arXiv subjects

Lili Du

Publications and source records attributed to Lili Du.

At least 37 records · Page 2Linked to original sources

arXiv2023

The non-existence of horizontally flat singularity for steady axisymmetric free surface flows near stagnation points

In a recent research on degenerate points of steady axisymmetric gravity flows with general vorticity, it has been shown that the possible asymptotics near any stagnation point must be the "Stokes corner", the "horizontal cusp", or the "horizontal flatness" (Theorem 1.1, Du, Huang, Pu, Commun. Math. Phys., 400, 2137-2179, 2023). In this paper, we focus on the horizontally flat singularity and show that it is not possible, and therefore the "Stokes corner" and the "cusp" are the only possible asymptotics at the stagnation points. The basic idea of our proof relies on a perturbation of the frequency formula for the two-dimensional problem (Varvaruca, Weiss, Acta Math., 206, 363-403, 2011). Our analysis also suggests that, for steady axisymmetric rotational gravity flows, the singular asymptotic profiles at stagnation points are similar to the scenario observed in two-dimensional waves with vorticity (Varvaruca, Weiss, Ann. I. H. Poincare-AN, 29, 861-885, 2012).

math.AP↗

arXiv2023

Uniqueness of blowup at singular points for superconductivity problem

In this paper, we prove that the uniqueness of blowup at the maximum point of coincidence set of the superconductivity problem, mainly based on the Weiss-type and Monneau-type monotonicity formulas, and the proof of the main results in this paper is inspired the recent paper \cite{CFL22} by Chen-Feng-Li.

math.AP↗

arXiv2022

The primitive equations with magnetic field approximation of the 3D MHD equations

In our earlier work \cite{DLL}, we have shown the global well-posedness of strong solutions to the three-dimensional primitive equations with the magnetic field (PEM) on a thin domain. The heart of this paper is to provide a rigorous justification of the derivation of the PEM as the small aspect ratio limit of the incompressible three-dimensional scaled magnetohydrodynamics (SMHD) equations in the anisotropic horizontal viscosity and magnetic field regime. For the case of $H^1$-initial data case, we prove that global Leray-Hopf weak solutions of the three-dimensional SMHD equation strongly converge to the global strong solutions of the PEM. In the $H^2$-initial data case, the strong solution of the SMHD can be extended to be a global one for small $\v$. As a consequence, we observe that the global strong solutions of the SMHD strong converge to the global strong solutions of the PEM. As a byproduct, the convergence rate is of the same order as the aspect ratio parameter.

math.AP↗

arXiv2022

Cooperative Trajectory Control for Synchronizing the Movement of Two Connected and Autonomous Vehicles Separated in a Mixed Traffic Flow

When connected and autonomous vehicles (CAVs) are widely used in the future, we can foresee many essential applications, such as platoon formation and autonomous police patrolling, which need two CAVs, originally separated in a mixed traffic flow involving CAVs and human-drive vehicles (HDVs), to quickly approach each other and then keep a stable car-following mode. The entire process should not jeopardize surrounding traffic safety and efficiency. The existing literature has not studied this CAV synchronization control well, and this study seeks to partially make up this gap.

math.OC↗

arXiv2022

Feasibility Analysis and Proof of Model Predictive Control

Sequential and terminal constraint feasibility of the model predictive control (MPC) play important roles in ensuring MPC control continuity. This study thus investigates these two properties theoretically using an MPC model for vehicle platooning and eco-driving strategy at signalized intersections. This study is served as a supplement material for our paper coming soon.

math.OC↗

arXiv2022

Global well-posedness of the 3D Primitive Equations with magnetic field

In this paper, the three-dimensional primitive equations with magnetic field (PEM) are considered on a thin domain. We showed the global existence and uniqueness (regularity) of strong solutions to the three-dimensional incompressible PEM without any small assumption on the initial data. More precisely, there exists a unique strong solution globally in time for any given $H^2$-smooth initial data.

math.AP↗

arXiv2022

The free boundary of steady axisymmetric inviscid flow with vorticity I: near the degenerate point

