In this paper, we prove the Egorov theorem for paradlfferentlal operators based on the theory of para-Fourier integral operators. Obviously, this theorem is imporfant for the study on properlies of paradifferential op...In this paper, we prove the Egorov theorem for paradlfferentlal operators based on the theory of para-Fourier integral operators. Obviously, this theorem is imporfant for the study on properlies of paradifferential operators. Before proving this theorem, we introduce the concept of adjoint para-Fourier integral operators and establish some of its properties.展开更多
The asymptotic expansions are studied for the vorticity {ω ε(t,x)} to 2D incompressible Euler equations with initial vorticity ω ε 0(x)=ω 0(x)+εω 1 0x,φ 0(x)ε , where φ 0(x) satisfies d φ 0(x)≠0 on the sup...The asymptotic expansions are studied for the vorticity {ω ε(t,x)} to 2D incompressible Euler equations with initial vorticity ω ε 0(x)=ω 0(x)+εω 1 0x,φ 0(x)ε , where φ 0(x) satisfies d φ 0(x)≠0 on the support of ω 1 0(·,θ),θ∈T , and ω 0(x) (resp. ω 1 0(x,θ)) is sufficiently smooth and with compact support in R 2 (resp. R 2× T ). The limit, v(t,x) , of the corresponding velocity fields v ε(t,x) is obtained, which is the unique solution of (E) with initial vorticity ω 0(x) . Moreover, ω ε(t,x)=ω(t,x)+εω 1t,x,φ(t,x)ε+o(ε) in C([0,∞), L p([KX1] R 2)) for all 1≤p<∞ , where ω(t,x)= 1v 2(t,x)- 2v 1(t,x),ω 1(t,x,θ) and φ(t,x) satisfy some modulation equation and eikonal equation, respectively.展开更多
In this paper, the author applies the Egorov Theorem for paradifferential operators established by the theory of para-Fourier integral operators to reduce microlocally paradifferential operators with the principal typ...In this paper, the author applies the Egorov Theorem for paradifferential operators established by the theory of para-Fourier integral operators to reduce microlocally paradifferential operators with the principal type. This reduction is used to study the propagation of singularities for solutions to nonlinear differential equations in the lower frequency range.展开更多
In this paper, the concept of para-Fourier integral operators is introduced by studying Fourier integral operators with irregular phase functions, and some basic properties of the para-Fourier integral operators are g...In this paper, the concept of para-Fourier integral operators is introduced by studying Fourier integral operators with irregular phase functions, and some basic properties of the para-Fourier integral operators are given.展开更多
基金Project supported by the National Natural Science Foundation of China
文摘In this paper, we prove the Egorov theorem for paradlfferentlal operators based on the theory of para-Fourier integral operators. Obviously, this theorem is imporfant for the study on properlies of paradifferential operators. Before proving this theorem, we introduce the concept of adjoint para-Fourier integral operators and establish some of its properties.
文摘The asymptotic expansions are studied for the vorticity {ω ε(t,x)} to 2D incompressible Euler equations with initial vorticity ω ε 0(x)=ω 0(x)+εω 1 0x,φ 0(x)ε , where φ 0(x) satisfies d φ 0(x)≠0 on the support of ω 1 0(·,θ),θ∈T , and ω 0(x) (resp. ω 1 0(x,θ)) is sufficiently smooth and with compact support in R 2 (resp. R 2× T ). The limit, v(t,x) , of the corresponding velocity fields v ε(t,x) is obtained, which is the unique solution of (E) with initial vorticity ω 0(x) . Moreover, ω ε(t,x)=ω(t,x)+εω 1t,x,φ(t,x)ε+o(ε) in C([0,∞), L p([KX1] R 2)) for all 1≤p<∞ , where ω(t,x)= 1v 2(t,x)- 2v 1(t,x),ω 1(t,x,θ) and φ(t,x) satisfy some modulation equation and eikonal equation, respectively.
基金Project supported by the National Natural Science Foundation of China.
文摘In this paper, the author applies the Egorov Theorem for paradifferential operators established by the theory of para-Fourier integral operators to reduce microlocally paradifferential operators with the principal type. This reduction is used to study the propagation of singularities for solutions to nonlinear differential equations in the lower frequency range.
基金Project supported by the National Natural Science Foundation of China.
文摘In this paper, the concept of para-Fourier integral operators is introduced by studying Fourier integral operators with irregular phase functions, and some basic properties of the para-Fourier integral operators are given.
