Analysis of the Landau Solution
GRIN Verlag, 2011 - 80 pages
Bachelor Thesis from the year 2010 in the subject Mathematics - Analysis, grade: 1,0, Technical University of Darmstadt, language: English, abstract: In 1944 Lev D. Landau found a non-trivial solution to a stationary Navier-Stokes flow on R, which was symmetric around some axis and fulfilled the condition, that the velocity decayed linearly and the pressure quadratically in x, depending on a parameter -1 1. As it turns out, such a modified Landau solution is no longer a solution to a Navier-Stokes system - clearly not in the classical, but as well neither in the weak nor very weak sense. If d = 1, the velocity will be unbounded on the half line x1 = x2 = 0, x3 0. In this case, we have a bit more insight in the behaviour of the modified Landau solution, yet still no physically reasonable interpretation can be given in the whole space R . In any case, one can c"
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1)-homogeneous angle axis of symmetry axisymmetric behaviour cartesian coordinates classical Landau solution classical solution compute conditions u(x cone constant continuously differentiable converging critical half line critical Landau solution deﬁned deﬁnition denoted Dirac distribution discretely self-similar distribution vanishes domain Dominated Convergence Theorem dual space Euler Theorem Farwig ﬁrst flow ﬂuid force F formulae Furthermore holds true Homogeneous Functions inﬁnity integrand Jonas Sauer Landau parameter leading term Miura modiﬁed modified Landau solution Navier-Stokes equations Navier-Stokes system 2.21 oi-homogeneous opening semi-angle ordinary differential equation partial differential equations physical interpretation pressure principal value integral proof rotating satisﬁes sE0+ sense of distributions smooth solution solutions to partial spherical coordinates stereographic projection stream function Streamlines streamtube supercritical Landau solution Šverák test functions Theorem on Homogeneous total force Transport Theorem Tsai velocity weak solution whole space x3-axis zero force satisfying