Numerical investigations of non-uniqueness for the Navier-Stokes initial value problem in borderline spaces

We consider the Cauchy problem for the incompressible Navier-Stokes equations in $\mathbb{R}^3$ for a one-parameter family of explicit scale-invariant axi-symmetric initial data, which is smooth away from the origin and invariant under the reflection with respect to the $xy$-plane. Working in the class of axi-symmetric fields, we calculate numerically scale-invariant solutions of the Cauchy problem in terms of their profile functions, which are smooth. The solutions are necessarily unique for small data, but for large data we observe a breaking of the reflection symmetry of the initial data through a pitchfork-type bifurcation. By a variation of previous results by Jia & \v{S}ver\'ak (2013) it is known rigorously that if the behavior seen here numerically can be proved, optimal non-uniqueness examples for the Cauchy problem can be established, and two different solutions can exists for the same initial datum which is divergence-free, smooth away from the origin, compactly supported, and locally $(-1)$-homogeneous near the origin. In particular, assuming our (finite-dimensional) numerics represents faithfully the behavior of the full (infinite-dimensional) system, the problem of uniqueness of the Leray-Hopf solutions (with non-smooth initial data) has a negative answer and, in addition, the perturbative arguments such those by Kato (1984) and Koch & Tataru (2001), or the weak-strong uniqueness results by Leray, Prodi, Serrin, Ladyzhenskaya and others, already give essentially optimal results. There are no singularities involved in the numerics, as we work only with smooth profile functions. It is conceivable that our calculations could be upgraded to a computer-assisted proof, although this would involve a substantial amount of additional work and calculations, including a much more detailed analysis of the asymptotic expansions of the solutions at large distances.Comment: 31 pages, 19 figure

Design of single-purpose machine for washer stamping

Tato bakalářská práce se věnuje návrhu a konstrukci jednoúčelového stroje pro zalisování podložek. Je popsán rozbor výrobní operace, navrhnuto řešení a zvoleny technické parametry stroje. Dále je v CAD systému tento stroj navrhnut, jsou provedeny základní konstrukční výpočty a je zdůvodněna volba nakupovaných dílů.This bachelor’s thesis deals with design and construction of single-purpose machine for washer stamping. Manufacturing operation is described, solution is suggested and technical parameters of machine are selected. Then machine is designed in CAD system, basic construct calculations are performed and choice of purchased parts is justified.

Influence of thermal insulation on the building proposal heat

Tato diplomová práce se zabývá vlivem zateplení objektu na návrh zdroje tepla. Teoretická část popisuje historický vývoj stavebnictví, počátky zateplování a základní pojmy spojené s výpočtem tepelných ztrát. V úvodu praktické části jsou popsány možné způsoby návrhu vytápění a stavebně-technický popis novostavby bytového domu Cacovická. Hlavní část řeší „Vliv zateplení na návrh zdroje tepla“, který je proveden ve dvou odlišných variantách. Závěrem práce je porovnání obou variant a vyhodnocení nákladově nižší varianty.This thesis deals with the influence of building insulation to design of the heat source. The theoretical part describes the historici evolution of the construction, insulation origins and basic concepts associated with heat loss calculations. Introduction to the practical part describes possible wals of rating proposal and construction and technical description of the new residential building Cacovická. The main part deals with "Influence of thermal insulation on the building proposal heat" performed in two different variants. Finally work is a comparison of the two variants and evaluation of lower cost options.

Liouville theorems in unbounded domains for the time-dependent Stokes system

In this paper, We characterize bounded ancient solutions to the time-dependent Stokes system with zero boundary value in various domains, including the half space.Comment: 11 pages; final versio

A stability result for nonlinear Neumann problems under boundary variations

In this paper we study, in dimension two, the stability of the solutions of some nonlinear elliptic equations with Neumann boundary conditions, under perturbations of the domains in the Hausdorff complementary topology.Comment: 26 page