In connection with the recent proposal for possible singularity formation at the boundary for solutions of three-dimensional axisymmetric incompressible Euler’s equations (Luo and Hou, Proc. Natl. Acad. Sci. USA (2014)), we study models for the dynamics at the boundary and show that they exhibit a finite-time blowup from smooth data.

A new augmented method is proposed for elliptic interface problems with a piecewise variable coefficient that has a finite jump across a smooth interface. The main motivation is to get not only a second order accurate solution but also a second order accurate gradient from each side of the interface. Key to the new method is introducing the jump in the normal derivative of the solution as an augmented variable and rewriting the interface problem as a new PDE that consists of a leading Laplacian operator plus lower order derivative terms near the interface. In this way, the leading second order derivative jump relations are independent of the jump in the coefficient that appears only in the lower order terms after the scaling. An upwind type discretization is used for the finite difference discretization at the irregular grid points on or near the interface so that the resulting coefficient matrix is an M-matrix. A multigrid solver is used to solve the linear system of equations, and the GMRES iterative method is used to solve the augmented variable. Second order convergence for the solution and the gradient from each side of the interface is proved in this paper. Numerical examples for general elliptic interface problems confirm the theoretical analysis and efficiency of the new method.

. Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\times n$ system of conservation laws, each characteristic field being linearly degenerate or genuinely nonlinear. In this paper we explicitly define a functional $\Phi=\Phi(u,v)$, equivalent to the $\L^1$ distance, which is almost decreasing i.e., $$ \Phi\big( u(t),~v(t)\big)-\Phi\big( u(s),~v(s)\big)\leq \O(\ve)\cdot (t-s)\quad\hbox{for all}~~t&gt;s\geq 0,$$ for every pair of <i></i>-approximate solutions <i>u</i>, <i>v</i> with small total variation, generated by a wave front tracking algorithm. The small parameter <i></i> here controls the errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all non-physical waves in <i>u</i> and in <i>v</i>. From the above estimate, it follows that front-tracking approximations converge to a unique limit solution, depending Lipschitz continuously on the initial data, in the ${\vec L}^1$ norm. This provides a new proof of the existence of the

In this work, we consider the numerical solution of the nonlinear Schrödinger equation with a highly oscillatory potential (NLSE-OP). The NLSE-OP is a model problem which frequently occurs in recent studies of some multiscale dynamical systems, where the potential introduces wide temporal oscillations to the solution and causes numerical difficulties. We aim to analyze rigorously the error bounds of the splitting schemes for solving the NLSE-OP to a fixed time. Our theoretical results show that the Lie–Trotter splitting scheme is uniformly and optimally accurate at the first order provided that the oscillatory potential is integrated exactly, while the Strang splitting scheme is not. Our results apply to general dispersive or wave equations with an oscillatory potential. The error estimates are confirmed by numerical results.

Adapting the convex integration technique introduced in \cite{dLSz5} and subsequently developed in \cite{IsettVicol,IsettOh16,Buckmaster2013transporting}, we construct H\"{o}lder continuous weak solutions to the three dimensional Prandtl system and some other models with vertical viscosity.