In this paper, we investigate the singularity near the degenerate points of the steady axisymmetric flow with general vorticity of an inviscid incompressible fluid acted on by gravity and with a free surface. We called the points on the free boundary at which the gradient of the stream function vanishes as the degenerate points. The main results in this paper give the different classifications of the singularity near the degenerate points on the free surface. More precisely, we obtained that at the stagnation points, the possible profiles must be a Stokes corner, or a horizontal cusp, or a horizontal flatness. At the degenerate points on the symmetric axis except the origin, the wave profile must be a cusp. At the origin, the possible wave profiles must be a Garabedian pointed bubble, or a horizontal cusp, or a horizontal flatness.

math.AP↗

arXiv2021

Nonconvex, Fully Distributed Optimization based CAV Platooning Control under Nonlinear Vehicle Dynamics

CAV platooning technology has received considerable attention in the past few years, driven by the next generation smart transportation systems. Unlike most of the existing platooning methods that focus on linear vehicle dynamics of CAVs, this paper considers nonlinear vehicle dynamics and develops fully distributed optimization based CAV platooning control schemes via the model predictive control (MPC) approach for a possibly heterogeneous CAV platoon. The nonlinear vehicle dynamics leads to several major difficulties in distributed algorithm development and control analysis and design. Specifically, the underlying MPC optimization problem is nonconvex and densely coupled. Further, the closed loop dynamics becomes a time-varying nonlinear system subject to external perturbations, making closed loop stability analysis rather complicated. To overcome these difficulties, we formulate the underlying MPC optimization problem as a locally coupled, albeit nonconvex, optimization problem and develop a sequential convex programming based fully distributed scheme for a general MPC horizon. Such a scheme can be effectively implemented for real-time computing using operator splitting methods. To analyze the closed loop stability, we apply various tools from global implicit function theorems, stability of linear time-varying systems, and Lyapunov theory for input-to-state stability to show that the closed loop system is locally input-to-state stable uniformly in all small coefficients pertaining to the nonlinear dynamics. Numerical tests on homogeneous and heterogeneous CAV platoons demonstrate the effectiveness of the proposed fully distributed schemes and CAV platooning control.

math.OC↗

arXiv2021

A Charging-as-a-Service Platform for Charging Electric Vehicles On the Move: New Vehicle Routing Model and Solution

Range anxiety has been the main challenge for the mass-adoption of electric vehicles (EVs). The emerging mobile electric-vehicle-to-electric-vehicle (mE2) charging technology, which allows an EV with the extra battery to charge the other EV on the move, offers a promising solution. Even though the physical feasibility of this new technology has been confirmed, we still face many operation problems. For example, how can we efficiently pair and route an electricity provider (EP) to a demand (ED) without introducing an extra detour? Motivated by this view, this study develops a Charging-as-a-Service (CaaS) platform, which seeks to dispatches the commercial EPs to serve the EDs, for cultivating this emerging charging service towards the low EV penetration environment. Mathematically, the CaaS platform is modeled as a vehicle routing problem (i.e., mE2-VRP), which optimally dispatches the EPs to approach and serve the EDs on the move while minimizing the EP fleet size and fulfilling all service requests. To adapt the CaaS platform to the online application in practice, we develop the Clustering-aided Clarke and Wright Savings (CCWS) algorithm to efficiently decompose and then solve the large-scale mE2-VRP by parallel computing. Our numerical experiments built upon citywide (Chicago) and statewide (Florida) found that the CCWS algorithm outperforms existing commercial solvers, and it enables us to investigate the performance of the CaaS platform under a realistic large-scale setup in a city or state. The CaaS performs better in low EV penetration markets while traffic congestion is mild and EDs require energy earlier. We can improve the performance by developing proper pricing strategies according to the EDs' energy requests and trip lengths.

math.OC↗

arXiv2021

Fully Distributed Optimization based CAV Platooning Control under Linear Vehicle Dynamics

This paper develops distributed optimization based, platoon centered CAV car following schemes, motivated by the recent interest in CAV platooning technologies. Various distributed optimization or control schemes have been developed for CAV platooning. However, most existing distributed schemes for platoon centered CAV control require either centralized data processing or centralized computation in at least one step of their schemes, referred to as partially distributed schemes. In this paper, we develop fully distributed optimization based, platoon centered CAV platooning control under the linear vehicle dynamics via the model predictive control approach with a general prediction horizon. These fully distributed schemes do not require centralized data processing or centralized computation through the entire schemes. To develop these schemes, we propose a new formulation of the objective function and a decomposition method that decomposes a densely coupled central objective function into the sum of several locally coupled functions whose coupling satisfies the network topology constraint. We then exploit the formulation of locally coupled optimization and operator splitting methods to develop fully distributed schemes. Control design and stability analysis is carried out to achieve desired traffic transient performance and asymptotic stability. Numerical tests demonstrate the effectiveness of the proposed fully distributed schemes and CAV platooning control.

