Lieb-Thirring integral inequalities and their applications to attractors of the Navier-Stokes equations

Abstract

To understand plasma phenomena which will occur in a wall-confined, shock-heated fusion system, experiments and computer simulations have been carried out. The thermonuclear fusion cycle, energy transfer from a 6 > 1 hot plasma to a wall, plasma confinement by means of walls and a magnetic dam, and some new strong shock wave physics results are described and reviewed.

Sharp constants in the Sobolev embedding theorem and a derivation of the Brezis-Gallouet interpolation inequality

“…provided it is understood that these operators are being used as differential operators "acting" on functions inḢ s , according to (8) and (10). For more details see Refs.…”

“…Nevertheless, in other situations the estimation of constants is an important exercise as they often contain geometric and number theoretic information which shed light on the overall significance and power of the inequalities themselves. [4][5][6][7][8][9][10][11][12][13] In addition, knowledge of these can be crucial for a sharp and detailed analysis of solutions of PDEs. [12][13][14][15][16][17][18] In this paper, we wish to provide a sharp explicit value for the constants appearing in the L ∞ norm in the Sobolev embedding theorem (SET); furthermore, we derive the Brezis-Gallouet inequality (BGI) (Ref.…”

The Damped-Driven 2D Navier–Stokes System on Large Elongated Domains

“…The spectral characterization of these projectors and the associated anisotropic Lieb-Thirring inequalities will be given in §3 (see [17]). For the moment we observe that if div w = 0, then div Nw = 0 and div Mw = 0.…”

“…The following theorem (see [17]) collects the Lieb-Thirring inequalities used in §2. It is important for us that for function in NH the constants in the corresponding inequalities are bounded by absolute constants as α → 0. the following inequalities hold:…”

“…It has later been shown in [31] that the rate of growth 1/α of the dimension of the global attractor is sharp, and the following explicit estimate for the dimension of the attractor for the Kolmogorov flow problem was obtained in [17] …”