We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and <i>C</i>  in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem.

We establish several gradient estimates for second-order divergence type parabolic and elliptic systems. The coefficients and data are assumed to be Hlder or Dini continuous in the time variable and all but one spatial variable. This type of system arises from the problems of linearly elastic laminates and composite materials. For the proof, we use Campanatos approach in a novel way. Non-divergence type equations under similar conditions are also discussed.

In this paper, we discuss error estimates associated with three different aggregation-diffusion splitting schemes for the Keller-Segel equations. We start with one algorithm based on the Trotter product formula, and we show that the convergence rate is CΔt, where Δt is the time-step size. Secondly, we prove the convergence rate CΔt2 for the Strang's splitting. Lastly, we study a splitting scheme with the linear transport approximation, and prove the convergence rate CΔt.

In this paper, we establish the existence theory for general system of hyperbolic conservation laws and obtain the uniform <i>L</i> _<i>1</i> boundness for the solutions. The existence theory generalizes the classical Glimm theory for systems, for which each characteristic field is either genuinely nonlinear or linearly degenerate in the sense of Lax. We construct the solutions by the Glimm scheme through the wave tracing method. One of the key elements is a new way of measuring the potential interaction of the waves of the same characteristic family involving the angle between waves. A new analysis is introduced to verify the consistency of the wave tracing procedure. The entropy functional is used to study the <i>L</i> _<i>1</i> boundedness.

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