By Hajer Bahouri
In contemporary years, the Fourier research equipment have expereinced a turning out to be curiosity within the examine of partial differential equations. particularly, these strategies in line with the Littlewood-Paley decomposition have proved to be very effective for the learn of evolution equations. the current e-book goals at proposing self-contained, country- of- the- artwork versions of these innovations with purposes to various periods of partial differential equations: shipping, warmth, wave and Schrödinger equations. It additionally deals extra subtle types originating from fluid mechanics (in specific the incompressible and compressible Navier-Stokes equations) or common relativity.
It is both directed to an individual with an outstanding undergraduate point of data in research or helpful for specialists who're wanting to recognize the convenience that one may possibly achieve from Fourier research whilst facing nonlinear partial differential equations.
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Extra info for Fourier Analysis and Nonlinear Partial Differential Equations
We prove the theorem only in the case where the integer part of s−d/2 is 0. As s is greater than d/2, writing u = 1B(0,1) u + (1 − 1B(0,1) )u, we get that u belongs to L1 (Rd ), and thus u is a bounded continuous function. 27) into low and high frequencies. The lowfrequency part of u is of course smooth. By Taylor’s inequality, we have |u ,A (x) −u ,A (y)| ≤ ∇u ,A L∞ |x − y|. Using the Fourier inversion formula and the Cauchy–Schwarz inequality, we get ∇u ,A L∞ ≤C Rd |ξ| |u ,A (ξ)| dξ 1 2 ≤C ≤ |ξ|≤CA C (1 − ρ) 1 2 |ξ| 2−2s A1−ρ u dξ ˙s H u ˙s H with ρ = s − d/2.
Let R be the radius of the ball B. We have u ,A − (u ,A )B L2 (B, dx ) |B| ≤ R ∇u ≤ CR ,A L∞ d Rd d |ξ|1− 2 |ξ| 2 |u ≤ CRA u d ˙ 2 H ,A (ξ)| dξ . We infer that dx ≤ CRA u |u − uB | |B| B 1 2 d ˙ 2 H + C(AR) −d 2 Choosing A = R−1 then completes the proof. |ξ|≥A |ξ|d |u(ξ)|2 dξ . 49. Let (k, ρ) be in N ×]0, 1]. The H¨older space C k,ρ (Rd ) (or C k,ρ , if no confusion is possible) is the space of C k functions u on Rd such that |∂ α u(x) − ∂ α u(y)| u C k,ρ = sup ∂ α u L∞ + sup < ∞. |x − y|ρ x=y |α|≤k Proving that the sets C k,ρ are Banach spaces is left as an exercise.
Xd ) 1 can be continuously extended from H s (Rd ) onto H s− 2 (Rd−1 ). 4 Nonhomogeneous Sobolev Spaces on Rd 43 Proof. We ﬁrst prove the existence of γ. Arguing by density, it suﬃces to ﬁnd a constant C such that ∀φ ∈ S , γ(φ) ≤C φ 1 H s− 2 Hs . 40) To achieve the above inequality, we may rewrite the trace operator in terms of a Fourier transform: φ(0, x ) = (2π)−d ei(x |ξ ) φ(ξ1 , ξ ) dξ1 dξ Rd ei(x |ξ ) (2π)−1 = (2π)1−d Rd−1 R φ(ξ1 , ξ ) dξ1 dξ . We thus have γ(φ)(ξ ) = (2π)−1 R 2 φ(ξ1 , ξ ) dξ1 .
Fourier Analysis and Nonlinear Partial Differential Equations by Hajer Bahouri