Self-Similar Blow-Up Profiles for a Reaction-Diffusion Equation with Strong Weighted Reaction

We study the self-similar blow-up profiles associated to the following second-order reaction-diffusion equation with strong weighted reaction and unbounded weight:\partial_{t}u=\partial_{xx}(u^{m})+|x|^{\sigma}u^{p},posed for {x\in\mathbb{R}}, {t\geq 0}, where {m>1}, {0<p<1} and {\sigma>\frac{2(1-p)}{m-1}}. As a first outcome, we show that finite time blow-up solutions in self-similar form exist for {m+p>2} and σ in the considered range, a fact that is completely new: in the already studied reaction-diffusion equation without weights there is no finite time blow-up when {p<1}. We moreover prove that, if the condition {m+p>2} is fulfilled, all the self-similar blow-up profiles are compactly supported and there exist two different interface behaviors for solutions of the equation, corresponding to two different interface equations. We classify the self-similar blow-up profiles having both types of interfaces and show that in some cases global blow-up occurs, and in some other cases finite time blow-up occurs only at space infinity. We also show that there is no self-similar solution if {m+p<2}, while the critical range {m+p=2} with {\sigma>2} is postponed to a different work due to significant technical differences.

A Nitsche-eXtended Finite Element Method for Distributed Optimal Control Problems of Elliptic Interface Equations

This paper analyzes an interface-unfitted numerical method for distributed optimal control problems governed by elliptic interface equations. We follow the variational discretization concept to discretize the optimal control problems and apply a Nitsche-eXtended finite element method to discretize the corresponding state and adjoint equations, where piecewise cut basis functions around the interface are enriched into the standard linear element space. Optimal error estimates of the state, co-state and control in a mesh-dependent norm and the {L^{2}} norm are derived. Numerical results are provided to verify the theoretical results.

Extension of lattice Boltzmann flux solver for simulation of compressible multi-component flows

The lattice Boltzmann flux solver (LBFS), which was presented by Shu and his coworkers for solving compressible fluid flow problems, is extended to simulate compressible multi-component flows in this work. To solve the two-phase gas–liquid problems, the model equations with stiffened gas equation of state are adopted. In this model, two additional non-conservative equations are introduced to represent the material interfaces, apart from the classical Euler equations. We first convert the interface equations into the full conservative form by applying the mass equation. After that, we calculate the numerical fluxes of the classical Euler equations by the existing LBFS and the numerical fluxes of the interface equations by the passive scalar approach. Once all the numerical fluxes at the cell interface are obtained, the conservative variables at cell centers can be updated by marching the equations in time and the material interfaces can be identified via the distributions of the additional variables. The numerical accuracy and stability of present scheme are validated by its application to several compressible multi-component fluid flow problems.

Surface plasma waves induced electron acceleration in a static magnetic field

The acceleration of an electron beam by surface plasma waves (SPW), in the presence of external magnetic field parallel to surface and perpendicular to direction of propagation of SPW has been studied. This wave propagating along the$\hat z$-axis is excited using Kretschmann geometry, having maximum amplitude at the metal–vacuum interface. Equations of motion have been solved for electron energy and trajectory. The electron gains and retains energy in the form of cyclotron oscillations due to the combined effect of the static magnetic field and SPW field. The energy gained by the beam increases with the strength of magnetic field and laser intensity. In the present scheme, electron beams can achieve ~15 KeV energy for the SPW amplitudeA1= 1.6 × 1011V/m, plasma frequency ωp= 1.3 × 1016rad/s and cyclotron frequency ωc/ωp= 0.003.