CPAA
Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux
Tong Li Hui Yin
Communications on Pure & Applied Analysis 2014, 13(2): 835-858 doi: 10.3934/cpaa.2014.13.835
This paper is concerned with the initial-boundary value problem for the generalized Benjamin-Bona-Mahony-Burgers equation in the half space $R_+$ \begin{eqnarray} u_t-u_{txx}-u_{xx}+f(u)_{x}=0,\quad t>0, x\in R_+,\\ u(0,x)=u_0(x)\to u_+, \quad as \ \ x\to +\infty,\\ u(t,0)=u_b. \end{eqnarray} Here $u(t,x)$ is an unknown function of $t>0$ and $x\in R_+$, $u_+\not=u_b$ are two given constant states and the nonlinear function $f(u)$ is assumed to be a non-convex function which has one or finitely many inflection points. In this paper, we consider $u_b
keywords: convergence rate. global stability Generalized Benjamin-Bona-Mahony-Burgers equation non-convex flux boundary layer solution
NHM
Critical thresholds in a quasilinear hyperbolic model of blood flow
Tong Li Sunčica Čanić
Networks & Heterogeneous Media 2009, 4(3): 527-536 doi: 10.3934/nhm.2009.4.527
Critical threshold phenomena in a one dimensional quasi-linear hyperbolic model of blood flow with viscous damping are investigated. We prove global in time regularity and finite time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the blood flow model. New results are obtained showing that the class of data that leads to global smooth solutions includes the data with negative initial Riemann invariant slopes and that the magnitude of the negative slope is not necessarily small, but it is determined by the magnitude of the viscous damping. For the data that leads to shock formation, we show that shock formation is delayed due to viscous damping.
keywords: blood flow quasi-linear hyperbolic system global regularity. Critical thresholds finite-time singularities
DCDS-B
Well-posedness theory of an inhomogeneous traffic flow model
Tong Li
Discrete & Continuous Dynamical Systems - B 2002, 2(3): 401-414 doi: 10.3934/dcdsb.2002.2.401
We study a traffic flow model with inhomogeneous road conditions such as obstacles. The model is a system of nonlinear hyperbolic equations with both relaxation and sources. The flux and the source terms depend on the space variable. Waves for such a system propagate in a more complicated way than those do for models with homogeneous road conditions.
The $L^1$ well-posedness theory for the model is established. In particular, we derive the continuous dependence of the solution on its initial data in $L^1$ topology. Moreover, the $L^1$-convergence to the unique zero relaxation limit is proved. Finally, the asymptotic states of a general solution whose initial data tend to constant states as $|x| \rightarrow +\infty$ are constructed.
keywords: $L^1$-contraction Inhomogeneous marginally stable zero relaxation limit.
NHM
Qualitative analysis of some PDE models of traffic flow
Tong Li
Networks & Heterogeneous Media 2013, 8(3): 773-781 doi: 10.3934/nhm.2013.8.773
We review our previous results on partial differential equation(PDE) models of traffic flow. These models include the first order PDE models, a nonlocal PDE traffic flow model with Arrhenius look-ahead dynamics, and the second order PDE models, a discrete model which captures the essential features of traffic jams and chaotic behavior. We study the well-posedness of such PDE problems, finite time blow-up, front propagation, pattern formation and asymptotic behavior of solutions including the stability of the traveling fronts. Traveling wave solutions are wave front solutions propagating with a constant speed and propagating against traffic.
keywords: traveling waves nonlocal flux blowup criteria Arrhenius look-ahead dynamics Traffic flow shock waves asymptotic behavior chaos.
DCDS-B
On a quasilinear hyperbolic system in blood flow modeling
Tong Li Kun Zhao
Discrete & Continuous Dynamical Systems - B 2011, 16(1): 333-344 doi: 10.3934/dcdsb.2011.16.333
This paper aims at an initial-boundary value problem on bounded domains for a one-dimensional quasilinear hyperbolic model of blood flow with viscous damping. It is shown that, for given smooth initial data close to a constant equilibrium state, there exists a unique global smooth solution to the model. Time asymptotically, it is shown that the solution converges to the constant equilibrium state exponentially fast as time goes to infinity due to viscous damping and boundary effects.
keywords: initial-boundary value problem global existence long-time behavior. Hyperbolic balance laws blood flow classical solution
NHM
Shock formation in a traffic flow model with Arrhenius look-ahead dynamics
Dong Li Tong Li
Networks & Heterogeneous Media 2011, 6(4): 681-694 doi: 10.3934/nhm.2011.6.681
