RANDOM PROPERTIES BENOIT PAUSADER

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1 RANDOM PROPERTIES BENOIT PAUSADER. Quasilinear problems In general, one consider the following trichotomy for nonlinear PDEs: A semilinear problem is a problem where the highest-order terms appears linearly with coefficients independent of the solution: e.g. u = u u + u, ( t + xxx ) u = u. A quasilinear problem is a problem where the highest-order term appears linearly, but with a coefficient that depends on fewer derivative of the solution, e.g. i {A ij (u) j u} = k, ( tt ) u = u u. A fully nonlinear problem is a problem where the highest order derivative appears in a genuinely nonlinear fashion, e.g. det u = k. However, I do not find this taxonomy completely satisfactory in that one has to be careful as to what the highest order derivatives are, e.g. u t + xx u = u In many cases, a fully nonlinear problem can be reduced to a quasilinear one by a simple differentiation. Let us look at the example det u = αu = u u ( u), u : R R. (.) {Jac} Taking one more derivative and introducing the co-hessian matrix: A := Com( u) or ( ) u A := u u u we may recast (.) as A ij ij ( k u) = αu k u which is a quasilinear problem (this is then an elliptic problem in a neighboorhod of a solution u which is convex). We caution however that (i) this is only possible in a relatively smooth setting. If e.g. one considers coefficients which are not smooth, taking another derivative might not be an option, (ii) taking another derivative might also not be appropriate in the presence of boundary conditions, (iii) this forces to consider systems instead of scalar functions and tools such as the maximum principle might not be nicely preserved. Date: Monday st September, 05, 0:7. in all the examples below u is the unknown function, and k is a known function.

2 BENOIT PAUSADER However,this trick or a variation works in many cases. In fact a big part of the work on the water-wave system is to make it amenable to such a trick (this involves extracting the right notion of derivative, see later). This trick is related to the following estimates/heuristics D α (f(u)) = f (u)d α u + smoother uv H s u L v H s + u L v H s. Stationary phase This section is very standard and we will be sketchy. We refer to books of Stein for more details. Many terms that will appear later will be transformable into integrals of the form I(λ) = e iλφ(x) a(x)dx R d where a Cc (R d ) is a smooth amplitude and Φ C (R d \ {0}) is a smooth phase. We wish to obtain estimates on the size of I(λ) as λ. The idea is that since e ix is oscillating with mean 0, as λ becomes large, the integration over the argument in the exponential and the integration over the argument in the amplitude become independent and the fact that e ix is mean zero yields smallness though some form of Fubini. One can first remark that, indeed, if Φ > 0 on the support of a, one has rapid decay (this is a slight generalization of the Riemann-Lebesgues Lemma if Φ(x) = x): Lemma.. Assume that a C c (R d ), and that Φ, D α Φ D α on supp(a), then, for any N, there exists C N such that I(λ) C N ( + D) N+ ( Proof. We introduce L = Φ i Φ, sup D β a L β N ) λ N { L = div i Φ } Φ, e iλφ = λ LeiλΦ and using the last reproducing formula above, we find that I(λ) = e iλφ L adx λ R d = λ n e iλφ (L ) n adx R d and a crude estimate of the last term yields the claim. However, in most cases of interest the gradient vanishes on the support of integration. generic case is when the phase has nondegenrate critical points. Proposition.. Assume that Φ has one nondegenerate critical point x c on the support of a, that is Φ(x) = 0 x = x c, then det Φ(x c ) 0, I(λ) a,φ λ d. The

3 In fact, one has an asymptotic expansion where σ = sign( Φ(x c )). I(λ) = RANDOM PROPERTIES 3 e iσ π 4 e iφ(xc) a(x c ) + O(λ d+ (π) d det Φ(x c ) λ d ), Part of the analysis later will be to obtain a nonlinear variant of this, so we refer to the books of Stein for reference, see e.g. [, Chapter 8.]... Decay for solutions of Klein-Gordon equations. We consider solutions of the Klein- Gordon equation ( t + ) u = 0, u(t = 0) = f, t u(t = 0) = g where f and g are smooth: f, ĝ C c. Taking the Fourier transform, one can easily find the solution u(t) = cos(t )f + sin(t ) g, û(ξ, t) = eit + ξ + e it + ξ f(ξ) + eit + ξ e it + ξ i ĝ(ξ) + ξ. thus we see that it suffices to consider the decay rate of solutions of the form u(t) = F e it + ξ ϕ(ξ) where ϕ Cc. Writing down the Fourier transform, one finds [x ξ+t ] + ξ u(x, t) = c ϕ(ξ)dξ and we can rewrite the phase as R d e i tφ, Φ = + ξ + x t ξ we can then compute that ξ Φ = ξ + ξ + x t, det Φ = [ + ξ ] d+. Thus for each x and t fixed, Φ has at most critical point, which is nondegenerate and one obtains that u L t d which, as we have seen is in fact the optimal decay rate... Dispersive equations. only if x t and for frequency ξ = x/t. x t