math.OC↗

arXiv2021

Steady collision of two jets issuing from two axially symmetric channels

In the classical survey (Chapter 16.2, {\it Mathematics in industrial problem}, Vol. 24, Springer-Verlag, New York, 1989), A. Friedman proposed an open problem on the collision of two incompressible jets emerging from two axially symmetric nozzles. In this paper, we concerned with the mathematical theory on this collision problem, and establish the well-posedness theory on hydrodynamic impinging outgoing jets issuing from two coaxial axially symmetric nozzles. More precisely, we showed that for any given mass fluxes $M_1>0$ and $M_2<0$ in two nozzles respectively, that there exists an incompressible, inviscid impinging outgoing jet with contact discontinuity, which issues from two given semi-infinitely long axially symmetric nozzles and extends to infinity. Moreover, the constant pressure free stream surfaces of the impinging jet initiate smoothly from the mouths of the two nozzles and shrink to some asymptotic conical surface. There exists a smooth surface separating the two incompressible fluids and the contact discontinuity occurs on the surface. Furthermore, we showed that there is no stagnation point in the flow field and its closure, except one point on the symmetric axis. Some asymptotic behavior of the impinging jet in upstream and downstream, geometric properties of the free stream surfaces are also obtained. The main results in this paper solved the open problem on the collision of two incompressible axially symmetric jets in [24].

math.AP↗

arXiv2020

Oblique Injection of incompressible ideal fluid from a slot into a free stream

This paper deals with a two-phase fluid free boundary problem in a slot-film cooling. We give two well-posedness results on the existence and uniqueness of the incompressible inviscid two-phase fluid with a jump relation on free interface. The problem formulates the oblique injection of an incompressible ideal fluid from a slot into a free stream. From the mathematical point of view, this work is motivated by the pioneer work [13] by A. Friedman, in which some well-posedness results are obtained in some special case. Furthermore, A. Friedman proposed an open problem in [14] on the existence and uniqueness of the injection flow problem for more general case. The main results in this paper solve the open problem and establish the well-posedness results on the physical problem.

math.AP↗

arXiv2020

Existence and uniqueness of axially symmetric compressible subsonic jet impinging on an infinite wall

This paper is concerned with the well-posedness theory of the impact of a subsonic axially symmetric jet emerging from a semi-infinitely long nozzle, onto a rigid wall. The fluid motion is described by the steady isentropic Euler system. We showed that there exists a critical value $M_{cr}>0$, if the given mass flux is less than $M_{cr}$, there exists a unique smooth subsonic axially symmetric jet issuing from the given semi-infinitely long nozzle and hitting a given uneven wall. The surface of the axially symmetric impinging jet is a free boundary, which detaches from the edge of the nozzle smoothly. It is showed that a unique suitable choice of the pressure difference between the chamber and the atmosphere guarantees the continuous fit condition of the free boundary. Moreover, the asymptotic behaviors and the decay properties of the impinging jet and the free surface in downstream were also obtained. The main results in this paper solved the open problem on the well-posedness of the compressible axially symmetric impinging jet, which has proposed by A. Friedman in Chapter 16 in [FA2]. The key ingredient of our proof is based on the variational method to the quasilinear elliptic equation with the Bernoulli's type free boundaries.

math.AP↗

arXiv2020

A free boundary problem for semi-linear elliptic equation and its applications

In this paper, we consider a free boundary problem of a semilinear nonhomogeneous elliptic equation with Bernoulli's type free boundary. The existence and regularity of the solution to the free boundary problem are established by use of the variational approach. In particular, we establish the Lipschitz continuity and non-degeneracy of a minimum, and regularity of the free boundary. As a direct and important application, the well-posedness results on the steady, incompressible inviscid jet and cavitational flow with general vorticity are also obtained in this paper.

math.AP↗

arXiv2019

Interference and Efficient Transmission Range via V2V Communication at Roads Traffic Intersections

