Download Analysis of Singularities for Partial Differential Equations by Shuxing Chen PDF

By Shuxing Chen

The booklet offers a finished evaluation at the idea 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 integrated, resembling the propagation of singularities of ideas to nonlinear equations, singularity index and formation of shocks

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Example text

In the discussion of the singularity analysis for the operators with constant multiple characteristics the operators have to satisfy the Levi condition, which is a restriction on the lower order terms. For instance, if P is an operator of second order with multiple characteristics, the symbol of P is p(x, ξ) = q(x, ξ)2 + p1 (x, ξ), where q and p1 are the symbols of first order, then Levi condition means that the symbol p1 − 1 i n j=1 ∂q ∂q ∂xj ∂ξj is of order zero on q = 0. Therefore, there is a symbol h of order zero and a corresponding operator H, such that p1 = h1 q + 1 i n j=1 ∂q ∂q .

4. If U is a solution of Eq. 53)  σ1 (x, Dx ) .. σN0 (x, Dx ) e+ (x, Dx ) e− (x, Dx )    .   The principal symbol of σj (x, Dx ) is λj (x, ξ ), while e+ (x, Dx ), e− (x, Dx ) have complex symbols of first order, R is a matrix of pseudodifferential operators of order −∞. Therefore, in order to derive a result on singularity reflection for general cases we have to discuss the regularity of solutions to boundary value problems for elliptic systems. 13. 54) W + (0) = h+ , where E + has its principal symbol e+ (x, ξ ) satisfying Im(spec(e+ )) ≥ c0 |ξ|, + c0 > 0.

In order to prove Eq. 28) we write the symbols of σ and Q as q(x, ξ , ξn ) ∼ qm−1 + qm−2 + · · · , σ(x, ξ ) ∼ λ(x, ξ ) + σ0 + σ−1 + · · · , where qj is a homogeneous function of ξ with degree j, and is a polynomial of ξn , σ is a homogeneous function of ξ with degree 1. To determine these symbols we use the asymptotic expansion of corresponding symbols in Eq. 28). Comparing the homogeneous terms with degree m − 1 we have n−1 −σ0 qm−1 + Dxn qm−1 − j=1 (∂ξj λ)Dxj qm−1 + (ξn − λ)qm−1 = pm−1 . 29) Let ξn = λ(x, ξ ), in view of qm−1 = 0 we have   n−1 1  σ0 = Dxn qm−1 − (∂ξj λ)Dxj qm−1 − pm−1  qm−1 j=1 .

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