We study linear and nonlinear Schrodinger equations defined by fractal measures. Under the assumption that the Laplacian has compact resolvent, we prove that there exists a uniqueness weak solution for a linear Schrodinger equation, and then use it to obtain the existence and uniqueness of weak solution of a nonlinear Schrodinger equation. We prove that for a class of self-similar measures on R with overlaps, the Schrodinger equations can be discretized so that the finite element method can be applied to obtain approximate solutions. We also prove that the numerical solutions converge to the actual solution and obtain the rate of convergence.

We study the averaged behavior of interfaces moving with oscillatory normal velocity that is periodic in time and stationary ergodic in space. This problem can be interpreted as a homogenization problem of a Hamilton-Jacobi equation with a positively homogeneous and time dependent Hamiltonian. In an earlier work, we studied the setting when the environment is periodic in space and random in time, and established homogenization results through the averaging of reachable sets. In the present setting, we show that the minimal travel time between two spatial points, which now depends also on a starting time, has a deterministic averaging limit that is independent of the starting time. This averaged travel time function is then shown to determine the effective behavior of the moving interfaces.

As a typical example of hyperbolic conservation laws, the system of Euler equations representing the conservation of mass, momentum and energy has three basic wave patterns. They are two nonlinear waves called shock and rarefaction wave, and one linearly degenerate wave called contact discontinuity. These basic wave patterns have their counterparts both in other physical models for gas motions in equilibrium involving viscosity and heat conductivity, and gas motion in non-equilibrium. The stability of these basic wave patterns in the system of Navier-Stokes equations and the Boltzmann equation has been an active research topic. Even though the stability of the two nonlinear wave patterns has been extensively studied, the stability of the linearly degenerate contact wave was not solved until recently. In this paper, we will briefly present our recent results in [32] on the stability of the contact wave patterns for both the Navier-Stokes equations and the Boltzmann equation.

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The weak well-posedness of strong damping wave equations defined by fractal Laplacians is proved by using Galerkin method. These fractal Laplacians are defined by self-similar measures with overlaps, such as the well-known infinite Bernoulli convolution associated with the golden ratio, the three-fold convolution of the Cantor measure, and a class of self-similar measures that we call essentially of finite type. In general, the structure of self-similar measures with overlap are complicated and intractable. However, some important information about the structure of the three measures above can be obtained. We make use of these information to set up a framework for one-dimensional measures to discretize the equations, and use the finite element and central difference methods to obtain numerical approximations of the weak solutions. We also show that the numerical solutions converge to the actual solution and obtain the rate of convergence.