By Michael Beals

This e-book constructed from a sequence of lectures I gave on the Symposium on Nonlinear Microlocal research held at Nanjing college in October. 1988. Its goal is to offer an summary of using microlocal research and commutators within the research of options to nonlinear wave equations. The susceptible singularities within the options to such equations behave as much as a definite quantity like these found in the linear case: they propagate alongside the null bicharacteristics of the operator. nevertheless. examples showing singularities now not found in the linear case is usually developed. i've got attempted to provide a crossection of either the regularity effects and the singular examples. for difficulties at the inside of a website and on domain names with boundary. the most emphasis is at the case of a couple of house dimen sion. on the grounds that that case is taken care of in nice element within the paper of Rauch-Reed 159]. the consequences offered the following have for the main half seemed somewhere else. and are the paintings of many authors. yet a couple of new examples and proofs are given. i've got tried to point the basic principles in the back of the arguments. in order that just some of the implications are proved in complete aspect. it truly is was hoping that the crucial notions of the extra technical proofs showing in learn papers may be illuminated via those less complicated cases.

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**Example text**

In particular. it E 12(Z~) of arbitrarily small norm may be chosen such that. with v - I(jl)-' a; Jf. Ov-O. vl/_o -80' d1vl/_0 - 8,. 8,) - (O). I;Bp,·v 3 1! H3s-n~2~"(Itn~') for any j. 8). If (Ilk) is a collection of seminorms on M. define C;,k,J1I - (fJ E M: III;A(fJ)IIu3s-n+2+c ~ mlfJlk)' It is easily seen that Cj,k,J1I is closed in M. 8), ~ f! C;,k,J1I' and therefore q,k,J1I is nowhere dense in M. 9) I;BfJv3 f! H3s-n+2+t(Rn+l)for any j. 1 1x Rn). 9) that Ou-fJu 3, « that is, u is singular on the truncated solid light cone I,I): III S I, 0 S I ~ I).

F). (Since the nonlinear function depends only on u and not on Ou, microlocal regularity is propagated up to order r < 2 s - (n. ) By the algebra property. A F). /( F). 1 0 and its eItension to functions of differing regularities that to '" II. 0. Eli H26+512+2rm,(f). 3), u i H26+512+2tJJ1Af) as long as r < 114. In particular, the singular support of u contains the forward characteristic (x - 0, t ) OJ. 2. 1. let jJ E coo(12), fJ '"' O. I ~ -112. fJ .. 1, t 2 O. For 80y s > 1 there is a choice of Cauchy data (go.

AInu), then the chain rule and the expressions for the commutators imply that there are smooth vector valued functions g and h and there is a smooth vector valued differential operator rill -I( I,I,O) such that Pill ( I,I,J)){! - g( I,I,U, ... ,j)IIl-1 u) + rlll_IU,I,j))lf. By assumption, if k l 1. [! E HS/««(t < 0)). Schauder's Lemma and the linear energy inequality then imply that [! r). 2 used in place of Schauder's Lemma. D. Now suppose that a solution is known to be conormal in the past with respect to a pair of characteristic hypersurfaces which intersect transversally in the future.