We consider a nonlocal traffic flow model with Arrhenius look-ahead dynamics. We provide a complete local theory and give the blowup alternative of solutions to the conservation law with a nonlocal flux. We show that the finite time blowup of solutions must occur at the level of the first order derivative of the solution. Furthermore, we prove that finite time singularities do occur for several types of physical initial data by analyzing the solutions on different characteristic lines. These results are new and are consistent with the blowups observed in previous numerical simulations on the nonlocal traffic flow model [6].
keywords: Traffic flow nonlocal flux shock waves Arrhenius look-ahead dynamics blowup criteria.
DCDS
Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems
Tong Li Anthony Suen
Discrete & Continuous Dynamical Systems - A 2016, 36(2): 861-875 doi: 10.3934/dcds.2016.36.861
We prove the global-in-time existence of intermediate weak solutions of the equations of chemotaxis system in a bounded domain of $\mathbb{R}^2$ or $\mathbb{R}^3$ with initial chemical concentration small in $H^1$. No smallness assumption is imposed on the initial cell density which is in $L^2$. We first show that when the initial chemical concentration $c_0$ is small only in $H^1$ and $(n_0-n_\infty,c_0)$ is smooth, the classical solution exists for all time. Then we construct weak solutions as limits of smooth solutions corresponding to mollified initial data. Finally we determine the asymptotic behavior of the global solutions.
keywords: chemotaxis energy estimates intermediate weak solution asymptotic behavior. global existence Keller-Segel model
DCDS
Critical thresholds in a relaxation system with resonance of characteristic speeds
Tong Li Hailiang Liu
Discrete & Continuous Dynamical Systems - A 2009, 24(2): 511-521 doi: 10.3934/dcds.2009.24.511
We study critical threshold phenomena in a dynamic continuum traffic flow model known as the Payne and Whitham (PW) model. This model is a quasi-linear hyperbolic relaxation system, and when equilibrium velocity is specifically associated with pressure, the equilibrium characteristic speed resonates with one characteristic speed of the full relaxation system. For a scenario of physical interest we identify a lower threshold for finite time singularity in solutions and an upper threshold for the global existence of the smooth solution. The set of initial data leading to global smooth solutions is large, in particular allowing initial velocity of negative slope.
keywords: singularity formation traffic flow. Critical thresholds quasi-linear relaxation model global regularity
NHM
Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model
Tong Li Kun Zhao
Networks & Heterogeneous Media 2011, 6(4): 625-646 doi: 10.3934/nhm.2011.6.625
This paper is concerned with an initial-boundary value problem on bounded domains for a one dimensional quasilinear hyperbolic model of blood flow with viscous damping. It is shown that $L^\infty$ entropy weak solutions exist globally in time when the initial data are large, rough and contains vacuum states. Furthermore, based on entropy principle and the theory of divergence measure field, it is shown that any $L^\infty$ entropy weak solution converges to a constant equilibrium state exponentially fast as time goes to infinity. The physiological relevance of the theoretical results obtained in this paper is demonstrated.
keywords: Large data Large-time behavior. Initial-boundary value problem Blood flow Entropy weak solution Hyperbolic balance laws Global existence
DCDS-B
Global existence and large time behavior of a 2D Keller-Segel system in logarithmic Lebesgue spaces
Chao Deng Tong Li
Discrete & Continuous Dynamical Systems - B 2018, 22(11): 1-13 doi: 10.3934/dcdsb.2018093

This paper is devoted to the global analysis for the two-dimensional parabolic-parabolic Keller-Segel system in the whole space. By well balanced arguments of the $L^1$ and $L^∞$ spaces, we first prove global well-posedness of the system in $L^1× L^∞$ which partially answers the question posted by Kozono et al in [19]. For the case $μ_0>0$, we make full use of the linear parts of the system to get the improved long time decay property. Moreover, by using the new formulation involving all linear parts, introducing the logarithmic-weight in time to modify the other endpoint space $L^∞× L^∞$, and carefully decomposing time into several pieces, we are able to establish the global well-posedness and large time behavior of the system in $L^∞_{ln}× L^∞$.

keywords: The Keller-Segel model of chemotaxis 2D parabolic system global well-posedness large time behavior logarithmic Lebesgue spaces

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