Seiji Ukai Chao-Jiang Xu Tong Yang
In this paper, we consider the Cauchy problem for the non-cutoff Boltzmann equation in the soft potential case. By using a singular change of velocity variables before and after collision, we prove the uniqueness of weak solutions to the Cauchy problem in the space of functions with polynomial decay in the velocity variable.
keywords: uniqueness of solution. Boltzmann equation singular change of velocity variables
KRM
Bounded solutions of the Boltzmann equation in the whole space
Radjesvarane Alexandre Yoshinori Morimoto Seiji Ukai Chao-Jiang Xu Tong Yang
We construct bounded classical solutions of the Boltzmann equation in the whole space without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, we show that if the initial data is non-negative and belongs to a uniformly local Sobolev space in the space variable and a standard Sobolev space with Maxwellian type decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a unique non-negative local solution in the same function space, both for the cutoff and non-cutoff collision cross section with mild singularity. The known solutions such as solutions on the torus (space periodic solutions) and in the vacuum (solutions vanishing at the spatial infinity), and solutions in the whole space having a limit equilibrium state at the spatial infinity are included in our category.
keywords: local existence locally uniform Sobolev space spatial behavior at infinity pseudo-differential calculus. Boltzmann equation
DCDS
Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff
Yoshinori Morimoto Seiji Ukai Chao-Jiang Xu Tong Yang
Most of the work on the Boltzmann equation is based on the Grad's angular cutoff assumption. Even though the smoothing effect from the singular cross-section without the angular cutoff corresponding to the grazing collision is expected, there is no general mathematical theory especially for the spatially inhomogeneous case. As a further study on the problem in the spatially homogeneous situation, in this paper, we will prove the Gevrey smoothing property of the solutions to the Cauchy problem for Maxwellian molecules without angular cutoff by using pseudo-differential calculus. Furthermore, we apply similar analytic techniques for the Sobolev space regularity to the nonlinear equation, and prove the smoothing property of solutions for the spatially homogeneous nonlinear Boltzmann equation with the Debye-Yukawa potential.
keywords: Debye-Yukawa potential Gevrey hypoellipticity Boltzmann equation non-cutoff cross-sections.
KRM
Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators
Nicolas Lerner Yoshinori Morimoto Karel Pravda-Starov Chao-Jiang Xu
We prove that the linearized non-cutoff Boltzmann operator with Maxwellian molecules is exactly equal to a fractional power of the linearized Landau operator which is the sum of the harmonic oscillator and the spherical Laplacian. This result allows to display explicit sharp coercive estimates satisfied by the linearized non-cutoff Boltzmann operator for both Maxwellian and non-Maxwellian molecules.
keywords: spectral analysis anisotropy microlocal analysis. Boltzmann operator Landau operator
KRM
Propagation of Gevrey regularity for solutions of Landau equations
Hua Chen Wei-Xi Li Chao-Jiang Xu
By using the energy-type inequality, we obtain, in this paper, the result on propagation of Gevrey regularity for the solution of the spatially homogeneous Landau equation in the cases of Maxwellian molecules and hard potential.
keywords: Landau equation Boltzmann equation Gevrey regularity.

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