By Shuxing Chen

The ebook presents a complete evaluate at the thought on research of singularities for partial differential equations (PDEs). It incorporates a summarization of the formation, improvement and major effects in this subject. a few of the author's discoveries and unique contributions also are incorporated, equivalent to the propagation of singularities of suggestions to nonlinear equations, singularity index and formation of shocks

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**Example text**

Then the projection of bicharacteristic strips through the origin on the space (x, y, z) is L1 : y = x tan θ, L2 : y = −x tan θ, c−1 + x tan θ L3 : y = , −2 −2 2 c− + (c−2 − − c+ ) tan θ L4 : y = −c−1 + x tan θ −2 −2 2 c−2 − + (c− − c+ ) tan θ , August 12, 2010 42 15:49 World Scientific Book - 9in x 6in singularities Analysis of Singularities for Partial Differential Equations where L2 is the equation of the incident ray. 14, we take i1 = 1, i2 = 3, then BS −1 = λ λ λe+ − λe+ −λ −λ λe− − λe− . The first column and the third column of the matrix forms a submatrix with rank 2.

62) where A= Its principal symbol is a= Λ −1 (c −2 Dt2 0 Λ 2 2 − Dy − Dz ) 0 0 λ λ−1 (c−2 τ 2 − η 2 − ζ 2 ) 0. . Obviously, the eigenvalue of a is ±(c−2 Dt2 − Dy2 − Dz2 )1/2 . For the point 3 (t, y, z, τ, η, ζ) on T ∗ (Rtyz ) we have: 2 2 1/2 If τ < c(η + ζ ) , then for any ξ, the symbol τ 2 − c2 (ξ 2 + η 2 + ζ 2 ) = 0, then u is microlocally regular at (τ, ξ, η, ζ). Hence u|x=s is microlocally regular for any s > 0. By using the fact that the eigenvalue of a is imaginary as τ < c(η 2 + ζ 2 )1/2 , then u|x=0 is also microlocally regular at (τ, η, ζ).

If (t0 , x0 ; τ0 , ξ0 ) ∈ W F (u), then (t0 , x0 ; τ0 , ξ0 ) ∈ W F (Au). The proof of these two theorem can be find in [40],[74]. 1 describes the singularity of a given function in C ∞ category. In some cases one needs to describe the singularity for a given function with finite order. For instance, for any given real number s, one can say “a distribution u is H s smooth”, which simply means u ∈ H s . Furthermore, one can also say that u has singularity in the sense of H s , if u is not in H s .