Vehicle-to-Vehicle (V2V) communication technology has dramatically promoted many promising applications to enhance traffic safety, mobility, and sustainability. However, However, we still lack the understanding of some fundamental properties of V2V technology under urban traffic conditions, such as interference at traffic intersections. Motivated by this view, this study develops the mathematical formulations to capture the worst-case interference at traffic intersections, considering the macroscopic traffic flow conditions and critical road geometric features including intersection diameter D, and intersection angle α. Built upon these formulations, we develop a mathematical model to approximate a conservative transmission range to sustain the successful V2V transmission at a traffic intersection. Our experiments illustrate that the proposed analytical formulations can provide accurate approximations for the interference and the corresponding transmission range at orthogonal (non-orthogonal) traffic intersections under various traffic congestion levels. Furthermore, this study conducted other experiments to understand how intersection geometric features (such as (D, α)) impact V2V communication at traffic intersections. The results illustrate that severer interference and smaller transmission range occur at a smaller intersection (with smaller diameter D) under heavy traffic congestion level. And the orthogonal intersection gives critical thresholds (such as severest interference and minimum transmission range) under all different traffic conditions, which help in understanding the V2V communication performance at an urban traffic intersection. These findings will potentially help to develop efficient MAC algorithms adaptive to urban traffic conditions, and further support various ITS applications using V2V communication. interference and transmission range.

eess.SP↗

arXiv2019

Coordination Group Formation for OnLine Coordinated Routing Mechanisms

This study considers that the collective route choices of travelers en route represent a resolution of their competition on network routes. Well understanding this competition and coordinating their route choices help mitigate urban traffic congestion. Even though existing studies have developed such mechanisms (e.g., the CRM [1]), we still lack the quantitative method to evaluate the coordination penitential and identify proper coordination groups (CG) to implement the CRM. Thus, they hit prohibitive computing difficulty when implemented with many opt-in travelers. Motived by this view, this study develops mathematical approaches to quantify the coordination potential between two and among multiple travelers. Next, we develop the adaptive centroid-based clustering algorithm (ACCA), which splits travelers en route in a local network into CGs, each with proper size and strong coordination potential. Moreover, the ACCA is statistically secured to stop at a local optimal clustering solution, which balances the inner-cluster and inter-cluster coordination potential. It can be implemented by parallel computation to accelerate its computing efficiency. Furthermore, we propose a clustering based coordinated routing mechanism (CB-CRM), which implements a CRM on each individual CG. The numerical experiments built upon both Sioux Falls and Hardee city networks show that the ACCA works efficiently to form proper coordination groups so that as compared to the CRM, the CB-CRM significantly improves computation efficiency with minor system performance loss in a large network. This merit becomes more apparent under high penetration and congested traffic condition. Last, the experiments validate the good features of the ACCA as well as the value of implementing parallel computation.

physics.soc-ph↗

arXiv2018

Incompressible impinging jet flow with gravity

In this paper, we investigate steady two-dimensional free-surface flows of an inviscid and incompressible fluid emerging from a nozzle, falling under gravity and impinging onto a horizontal wall. More precisely, for any given atmosphere pressure $p_{atm}$ and any appropriate incoming total flux $Q$, we establish the existence of two-dimensional incompressible impinging jet with gravity. The two free surfaces initiate smoothly at the endpoints of the nozzle and become to be horizontal in downstream. By transforming the free boundary problem into a minimum problem, we establish the properties of the flow region and the free boundaries. Moreover, the asymptotic behavior of the impinging jet in upstream and downstream is also obtained.

math.AP↗

arXiv2014

Two Dimensional Subsonic Euler Flow. Past a Wall or a Symmetric Body

The existence and uniqueness of two dimensional steady compressible Euler flows past a wall or a symmetric body are established. More precisely, given positive convex horizontal veloicty in the upstream, there exists a critical value $ρ_{cr}$ such that if the incoming density in the upstream is larger than $ρ_{cr}$, then there exists a subsonic flow past a wall. Furthermore, $ρ_{cr}$ is critical in the sense that there is no such subsonic flow if the density of the incoming flow is less than $ρ_{cr}$. The subsonic flows possess large vorticity and positive horizontal velocity above the wall except at the corner points on the boundary. Moreover, the existence and uniqueness of a two dimensional subsonic Euler flow past a symmetric body are also obtained when the incoming velocity field is a general small perturbation of a constant velocity field and the density of the incoming flow is larger than a critical value. The asymptotic behavior of the flows is obtained with the aid of some integral estimates for the velocity field and its far field states.

math.AP↗