Outline: Nonlinear partial differential equations (PDEs) have proven extremely successful in modeling the physical world, ranging from fluid mechanics, nonlinear acoustics and traffic flow. However, seemingly innocent looking PDEs can quickly give rise to nasty and intractable phenomena. Even the simplest looking example of a nonlinear PDE, Burgers’ equation, is such that its solutions become discontinuous in finite time. This project is devoted to exploring the family of scalar hyperbolic conservation laws, of which Burgers’ equation is an example. This will include developing the method of characteristics, understanding the regularizing effect of added viscosity, and performing the existence and uniqueness proof of entropy solutions in the L1 setting.
Sources: - Front Tracking for Hyperbolic Conservation Laws, Helge Holden, Nils Henrik Risebro,
- Burgers’ equation: https://en.wikipedia.org/wiki/Burgers%27_equation
- Conservation laws: https://en.wikipedia.org/wiki/Conservation_law