Singular and regular solutions of a non-linear parabolic system

We study a dissipative nonlinear equation modelling certain features of the Navier-Stokes equations. We prove that the evolution of radially symmetric compactly supported initial data does not lead to singularities in dimensions $n\leq 4$. For dimensions $n>4$ we present strong numerical evidence supporting existence of blow-up solutions. Moreover, using the same techniques we numerically confirm a conjecture of Lepin regarding existence of self-similar singular solutions to a semi-linear heat equation.Comment: 16 page

Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres

Using mixed analytical and numerical methods we investigate the development of singularities in the heat flow for corotational harmonic maps from the $d$-dimensional sphere to itself for $3\leq d\leq 6$. By gluing together shrinking and expanding asymptotically self-similar solutions we construct global weak solutions which are smooth everywhere except for a sequence of times $T_1<T_2<...<T_k<\infty$ at which there occurs the type I blow-up at one of the poles of the sphere. We show that in the generic case the continuation beyond blow-up is unique, the topological degree of the map changes by one at each blow-up time $T_i$, and eventually the solution comes to rest at the zero energy constant map.Comment: 24 pages, 8 figures, minor corrections, matches published versio

Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For the semilinear heat equation $u_t= \Delta u+ u^p$, various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure

Математическое моделирование тепловых режимов космических аппаратов негерметичного исполнения

A description of the software complex for modeling the thermal conditions of non-hermetic spacecrafts in circular and elliptical orbits is given. The software complex structure, mathematical models of thermal processes in the presence of thermal control system are presented. For thermal processes calculation the method of lumped parameters is used. The results of thermal stabilization for model of orbital optical device are given. The software complex can be used in the design and development of thermal control systems of nonhermetic spacecraft, and in the determination of the causes of emergency situations in these systems in orbit as well.Предлагается описание программного комплекса для моделирования тепловых режимов космических аппаратов негерметичного исполнения в условиях полета по круговым и эллиптическим орбитам. Приводятся состав и структура программного комплекса, математические модели тепловых процессов при наличии системы обеспечения тепловых режимов космических аппаратов. Для расчета тепловых процессов используется метод сосредоточенных параметров. Даются результаты вычислительных экспериментов по термостабилизации макета орбитального оптического прибора. Программный комплекс может быть использован при проектировании и создании систем обеспечения тепловых режимов космических аппаратов негерметичного исполнения, а также при определении причин нештатных ситуаций в данных системах на орбите

New CC0\pi\ GENIE Model Tune for MicroBooNE

A novel tune has been made for the MicroBooNE experiment. The fit uses 4 new parameters within the GENIE v3.0.6 Monte Carlo program. Charged current pionless data from the T2K experiment was used. New uncertainties were obtained. These results will be used in future MicroBooNE analyses.Comment: 24 pages, 14 figure