RESEARCH INTERESTS
GENERAL INTRODUCTION
(\u-\xi)\rho'&+ \rho \u' = 0,\cr (\u-\xi)\rho \u'+& p(\rho)' = \eps \u'',\cr (1) \rho(\pm\infty)=\rho_\pm&,\u(\pm\infty)=\u_\pm.
u_t+uu_x=\mu u_{xx},\quad u(x,0)=u_0(x), (2)
which causes the metastability of the problem.
For a fixed small viscosity $\mu >0$ the solution $u(\cdot,t)$ evolves into
a diffusion wave of a single hump structure (see [H]).
On the other hand, for a fixed large time $t>>1$,
the solution converges to an N-wave as $\mu \to 0$. To capture the general
mechanism for the behavior effectively we consider self-similar variables
$s=\ln(t),\xi=x/\sqrt{t}$ and $w(\xi,s)=\sqrt{t}u(x,t)$,
and transform the Burgers equation to
w_s+{1\over2}(w^2-\xi w)_\xi=\mu w_{\xi\xi}. (3)
It allows to estimate the time
of evolution from an N-wave to the final stage of a diffusion wave.
Standard hyperbolic theories of $L^{1}$-contraction and Oleinik inequality
for the self-similar Burgers (3) are considered.
We show how to use a special Lyapunov function to establish the asymptotic
in time profile (for $\mu$ fixed) of a diffusion wave.
u_t+f(u)_x=0, (4)
u_t=\mu u_{xx}, (5)
u_t+f(u)_x=\mu u_{xx}, (6)
|A\sin(x), -\pi\le x\le 0,
u(x,0)=-|B\sin(x), 0\le x\le \pi, (7)
| 0, otherwise,
where $B>A>0$. Let $A(t)$ and $B(t)$ be the area of
the negative part and the positive part of solutions at time $t$.
It is well known that if the flux of (4) is convex, $A(t),B(t)$ are invariant
variables for the solution of (4,7). On the other hand one of them converges
to zero for the solution of (5,7). To answer how $A(t)$ and $B(t)$
evolve for the problem (6,7) it is needed to understand the correlation
of the convection and the diffusion terms closely.
A(t)\le\eps,\quad \text{for all } t\ge T(\eps ,\mu ), (8)
for given $\mu,\eps>0$. Our first result of this project comes
from posing this question to the heat equation. We show that
$T(\eps,\mu)=O(\ln(\eps)/\mu)$, and the convergence to diffusive N-waves
has been studied, and it is transfered to Burgers under the Cole-Hopf
transformation, see [5]. We are pursuing the idea for a general case.
(u+u^p)_t+u_x=u_{xx}, u(x,0)=u_0(x), p>0. (9)
This model has been suggested in [DDG]. This topic is in progress.
u_t+uu_x=u,\quad u(x,0)=u_0(x) (10)with mean zero periodic initial data, which is a simple model of roll waves. It is easily verified that the total mass over a period $M(t)=\int_0^L u(x,t)dx$ satisfies $M(t)=e^tM(0)$. So the total mass increases exponentially if the initial total mass is not identically zero. In this case even a small rounding off error of numerical computations may grow up and eventually breaks down the solution. That is the reason for the instability observed in [JK]. Analytic behavior of (10) is considered together with numerical solutions in [3]. We suggest interface method to provide numerically stable solutions for the problem.
u_t+f'(u)u_x=\mu u_{xx}+f'(u). (11)
Numerical solutions of this example is studied in [SW].
I have interest for the case with highly oscillating initial data.
In the case we can observe different rolls of these three terms clearly.
This topic is in preparation.
Project 2, Piecewise self-similar solutions($L^1$ solutions for inviscid problems)
u_t+f(u)_x=0, u(\cdot,0)\in L^1(\Bbb R), (12)is a function of $\xi=x/t$. Then a characteristic line should be a line that passes through the origin, $f'(u)=x/t$. Roughly speaking (see [4] for detail), a piecewise self-similar solution has a structure,
f'(u)=(x-z_k)/(t+t_k),s_{k-1}(t)\le x\le s_k(t), k=1,...,N, (13)
and zero outside. If initial data are given as this form, the solution of (12)
keeps this structure. This is the way to fully recover the self-similarity
of conservation laws.
u(x,t)=(f')^{-1}((x-z_k)/(t+t_k)),s_{k-1}(t)\le x\le s_k(t),k=1,...,N.(14)
(14) is valid if $f'$ is invertible, and it is under the hypothesis
f''(u)\ne 0. (H)
v_t+((v^2+w^2)v)_x\;=\;\eps v_{xx},
w_t+((v^2+w^2)w)_x=\eps w_{xx}, (15)
we expect to observe similar relation between convection and diffusion terms
as the scalar case. Basically systems are different from scalar case
since there are interactions between the elements.
The inviscid case of the model is also interesting from the view point of
piecewise self-similar solutions, Project 2. Then (13) corresponds to
relations between eigen-values and eigen-vectors.
The focus of the project should be how to transfer the properties enjoyed
by self-similarity solutions of scalar equations to solutions
of the system. The study of the general hyperbolic system with respect to
self-similar solutions is the final goal of Project 2.