This paper studies existence of mild solution to a sharp cut off model for contact driven tumor growth.Analysis is based on application of the Crandall-Liggett theorem for w-quas-contractive semigroups on the Banach s...This paper studies existence of mild solution to a sharp cut off model for contact driven tumor growth.Analysis is based on application of the Crandall-Liggett theorem for w-quas-contractive semigroups on the Banach space L^(1)(Ω).Furthermore,numerical computations are provided which compare the sharp cut off model with the tumor growth model of Perthame,Quiros,and Vazquez[13].展开更多
In [16] a visco-elastic relaxation system, called the relaxed Burnett system, was proposed by Jin and Slemrod as a moment approximation to the Boltzmann equation. The relaxed Burnett system is weakly parabolic, has a ...In [16] a visco-elastic relaxation system, called the relaxed Burnett system, was proposed by Jin and Slemrod as a moment approximation to the Boltzmann equation. The relaxed Burnett system is weakly parabolic, has a linearly hyperbolic convection part, and is endowed with a generalized entropy inequality. It agrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman-Enskog expansion. We develop a one-dimensional non-oscillatory numerical scheme based on the relaxed Burnett system for the Boltzmann equation. We compare numerical results for stationary shocks based on this relaxation scheme, and those obtained by the DSMC (Direct Simulation Monte Carlo), by the Navier-Stokes equations and by the extended thermodynamics with thirteen moments (the Grad equations). Our numerical experiments show that the relaxed Burnett gives more accurate approximations to the shock profiles of the Boltzmann equation obtained by the DSMC, for a range of Mach numbers for hypersonic flows, than those obtained by the other hydrodynamic systems.展开更多
基金This work was supported in part by National Research Foundation of Korea(NRF-2017R1A2B2010398)The authors thank Profs.L.C.Evans and W.Strauss for their valuable suggestions.
文摘This paper studies existence of mild solution to a sharp cut off model for contact driven tumor growth.Analysis is based on application of the Crandall-Liggett theorem for w-quas-contractive semigroups on the Banach space L^(1)(Ω).Furthermore,numerical computations are provided which compare the sharp cut off model with the tumor growth model of Perthame,Quiros,and Vazquez[13].
基金Supported by NSF grant DMS-0196106 Supported by NSF grant DMS-9803223 and DMS-00711463.
文摘In [16] a visco-elastic relaxation system, called the relaxed Burnett system, was proposed by Jin and Slemrod as a moment approximation to the Boltzmann equation. The relaxed Burnett system is weakly parabolic, has a linearly hyperbolic convection part, and is endowed with a generalized entropy inequality. It agrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman-Enskog expansion. We develop a one-dimensional non-oscillatory numerical scheme based on the relaxed Burnett system for the Boltzmann equation. We compare numerical results for stationary shocks based on this relaxation scheme, and those obtained by the DSMC (Direct Simulation Monte Carlo), by the Navier-Stokes equations and by the extended thermodynamics with thirteen moments (the Grad equations). Our numerical experiments show that the relaxed Burnett gives more accurate approximations to the shock profiles of the Boltzmann equation obtained by the DSMC, for a range of Mach numbers for hypersonic flows, than those obtained by the other hydrodynamic systems.
