By Claude Bardos, François Golse (auth.), James Glimm, Andrew J. Majda (eds.)

This IMA quantity in arithmetic and its purposes MULTIDIMENSIONAL HYPERBOLIC difficulties AND COMPUTATIONS is predicated at the lawsuits of a workshop which used to be a vital part ofthe 1988-89 IMA software on NONLINEAR WAVES. we're thankful to the clinical dedicate tee: James Glimm, Daniel Joseph, Barbara Keyfitz, Andrew Majda, Alan Newell, Peter Olver, David Sattinger and David Schaeffer for making plans and imposing a thrilling and stimulating year-long application. We specially thank the paintings store Organizers, Andrew Majda and James Glimm, for bringing jointly a few of the significant figures in numerous examine fields attached with multidimensional hyperbolic difficulties. A vner Friedman Willard Miller PREFACE a first-rate target of the IMA workshop on Multidimensional Hyperbolic difficulties and Computations from April 3-14, 1989 used to be to stress the interdisciplinary nature of up to date study during this box concerning the combo of rules from the speculation of nonlinear partial differential equations, asymptotic equipment, numerical computation, and experiments. The twenty-six papers during this quantity span a large cross-section of this learn together with a few papers at the kinetic idea of gases and vortex sheets for incompressible movement as well as many papers on platforms of hyperbolic conservation legislation. This quantity comprises a number of papers on asymptotic tools comparable to nonlinear geometric optics, a few articles making use of numerical algorithms comparable to larger order Godunov tools and entrance monitoring to actual difficulties besides comparability to experimental information, and likewise a number of attention-grabbing papers at the rigorous mathematical concept of concern waves.

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17), we have If {j = giT, we have a power of provided (1 < 1/4, and we have g of 1/3 - (1(3 > (1 - 4(1)/3 == a; this is positive 1F(',t)ILI :SCg a , some a>O. 20) as g -+ + v'· (v'· \l)* 1 and v' is bounded, so that vi E LT, r> 1. Also v' E Lb for b> 2 so that v'· v' E LT for r > 1. Thus each of the two terms makes sense. 22) by Sobolev's Lemma. *

The same definition was used in [7]. 1. We can now state the convergence result for classical weak solutions. 27 THEOREM 1. Suppose an initial velocity field Vo is given which is locally in L2 of space. Assume that the vorticity Wo = \7 X Vo is in LP for some p > 1 and has bounded support. For each E > 0, let fj, XJ(t), ve(x, t)be determined by the vortex blob approximation described above for 0 ~ t ~ T. Assume the parameters are chosen so that 5(E) = E" for some a with 0 < a < 1/4, and h(E) ~ CE 4 exp( -COC 2 ) for a certain constant Co and any C.

CI> = 0. 21) is established, and the proof of the Lemma is complete. We shall also need an estimate for vi. 15) that v' is bounded in LP*, p* > 2. Thus v' . v' E L r with r = p* /2 > 1. • ° provided divif! = 0, if! E W·,q, s 2': 1, and s > 2/q. 26) holds for arbitrary if! E W·,q. 27) Iv:lw- •. r ::; e, r = p* /2. Finally we verify that a classical weak solution is obtained in the limit c -+ 0. 2 in [7]. 28) v' tS uniformly bounded in U· (R2 x (0, T)) 33 with p' = 2p/(2 - p) > 2. We may select a subsequence en -> 0 so that v' converges weakly in LP' to a limit v.