By L. L. Bonilla, A. Carpio, J. M. Vega, S. Venakides
The articles of this ebook are written through top specialists in partial differential equations and their purposes, who current overviews right here of contemporary advances during this large region of arithmetic. The formation of shocks in fluids, sleek numerical computation of turbulence, the breaking of the Einstein equations in a vacuum, the dynamics of defects in crystals, results because of entropy in hyperbolic conservation legislation, the Navier-Stokes and different limits of the Boltzmann equation, occupancy occasions for Brownian movement in a dimensional wedge, and new equipment of examining and fixing integrable structures are a few of this volume's matters. The reader will locate an exposition of significant advances with no lot of technicalities and with an emphasis at the simple principles of this box.
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Extra info for Recent Advances in Nonlinear Partial Differential Equations and Applications
Example text
3. Finally, the result whose proof is closest in spirit to ours and which has played the main motivating role in developing our approach, is the proof of global regularity of solutions of the Yang-Mills equations in M 3+1 by Eardley and Moncrief, see [EMI]. To explain the connection of their result to ours we review below its main ideas. Recall that the curvature tensor Fap = daAp — dpAa + [Aa, Ap] of a Yang Mills connection Aadxa, with values in the Lie algebra su(N) is a critical point of the Yang-Mills functional YM[F] = [ JR3+1 Tr(*FAF) and verifies the wave equation, (12) •(A)F = F*F, where • (A) = m^DJ^D^ denotes the covariant wave operator, D^ D'AlembertianinR 3 + 1 , •(A) = d^ + [A^, •].
We first establish a theorem, the fundamental energy estimate, which applies to a solution of the homogeneous wave equation in the acoustical spacetime, in particular to any first order variation. The proof of this theorem relies on certain bootstrap assumptions on the acoustical entities. The most crucial of these assumptions concern the behavior of the function \i. These assumptions are established later on on the basis of the final set of bootstrap assumptions, which consists only of pointwise estimates for the variations up to certain order.
In the present case however, the structure, which is here the acoustical structure, degenerates as shocks begin to form, and the precise way in which this degeneracy occurs must be guessed beforehand and established in the course of the argument of the mathematical proof. The fact that the underlying structure degenerates implies that our estimates are no longer even locally equivalent to standard energy estimates, which would of necessity have to fail when shocks appear. We first establish a theorem, the fundamental energy estimate, which applies to a solution of the homogeneous wave equation in the acoustical spacetime, in particular to any first order variation.