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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Additional resources for Analysis of Singularities for Partial Differential Equations
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 .