## MBI Publications

### MBI Publications for Kun Zhao (18)

• K. Zhao
Long-time behavior of a model for viscous two-phase fluid flows under the influence of gravitational force
(In Preparation)

#### Abstract

• S. Dai, D. Li, C. Xue and K. Zhao
On the mechanical origin of scratch wound healing: A particle based model
(In Preparation)

#### Abstract

• D. Li, R. Pan and K. Zhao
Quantitative decay of a hybrid type chemotaxis model with large data
(In Preparation)

#### Abstract

• J. Lowengrub, E. Titi and K. Zhao
Understand solid tumor growth by a diffuse interface model
(In Preparation)

#### Abstract

• S. Dai, D. Li and K. Zhao
A cell-based model for scratch wound healing
(In Preparation)

#### Abstract

• K. Zhao
On the isothermal compressible Euler equations with frictional damping
Communications in Mathematical AnalysisVol. 9 (2010) pp. 77-97

#### Abstract

This paper aims at initial-boundary value problems(IBVP) for the isothermal compressible Euler equations with damping on bounded domains. We first prove global existence and uniqueness of classical solutions for smooth initial data. Time asymptotically, it is shown that the density converges to its average over the domain and the momentum vanishes as time tends to infinity. Due to diffusion and boundary effects, the convergence rate is shown to be exponential. Second, based on the entropy principle, it is shown that similar results hold for $L^infty$ entropy weak solutions.
• K. Zhao
2D inviscid heat conductive Boussinesq equations on a bounded domain
Michigan Mathematical JournalVol. 59 (2010) pp. 329-352

#### Abstract

• K. Zhao and T. Li
Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model
Network and Heterogeneous MediaVol. 6, 2011 No. 4 (2011) pp. 625-646

#### Abstract

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 L1 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 L1 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.
• K. Zhao
Large time behavior for Cahn-Hilliard-Boussinesq equations on bounded domains
Electronic Journal of Differential EquationsVol. 2011 No. 46 (2011) pp. 1-21

#### Abstract

We study the asymptotic behavior of classical solutions to an

initial-boundary value problem (IBVP) for a coupled

Cahn-Hilliard-Boussinesq system on bounded domains with

large initial data. A sufficient condition is established

under which the solutions decay exponentially to constant

states as time approaches infinity.
• D. Li, T. Li and K. Zhao
On a hyperbolic-parabolic system modeling repulsive chemotaxis
Mathematical Models and Methods in Applied SciencesVol. 21 (2011) pp. 1631-1650

#### Abstract

• T. Li, K. Zhao, R. Pan and R. Pan
Global dynamics of a chemotaxis model on bounded
SIAM Journal on Applied Mathematics (2012)

#### Abstract

We prove global existence and qualitative behavior of classical solutions for
a hyperbolic-parabolic system describing chemotaxis on bounded domains. It is shown
that classical solutions to the initial-boundary value problem of the one-dimensional
model exist globally in time for large initial data and the solutions converge to constant
equilibrium states exponentially in time, which rigorously demonstrates the collapsing of
cell populations in chemotaxis. Moreover, similar results are established for the multi-
dimensional model when the initial data are small.
• K. Zhao
Long-time dynamics of a coupled Cahn-Hilliard-Boussinesq system
Communications in Mathematical SciencesVol. 10 (2012)

#### Abstract

We study large-time asymptotic behavior of classical solutions to an initial-boundary

value problem (IBVP) for a coupled Cahn-Hilliard-Boussinesq system on a bounded domain. Sufficient

conditions are established under which classical solutions converge exponentially to constant

states as time goes to infinity due to diffusion and boundary effects.
• K. Zhao
Large time behavior of density-dependent incompressible Navier-Stokes equations on bounded domains
Journal of Mathematical Fluid Mechanics (2012)

#### Abstract

The large-time asymptotic behavior of classical solutions to the density-dependent incompressible NavierÃ¢â?¬â??Stokes

equations driven by an external force on bounded domains in 2-D is studied. It is shown that the velocity field and its

first-order derivatives converge to zero as time goes to infinity for large initial data and external forces.
• M. Lai, R. Pan, K. Zhao and .
Initial boundary value problem for 2D viscous Boussi- nesq equations
Rational Mechanics and AnalysisVol. 199 (2012) pp. 739-760

#### Abstract

We study the initial boundary value problem of 2D viscous Boussinesq equations

over a bounded domain with smooth boundary. We show that the equations have

a unique classical solution for H3 initial data and no-slip boundary condition. In

addition, we show that the kinetic energy is uniformly bounded in time.
• T. Li and K. Zhao
On a quasilinear hyperbolic system in blood flow modeling
Discrete and Continuous Dynamical SystemsVol. 16 No. Series B (2012) pp. 333-344

#### Abstract

This paper aims at an initial-boundary value problem on bounded

domains for a one-dimensional quasilinear hyperbolic model of blood

ow 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 eff ects.
• K. Zhao
Global regularity for a coupled Cahn-Hilliard-Boussinesq system on bounded domains
Quarterly of Applied MathematicsVol. 69 (2012) pp. 331-356

#### Abstract

We study large-time asymptotic behavior of classical solutions to an initial-boundary

value problem (IBVP) for a coupled Cahn-Hilliard-Boussinesq system on a bounded domain. Sufficient

conditions are established under which classical solutions converge exponentially to constant

states as time goes to infinity due to diffusion and boundary effects.
• T. Li, R. Pan and K. Zhao
Global dynamics of a chemotaxis model on bounded domains with large data
SIAM Journal on Applied MathematicsVol. 72 No. 1 (2012)

#### Abstract

• S. Dai, D. Li and K. Zhao
Finite-time quenching of competing species with constrained border evaporation
DCDS-B (2012) (Submitted)