Viscous conservation laws with boundary layers.

Wang Jing.Thesis (M.Phil.)--Chinese University of Hong Kong, 2005.Includes bibliographical references (leaves 55-59).Abstracts in English and Chinese.Acknowledgments --- p.iAbstract --- p.iiIntroduction --- p.3Chapter 1 --- Formulation of the Problem --- p.10Chapter 1.1 --- Reformulated Navier-Stokes Equations --- p.10Chapter 1.2 --- Linearized Problems --- p.15Chapter 2 --- Construction of the Approximate Solution --- p.19Chapter 2.1 --- Two-scale Asymptotic Expansions --- p.19Chapter 2.2 --- Determination of Each Inner and Boundary Terms --- p.22Chapter 2.3 --- Truncation Terms --- p.31Chapter 3 --- Estimates of the Error Term of the Approximate Solution and Main Results --- p.33Chapter 3.1 --- Error Equations --- p.33Chapter 3.2 --- Energy Estimates --- p.36Chapter 3.2.1 --- BasicL2 Estimates --- p.36Chapter 3.2.2 --- Tangential Derivatives Estimates --- p.38Chapter 3.2.3 --- Normal Derivatives Estimates --- p.49Chapter 3.3 --- Pointwise Estimates --- p.52Bibliography --- p.5

Boundary Effect on Asymptotic Behaviour of Solutions to the p-System with Linear Damping

AbstractWe consider the asymptotic behaviour of solutions to the p-system with linear damping on the half-line R+=(0, ∞),vt−ux=0,ut+p(v)x=−αu, with the Dirichlet boundary condition u|x=0=0 or the Neumann boundary condition ux|x=0=0. The initial date (v0, u0)(x) has the constant state (v+, u+) at x=∞. L. Hsiao and T.-P. Liu [Commun. Math. Phys.143 (1992), 599–605] have shown that the solution to the corresponding Cauchy problem behaves like diffusion wave, and K. Nishihara [J. Differential Equations131 (1996), 171–188; 137 (1997), 384–395] has proved its optimal convergence rate. Our main concern in this paper is the boundary effect. In the case of null-Dirichlet boundary condition on u, the solution (v, u) is proved to tend to (v+, 0) as t tends to infinity. Its optimal convergence rate is also obtained by using the Green function of the diffusion equation with constant coefficients. In the case of null-Neumann boundary condition on u, v(0, t) is conservative and v(0, t)≡v0(0) by virtue of the first equation, so that v(x, t) is expected to tend to the diffusion wave v(x, t) connecting v0(0) and v+. In fact the solution (v, u)(x, t) is proved to tend to (v(x, t), 0). In the special case v0(0)=v+, the optimal convergence rate is also obtained. However, this is not known in the case v0(0)≠v+

Asymptotic behavior of solutions for damped wave equations with non-convex convection term on the half line

We study the asymptotic stability of nonlinear waves for damped wave equations with a convection term on the half line. In the case where the convection term satisfies the convex and sub-characteristic conditions, it is known by the work of Ueda [7] and Ueda-Nakamura-Kawashima [10] that the solution tends toward a stationary solution. In this paper, we prove that even for a quite wide class of the convection term, such a linear superposition of the stationary solution and the rarefaction wave is asymptotically stable. Moreover, in the case where the solution tends to the non-degenerate stationary wave, we derive that the time convergence rate is polynomially (resp. exponentially) fast if the initial perturbation decays polynomially (resp. exponentially) as x → ∞. Our proofs are based on a technical L 2 weighted energy method

Suppression of blow-up in 3-D Keller-Segel model via Couette flow in whole space

In this paper, we study the 3-D parabolic-parabolic and parabolic-elliptic Keller-Segel models with Couette flow in $\mathbb{R}^3$. We prove that the blow-up phenomenon of solution can be suppressed by enhanced dissipation of large Couette flows. Here we develop Green's function method to describe the enhanced dissipation via a more precise space-time structure and obtain the global existence together with pointwise estimates of the solutions. The result of this paper shows that the enhanced dissipation exists for all frequencies in the case of whole space and it is reason that we obtain global existence for 3-D Keller-Segel models here. It is totally different from the case with the periodic spatial variable $x$ in [2,10]. This paper provides a new methodology to capture dissipation enhancement and also a surprising result which shows a totally new mechanism.Comment: 22 pag