We prove uniform gradient and diameter estimates for a family of geometric complex Monge–Ampere equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge–Ampere equations. We also prove a uniform diameter estimate for collapsing families of twisted Kahler–Einstein metrics on Kahler manifolds of nonnegative Kodaira dimensions.

For the Boltzmann equation with an external potential force depending only on the space variables, there is a family of stationary solutions, which are local Maxwellians with space dependent density, zero velocity and constant temperature. In this paper, we will study the nonlinear stability of these stationary solutions by using the energy method. The analysis combines the analytic techniques used for the conservation laws using the fluid-type system derived from the Boltzmann equation (cf. [14]) and the dissipative effects on the fluid and non-fluid components of the Boltzmann equation through the celebrated H-theorem. To our knowledge, this is the first result on the global classical solutions to the Boltzmann equation with external force and non-trivial large time behavior in the whole space.

In this paper, we combine the method of constructing the compensating function introduced by Kawashima and the standard energy method for the study on the Landau equation with external forcing. Both the global existence of solutions near the time asymptotic states which are local Maxwellians and the optimal convergence rates are obtained. The method used here has its own advantage for this kind of studies because it does not involve the spectrum analysis of the corresponding linearized operator.

The solvability in Sobolev spaces is proved for divergence form complex-valued higher order parabolic systems in the whole space, on a half-space, and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable in one spacial direction and have small mean oscillations in the orthogonal directions on each small cylinder. The directions in which the coefficients are only measurable vary depending on each cylinder. The corresponding elliptic problem is also considered.

In this paper we study the existence assertion of the initial boundary value problem for the equation $\frac{\partial u}{\partial t} = \Delta e^{-\Delta u}$. This problem arises in the mathematical description of the evolution of crystal surfaces. Our investigations reveal that the exponent in the equation can have a singular part in the sense of the Lebesgue decomposition theorem, and the exponential nonlinearity somehow “cancels” it out. The net result is that we obtain a solution $u$ that satisfies the equation and the initial boundary conditions in the almost everywhere (a.e.) sense.