In this paper we study self-similar solutions to the singular and degenerate diffusion equation u t=(|(p (u)) x| -2 (p (u)) x) x,-< x<+, t> 0, where 1< < 2. The existence and uniqueness for the solutions are established. In addition, the asymptotic behavior is investigated.

In this paper, we establish the optimal error estimate of the particle method for a family of nonlinear evolutionary partial differential equations, or the so-called b-equation. The b-equation, including the Camassa-Holm equation and the Degasperis-Procesi equation, has many applications in diverse scientific fields. The particle method is an approximation of the b-equation in Lagrangian representation. We also prove short-time existence, uniqueness and regularity of the Lagrangian representation of the b-equation.

We study the front dynamics of solutions of the initial value problem of the Burgers equation with initial data being the viscous shock front plus the white noise perturbation. In the sense of distribution, the solutions propagate with the same speed as the unperturbed front, however, the front location is random and satisfies a central limit theorem with the variance proportional to the time<i>t</i>, as<i>t</i> goes to infinity. With probability arbitrarily close to one, the front width is<i>O</i>(1) for large time.

Diffusion-generated motion by mean curvature is a simple algorithm for producing motion by mean curvature of a surface, in which the motion is generated by alternately diffusing and renormalizing a characteristic function. In this paper, we generalize diffusion-generated motion to a procedure that can be applied to the curvature motion of filaments, i.e., curves in <i>R</i> ^3, that may initially consist of a complex configuration of links. The method consists of applying diffusion to a complex-valued function whose values wind around the filament, followed by normalization. We motivate this approach by considering the essential features of the complex Ginzburg-Landau equation, which is a reaction-diffusion PDE that describes the formation and propagation of filamentary structures. The new algorithm naturally captures topological merging and breaking of filaments without fattening curves. We justify the new

In this paper, we give a systematic study of the boundary layer behavior for linear convection-diffusion equation in the zero viscosity limit. We analyze the boundary layer structures in the viscous solution and derive the boundary condition satisfied by the viscosity limit as a solution of the inviscid equation. The results confirm that the Neumann type of far-field boundary condition is preferred in the outlet and characteristic boundary condition. Under some appropriate regularity and compatibility conditions on the initial and boundary data, we obtain optimal error estimates between the full viscous solution and the inviscid solution with suitable boundary layer corrections. These results hold in arbitrary space dimensions and similar statements also hold for the strip problem. This model well describes the behavior at the far-field for many physical and engineering systems such as fluid dynamical equation and electro-magnetic equation. The results obtained here should provide some theoretical guidance for designing effective far-field boundary conditions.