BILINEAR ESTIMATES AND APPLICATIONS TO NONLINEAR WAVE EQUATIONS

By S. Klainerman, S. Selberg. 2001

159 citations recorded by the source corpus.

We undertake a systematic review of results proved in [26, 27, 30-32] 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.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8dba4af48becbdde851ad6d7c315f31e80f53825

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Almost sure well-posedness of the cubic nonlinear Schr\ (2009, background): “|bv(n,n2 +λ)|2 1 2dλ. 1 by Cauchy-Schwarz inequality. See (22) and (23) in [5]. Note that (4.14) is a standard representation for functions in Xs,b for b>1 2. For example, see Klainerman-Selberg [20]. • Case 1: First, we consider the case if any of uj is of type (II). From Cases (A)–(D), we assu”
Invariance of the Gibbs measure for the periodic quartic gKdV (2012, methodology): “For the factors u3, u4 ∈ X 1 2 −δ, 1 2 −δ T , we will use the following standard representation for functions in X (see, for example, Klainerman-Selberg [15]).”
Concentration Compactness for Critical Wave Maps (2009, background): “18) By the subcritical existence theory of Klainerman and Machedon, see [18]–[16] as well as [20], [21], the solution can now be extended smoothly beyond this time interval.”
Global regularity of wave maps, I: small critical Sobolev norm in high dimension (2000, background): “the result extends as well to large data; see [ 16 ], [19].”
Energy Dispersed Large Data Wave Maps in 2 + 1 Dimensions (2009, background): “The classical estimate here (see for instance [10]) has the form: (191) ‖ e±it|Dx|u((1)) k1 · e ±it|Dx|u((2)) k2 ‖L2t (L(2)x) .”
Global Regularity of Wave Maps¶II. Small Energy in Two Dimensions (2000, result): “ assumes some symmetry on the data and some curvature assumptions on the target manifold ([3], [27], [31], [32]), but is not known in general. For further discussion on these problems see, e.g. [14], [19], [26], [30]. A similar result, but with the Sobolev norm H˙ n/2 replaced a slightly smaller Bes”
Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class (2008, result): “We refer to [4], [6], [12], [13], [24], [16, Chapter 6], [10], [8] for surveys of the initial value problem for wave maps, which is of course the primary concern of this project.”
An improved local well-posedness result for the modified KdV equation (2003, background): “ is complete. The above lemma can easily be generalized to multilinear estimates. We shall not make use of this here, except in the case r= 2, where this is well known, see e. 4 g. Proposition 3.5 in [KS01]. A simple but important consequence of Lemma 1 is the embedding (6) Xr s,b⊂ C(R,Hcr s) for al”