Download Fourier Analysis and Nonlinear Partial Differential by Hajer Bahouri PDF

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.

Show description

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

Best calculus books

Calculus I with Precalculus, A One-Year Course, 3rd Edition

CALCULUS I WITH PRECALCULUS, brings you on top of things algebraically inside of precalculus and transition into calculus. The Larson Calculus application has been extensively praised through a new release of scholars and professors for its reliable and potent pedagogy that addresses the desires of a huge diversity of training and studying kinds and environments.

An introduction to complex function theory

This e-book offers a rigorous but undemanding advent to the speculation of analytic capabilities of a unmarried complicated variable. whereas presupposing in its readership a level of mathematical adulthood, it insists on no formal necessities past a legitimate wisdom of calculus. ranging from simple definitions, the textual content slowly and thoroughly develops the information of advanced research to the purpose the place such landmarks of the topic as Cauchy's theorem, the Riemann mapping theorem, and the theory of Mittag-Leffler might be taken care of with no sidestepping any problems with rigor.

A Course on Integration Theory: including more than 150 exercises with detailed answers

This textbook presents an in depth remedy of summary integration idea, building of the Lebesgue degree through the Riesz-Markov Theorem and in addition through the Carathéodory Theorem. additionally it is a few user-friendly homes of Hausdorff measures in addition to the elemental houses of areas of integrable features and traditional theorems on integrals counting on a parameter.

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}.

Download PDF sample

Rated 4.08 of 5 – based on 44 votes