Mathematical Analysis

By Shuxing Chen

ISBN-10: 9814304832

ISBN-13: 9789814304832

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

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Extra resources for Analysis of Singularities for Partial Differential Equations

Example text

1. 2) we know that the space H n/2 is invariant under nonlinear composition as s > n/2. 1 is valid for r = s. When r > s we take δ = min(s − n/2, 1), and let ρ satisfy s ≤ ρ − δ, ρ ≤ r. 1. Therefore, in the following discussion we may assume that the lemma has been true for r = ρ − δ. By differentiating the function f (x, u) we have D(f (x, u)) = g(x, u) + f (x, u)Du. 8) The first term in the right hand side g(x, u) ∈ H s ∩ H ρ−δ (x0 , ξ0 ), the second term has the form vDu, where v ∈ H s ∩ H ρ−δ (x0 , ξ0 ), and Du ∈ H s−1 ∩ H ρ−1 (x0 , ξ0 ).

The lemma means that under the assumptions on the indices r and s the space H r (x0 , ξ0 ) ∩ H s is close with respect to the nonlinear composition. The closeness is essential in the study of singularity analysis for solutions to nonlinear partial differential August 12, 2010 15:49 World Scientific Book - 9in x 6in singularities Singularity analysis for semilinear equations 51 equations and will also used later. 1. 2 (Schauder). (1) If u ∈ H s , v ∈ H t with s > n/2, 0 ≤ t ≤ s, then uv ∈ H t ; (2) If f (u) is a C ∞ function of u, u(x) ∈ H s , s > n/2, then f (u(x)) ∈ s H .

2 (Schauder). (1) If u ∈ H s , v ∈ H t with s > n/2, 0 ≤ t ≤ s, then uv ∈ H t ; (2) If f (u) is a C ∞ function of u, u(x) ∈ H s , s > n/2, then f (u(x)) ∈ s H . Proof. The theorem can be proved by different methods. Next we use the theory of paradifferential operators to give a brief proof. 2) where Tu is a linear operator from H t to H t , r1 (u, v) ∈ H s+t−n/2 ⊂ H t . Hence uv ∈ H t . 3) where Tf (u) is a linear operator from H t to H t , R(x) ∈ H 2s−n/2 ⊂ H t . Hence the right hand side of Eq.