By Guy Métivier
The major target is to offer on the point of newcomers a number of sleek instruments of micro-local research that are valuable for the mathematical learn of nonlinear partial differential equations. The middle of those notes is dedicated to a presentation of the para-differential recommendations, which mix a linearization technique for nonlinear equations, and a symbolic calculus which mimics or extends the classical calculus of Fourier multipliers. those equipment observe to many difficulties in nonlinear PDE’s resembling elliptic equations, propagation of singularities, boundary worth difficulties, shocks or boundary layers. even though, in those introductory notes, we've got selected to demonstrate the speculation on chosen and comparatively easy examples, which permit turning into acquainted with the ideas. They predicament the good posed-ness of the Cauchy challenge for platforms of nonlinear PDE's, to start with hyperbolic structures and secondly coupled platforms of Schrödinger equations which come up in quite a few types of wave propagation.
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Extra resources for Para-differential calculus and applications to the Cauchy problem for nonlinear systems
Sample text
The Aj are N × N constant matrices. 3) A(ξ) = ξ j Aj . 4) t ˆ + u ˆ(t, ξ) = e−itA(ξ) h(ξ) ei(t −t)A(ξ) fˆ(t , ξ)dt . 2 Strongly hyperbolic systems Following the general discussion, the problem is to give estimates for the exponentials e−itA(ξ) = eiA(−tξ) . The next lemma is immediate. 1. For the exponential eiA(ξ) to have at most a polynomial growth when |ξ| → ∞, it is necessary that for all ξ ∈ Rd the eigenvalues of A(ξ) are real. In this case, the system is said to be hyperbolic. 17) is satisfied.
3 says that for all ξ, A(ξ) has only real eigenvalues and can be diagonalized. This does not necessarily imply strong hyperbolicity: the existence of a uniform bound for the eigenprojectors for |ξ| = 1 is a genuine additional condition. 7. 1) is said to be smoothly diagonalizable is there are real valued λj (ξ) and projectors Πj (ξ) which are real analytic functions of ξ on the unit sphere, such that A(ξ) = λj (ξ)Πj (ξ). 8. 1) is smoothly diagonalizable, then it is strongly hyperbolic. 9. 1) is said to be strictly hyperbolic if for all ξ = 0, A(ξ) has N distinct real eigenvalues.
2 Operators with symbols in the Schwartz class As an introduction, we first study the case of operators defined by symbols in the Schwartz class. The results will be extended to more general symbols in the following sections. 1) is convergent and defines a function in the Schwarz class S (Rd ). 2) p(x, Dx )u (x) = (Fξ−1 p)(x, x − y) u(y)dy , where Fξ−1 p ∈ S (Rd × Rd ) denotes the inverse Fourier transform of p with respect to the variables ξ. 1. 3) p(x, ∂x )v, u S (Rd )×S (Rd = K, u ⊗ v S (Rr ×Rd )×S (Rr ×Rd ) .