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 Navierâ???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)

    Abstract

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