RESEARCH INTERESTS
Applied Mathematics, Nonlinear PDEs, Numerical Analysis
AMS subject classifications. 76N10,65M06(76M20,35L67,35L65)
GENERAL INTRODUCTION
My research has focused on viscosity approximations of conservation laws. In general the solution of non-linear conservation laws breaks down in finite time, and hence a weak solution is considered as a global solution. In that case the weak solution is not unique anymore and the solution is usually studied under appropriate admissibility criteria to find the physically meaningful one.
Another way to resolve the situation is to consider regularized equations. For example, if we add viscosity terms to the problems, they have smooth solutions and the zero viscosity limit will provide us the admissible weak solution. These regularized problems are actually realistic since they are models of physical phenomena. We may consider different kinds of regularizing terms to a specific problem. Some of them can be closer to the real wold model and the others can be more artificial for the sake of mathematical analysis. Different models provide different views of the solution and different levels of the difficulty of mathematical analysis. Even though it is expected that the sequence of solutions will converge to the same limit as the regularizing terms go to zero.
RESEARCH PROJECTS
Project 1 : Self Similar Approach for Hyperbolic Systems
In the thesis we consider the Riemann problem for the system of conservation laws of one dimensional isentropic gas dynamics in Eulerian coordinates. We know that the system has the self-similar property, i.e., the solution is a function of $x/t$, and hence it is natural to try to keep the property when we consider its regularized problem. As suggested by Dafermos in [4], and followed in many articles, we consider the system with a self-similar viscosity,
\rho_t+(\rho \u)_x = 0,
(\rho \u)_t + (\rho& \u^2 +p(\rho))_x = \eps tu_{xx}, (1)
\rho(\pm\infty,0)=&\rho_\pm,\quad\u(\pm\infty,0)=\u_\pm,
where $\rho_\pm,t\in \Bbb R^+$ and $\u_\pm,x,u,\rho\in \Bbb R$.
Since this problem is invariant under dilations, $(x,t)\to(ax,at)$,
the solution should be a function of invariant variable $\xi=x/t$.
We can easily transform the problem to ODEs under
the invariance variable,
(\u-\xi)\rho'&+ \rho \u' = 0, (\u-\xi)\rho \u'+& p(\rho)' = \eps \u'', (2) \rho(\pm\infty)=\rho_\pm&,\u(\pm\infty)=\u_\pm.
The problem is considered without size restrictions on the data. So our biggest concern is the appearance of the vacuum state. For the existence we apply the Leray--Schauder type degree theory which is applied under a-priori estimates. We also study the structure of solution and provide a good understanding of the structure near vacuum which is impossible in Lagrangian coordinates. This kind of techniques can be applied to different hyperbolic systems. Actually similar approach is in progress for the one-dimensional navier-stokes equations for compressible flow.
For a general N-equations case,
U_t+F(U)_x=\eps t(B U_x)_x, (3)local properties are studied successfully for the identity viscosity matrix $B=I$, [5]. We have considered the problem with more general viscosity matrix. In that case the effects of the viscosity is more complicate since we can not single out the dominant term for each characteristic fields and it makes the analysis harder. In this case the problem is really challenging and it will be my future work.
Works and Results:
[1] Yong Jung Kim, A self-similar viscosity approach for the Riemann problem in isentropic gas dynamics and the structure of the solutions (submitted to Quart. Appl. Math.,ps-file : http://www.math.wisc.edu/\~jkim/ps/isen.ps)
Project 2 : Evolution of Viscous Conservation Laws and Invariance Property
When we study the evolution of a dynamic system, one of the main interests is its asymptotic behavior. In particular we have interest in a Cauchy problem of conservation laws,
u^\eps_t+f(u^\eps)_x=\eps u^\eps_{xx} (4)
u^\eps(x,0)=u_0(x).
The long time behavior of the viscous Burgers equation, $f(u)=u^2/2$,
has been studied in various situations.
It is well known that for a fixed small viscosity $\eps>0$ the solution
$u^\eps(\cdot,t)$ evolves into an N-wave in the first stage and
finally gets the structure of a single hump as $t\to\infty$ (see [6],[7]).
It takes really long time to get the second stage for a reasonably small
$\eps>0$. On the other hand, for a fixed large time $t>0$,
the solution converges to an N-wave as $\eps\to 0$.
So $\eps\to0,t\to\infty$ limiting process produce different
structure if we change the order of the limit.
These studies are mostly based on the analysis of the explicit
solution of the Burgers equation, and it is not easy to see how the
system evolves. Furthermore it is not even possible to capture the general
mechanism for the behavior of this kind of problems.
We believe that we need to develop another method which is based on the equation, not on the explicit solution to resolve these difficulties. First we consider the invariance property of the dynamical system (4). For example, in the Burgers case, the equation is carried into itself by the well-known group of substitutions $x\to cx, t\to c^2t, u\to u/c$. The equation does not contain $t$ explicitly, and hence it is also invariant under the substitution of $t\to t+a$. So it is natural to study the problem in terms of variables $s=\ln(t),\xi=x/\sqrt{t}$ and $w(\xi,s)=\sqrt{t}u(x,t)$, which are invariant under these substitutions. If we take these variables, the Burgers equation is transformed to
w_s+({1\over2}w^2-{1\over2}\xi w)_\xi=\mu w_{\xi\xi}, (5)
which is also a conservation law with a flux which depends on the space
variables $\xi$. One of the biggest merit of the transformed equation is that it
has a family of non-zero equilibrium solutions. The original Burgers equation
has only one stable state which is identically zero and the solution
evolves into a zero function eventually. On the other hand the solution of
Equation (5) converges to one of the non-zero equilibrium solutions
as $s\to\infty$. This fact makes us possible to study the asymptotic behavior
in a strong sense and all the arguments can be simplified considerably.
