By Hajer Bahouri

In contemporary years, the Fourier research equipment have expereinced a turning out to be curiosity within the research of partial differential equations. particularly, these strategies in keeping with the Littlewood-Paley decomposition have proved to be very effective for the learn of evolution equations. the current ebook goals at featuring self-contained, kingdom- of- the- artwork types of these strategies with functions to varied sessions of partial differential equations: delivery, warmth, wave and Schrödinger equations. It additionally bargains extra refined versions originating from fluid mechanics (in specific the incompressible and compressible Navier-Stokes equations) or common relativity.

It is both directed to somebody with a superb undergraduate point of information in research or valuable for specialists who're wanting to recognize the ease that one may well achieve from Fourier research while facing nonlinear partial differential equations.

**Read Online or Download Fourier Analysis and Nonlinear Partial Differential Equations PDF**

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**Additional resources for Fourier Analysis and Nonlinear Partial Differential Equations**

**Sample text**

28. Let z be a nonzero complex number with nonnegative real part. Then, 2 π d2 − |ξ|2 e 4z F e−z|·| (ξ) = z def with z − 2 = |z|− 2 e−i 2 θ if z = |z|eiθ with θ ∈ [−π/2, π/2]. d d d Proof. Let us remark that for any ξ in Rd , the functions z −→ e−i(x|ξ) e−z|x| dx 2 z −→ and Rd π z d 2 e− |ξ|2 4z are holomorphic on the domain D of complex numbers with positive real part. 20) states that these two functions coincide on the intersection of the real line with D. Thus, they also coincide on the whole domain D.

67. There exist two constants, c and C, depending only on the d dimension d, such that for any function u ∈ H 2 (Rd ), we have exp c Rd |f (x)| f d2 2 − 1 dx ≤ C. H Proof. As usual, arguing by density and homogeneity, it suﬃces to consider the case where f is in S and satisﬁes f d2 = 1. 30) and the deﬁnition of nonhomogeneous Sobolev spaces, there exists some constant Cd (depending only on the dimension d) such that √ f L2p ≤ Cd p for all p ≥ 1. 41) For all x ∈ Rd , we may write exp c|f (x)|2 − 1 = p≥1 cp |f (x)|2p .

33) The estimate of T f in terms of the B˙ −σ norm is described by the following proposition. 46. For σ ∈ ]0, d], a constant C exists such that Tf B˙ −σ ≤ 2d−2σ f B˙ −σ +C f L1 . Proof. 32), we have λd−σ θ(λ·) (T f ) L∞ ≤ λd−σ θ Tf L∞ L1 ≤ λd−σ θ L∞ f L1 , we get sup λ−σ λd θ(λ·) (T f ) L∞ ≤ θ L∞ f L1 . 34) λ≤1 The case where λ is large (which corresponds to high frequencies) is more intricate. We ﬁrst estimate λd (θ(λ·) (T f ))(x) when x is not too close to T (Q), def namely, x ∈ Qc = {x ∈ Q / d(x, T (Q)) ≥ 1/8}.