Bilinear Estimates and Applications to Nonlinear Wave Equations

Abstract

We undertake a systematic review of results proved in [26,27,30,31,32] concerning local well-posedness of the Cauchy problem for certain systems of nonlinear wave equations, with minimal regularity assumptions on the initial data. Moreover we give a considerably simplified and unified treatment of these results and provide also complete proofs for large data. The paper is also intended as an introduction to and survey of current research in the very active area of nonlinear wave equations. The key ingredients throughout the survey are the use of the null structure of the equations we consider and, intimately tied to it, bilinear estimates.

Geometric renormalization of large energy wave maps

“…The Cauchy problem for wave maps has been extensively studied (see references); we refer the interested reader to the surveys in [23], [30], [50], [59], [69]. For arbitrary dimension n and arbitrary targets N, one can easily show by general energy methods (which indeed apply to any nonlinear wave equation, see e.g.…”

Global Regularity of Wave Maps from R2+1 to H2. Small Energy

“…The nature of these spaces is really revealed by means of the bilinear estimates which they satisfy, one of the most pivotal of which is (14). Another manifestation of this 'bilinear character' is expressed by the following lemma, which is proved exactly as in [19], in the 3-dimensional context (see also section 6, lemma 6.7):…”

“…First, we recall the homogeneous Besov analogues of the classical X s,b -spaces of Klainerman-Machedon: We introduce the Littlewood-Paley localizers P k which restrict frequency to dyadic size ∼ 2 k , k ∈ Z. More precisely, choosing a smooth nonnegative bump function m 0 (x) : R → R with support on 1 4 < x < 4 and satisfying 14 . Similarly, let Q j , j ∈ Z microlocalize to dyadic distance ∼ 2 j from the light cone.…”

“…We immediately note the 2nd pivotal bilinear inequality, which is essentially dual to (14): letting the notation and assumptions be like there, we have…”

The Cauchy problem for wave maps on a curved background

“…In the non-equivariant case, Klainerman and Machedon in [11], [12], [13], [14], and Klainerman and Selberg in [16], [17], established strong well-posedness in the subcritical norm H s × H s−1 (R d ) with s > d 2 by exploiting the null-form structure present in (1.7).…”