These equilibrium solutions also play a key role in the
estimates of the solution. We establish the existence, convergence and the
structure of the limit along the two different indexes $\eps$ and $s$ in [2].
This approach enables us to see how the solution evolves and provide us a
qualitative way to explain its asymptotic behavior.
Our current objective is to apply the method to a more general scalar conservation law or to a system of the similar asymptotic behavior. If the flux is a homogeneous function of order $\alpha>1$, i.e., $f(au)=a^\alpha f(u)$, then we can easily see that a non-viscous conservation law is invariant under the group of substitutions $x\to cx, t\to c^\alpha t, u\to u/c$. So $s=\ln(t),\xi=t^{-1/\alpha}x$ and $w(\xi,s) =t^{1/\alpha}u(x,t)$ are the invariant variables. Unlike the case of Burgers equation, viscous conservation law does not have invariance property if $\alpha \ne 2$. Even though these variables play the key role in the evolution of the solution.
This method is successful to a class of scalar conservation law, where flux functions are given by $f(u)=\sign(u)|u|^\alpha$ or $f(u)=|u|^\alpha$ in [3]. In particular cubic law $f(u)=u^3$ is an example of the case. Of course there should be a significant modification of the method, but the frame of the work is pretty similar. In fact the asymptotic behavior of non-viscous problems was studied in [8], under the Kruzkov entropy condition. We compare $\eps\to 0$ limit of the viscous problem with these solutions. We also study $t\to \infty$ limit for a fixed small $\eps>0$.
We expect that the above problems will provide techniques to deal with the more complicated system of magnetohydrodynamics,
v_t+((v^2+w^2)v)_x&=\eps v_{xx} (6)
w_t+((v^2+w^2)w)_x&=\eps w_{xx}.
This system has an odd flux which is homogeneous of order $3$.
So it has invariant variables which are same as the cubic law and
we may expect that the system has similar properties.
Basically systems are different from scalar case since there are interactions
between the elements. It is really an interesting and challenging
problem and we are going to concentrate on it.
In the sense of numerical computation this approach is very successful. Actually numerical schemes which are based on Equation (4) do not provide the asymptotic behavior well. The reason is that it takes really a long time to get the second stage of the single hump. In an example from [2] the second stage has been reached when $t\sim e^{22}=3584912846$. Furthermore the solution spreads out along the $x$-axis and we have to take the interval of $x$-axis very big to fulfill the computation. So it looks hopeless to finish it. Even if we can do the computation, it is possible that the numerical solution becomes identically zero before getting the asymptotic structure. On the other hand numerical schemes which are based on the invariant variables show us the exact long time behavior of the solution. The time variable $s=\ln(t)$ makes the process exponentially short and the space variable $\xi=x/\sqrt{t}$ makes the interval of computation small. We can see how the solution evolves from a numerical example in [2]. This method can compute long time behavior effectively for a large class of the dynamical systems.
Works and Results:
[2]Yong Jung Kim and Athanasios E. Tzavaras, Asymptotic behavior of the viscous Burgers equation under the variables of invariance property and numerical examples (in preparation, ps-file : http://www.math.wisc.edu/\~jkim/ps/burgers.ps)
[3] Yong Jung Kim and Athanasios E. Tzavaras, Asymptotic behavior of viscous scalar conservation laws with homogeneous flux (in progress)
{\bf Project 3: Conservation Laws with Source Terms and Numerical Schemes} This project is what I want to do in the near future. Here I explain the basic idea and the plan. We consider one dimensional conservation laws with source terms,
u_t+f(u,x,t)_x+\psi(u,x,t)=0, (7)where $x\in\Bbb R,t\in\Bbb R^+$ and $u=u(x,t),f,\psi\in\Bbb R^N$. For the scalar case, $N=1$, it has been studied in some cases. In most of the cases the flux $f$ is a function of $u$ only and the source term $\psi$ is usually a function of space variable $x$ as in [9], or $u$ as in [10]. In the previous project we also have seen an example that the flux depends on the space variable. We may develop a technique of change of variables which is based on the characteristics to simplify the equation. This approach was successful in some cases. For example, Burgers equation with a linear source term can be transformed to a homogeneous Burgers equation. It will be interesting to study the basic mechanism why the change of variables works in those cases. The main part of this project is to understand the equation as a two dimensional conservation law,
u_t+(f(u,x,t)+x\psi(u,x,t)-y\psi(u,x,t))_x =0, (8)where the solution $u=u(x,y,t)$ is a function of two space variables. Of course our interest is in the behavior of the solution along the diagonal $y=x$.
In the sense of numerical schemes this approach provides an easy way to develop conservative schemes for inhomogeneous conservation laws. For example, if we directly apply the Lax-Friedrichs scheme to Equation (8) and then substitute $x$ into $y$, we get a conservative method for the solution of Equation (7). One of the merit of this approach is that we can expand the analysis of conservative schemes to inhomogeneous cases. For example we can easily check that this kind of schemes always produce the weak solution in the sense of Lax-Wendroff. It is needed to continue the analysis of schemes for the stability convergence, entropy condition, etc. Note that some of the conservative schemes are not applied to Equation (8) directly. For example the Godunov scheme is not applied easily since the scheme depends on the fact that characteristics are strait lines, which is not the case of Equation (7).