4 4 BENOIT PAUSADER... dispersion relations. This generalizes to dispersive equations. Dispersive equations are equations whose solutions have the form { } u(t) = F e itω(ξ) û(t = 0) where ω is real and nondegenerate: det ω 0 The prototypical example is the Schrödinger equation one also has the Klein-Gordon equation (i t )u = 0, ω(ξ) = ξ, ( t + ) u = 0 generically. ω = Id ( ( + ξ 0 )) ( ) t + i 0 a = 0 + ξ b a = ( t + ξ )û(ξ), ω ± (ξ) = ±ω(ξ), ω ± (ξ) = [ + ξ ] 3 and the Airy equation b = ( t + + ξ )û(ξ), ξ ξ ξ ξ + (Id ξ + ξ ξ ξ ξ ) ( t + xxx ) u = 0, ω(ξ) = ξ 3, ω = 6ξ.... group velocity. An important fact about dispersive equations is that a solution localized in frequency around the frequency ξ 0 will travel at velocity close to the group velocity v g (ξ 0 ) = ω(ξ 0 ). this can be seen as a simple consequence of the stationary phase estimates. Indeed, consider a general solution u(t) = F { e itω(ξ) ϕ(ξ ξ 0 ) }. Taking the Fourier transform and changing variables η = ξ ξ 0, we may rewrite u(x, t) = e ix ξ e itω(ξ) ϕ(ξ ξ 0 )dξ R d = e ix ξ 0 e iω(ξ x 0) η] t ϕ(η)dη R d e it[ω(ξ 0+η) ω(ξ 0 )+ Observing that the term in front of the integral has modulus, we my discard it. Taking the gradient of the phase, we obtain [ η ω(ξ 0 + η) ω(ξ 0 ) + x ] t η = ω(ξ 0 + η) + x t Thus, we see that if x + t ω(ξ 0 ), then we have arbitrary decay of the amplitude. On the other hand, if x t ω(ξ 0 ), the stationary phase tells us that, in general u(x, t) t d. 3. Littlewood-Paley analysis 3.. Littlewood-Paley basis. In order to make the previous considerations more precise, we need a good way to localize in Fourier space. In our case we will have ω(ξ) = ω( ξ ) and we will focus on quantifying the energy ξ (the direction is also important but can usually be recovered by commuting with rotational vector fields). The easiest option would be to use exact projection in Fourier space. These projections are of the form π R : f F R Ff, R R d ξ. Unfortunately, the nonsmooth charater of the charateristic function induces all sorts of trouble (e.g. the stationary phase analysis requires smooth amplitudes). Instead we will be satisfied with almost projectors, the Littlewood-Paley projectors.

5 RANDOM PROPERTIES 5 Let ϕ C c (R) be a smooth function such that ϕ(x) = whenever x and ϕ(x) = 0 whenever x, and define φ 0 (x) = ϕ( x ) ϕ( x ), φ k (x) = φ 0 ( k x) We then observe that φ k (x), x 0, k Z φ k Cc (R d ) φ k is supported on k x k+. The family {φ k } k will be our smooth proxy for {{ k x k+ }} k. We define the Littlewood-Paley projectors P k f(ξ) = φ k (ξ) f(ξ). Then the family {P k } satisfies the following properties completeness f = k P kf, almost orthogonality P k P j = 0 if j k 5, almost eigenvectors: P k f L k P k f L. In addition, the formula f = k P kf allows to decompose f into a sum of analytic functions if the sum is convergent, we can work with a finite truncation and only consider smooth functions (although the various estimates will only be uniform in some norms). The above properties should be compared to the properties of the family of projector diagonalizing a self-adjoint compact operator A: {π j } j such that j π j = Id, π j π q = 0 if j q, Aπ j = λ j π j where λ j R is an eigenvalue. 3.. Nonlinear analysis. In order to use these projectors in a nonlinear context, it is useful to consider non-hilbertian norms. Two important properties are the Bernstein s estimates: P k f L p f L p, P k f L q k( d p d q ) P k f L p, p q, (3.) {Ber} P k f L p kn n P k f L p The first property follows from { } P k f = F φ k f = ˇφ k f, P k f L p ˇφ k L f L p, ˇφ k L = dk ˇφ0 ( k x) L = ˇφ 0 L <. For the second property, we first observe that, using the first property twice and interpolation, it remains only to show that P k f L dk P k f L. Fortunately, we know how to estimate the L L norm of a convolution operator. We introduce φ 0 = φ + φ 0 + φ which satisfies φ 0φ 0 = φ 0. We then introduce φ k (x) = φ 0 ( k x) and let p k be the associated projector. We then observe that P k f L = P k P kf L = ˇφ k P kf L ˇφ k L P kf L ˇφ k L = dk ˇφ 0 ( k ) L = dk ˇφ 0 L = Cdk

6 6 BENOIT PAUSADER The third property is direct once one observes that {( ) ξ n ( k ) n P k f = F k φ 0 ( ξ } { k ) f(ξ) = F χ( ξ } k ) f(ξ), χ(x) = x n φ 0 (x) and χ has similar properties as φ 0. Sometimes it is useful to also consider integrated version: which satisfies similar properties P k = l k P l ; P k f = kd ϕ( k ) f, P k f L p f L p, P k f L q k( d p d q ) P k f L p, p q, n P k f L p kn P k f L p, n 0. {Sob} Littlewood-Paley multipliers allow to get nonlinear estimates efficiently. We give a few applications Proposition 3. (Sobolev (subcritical)). Assume that p q, q > p s d, then f L q f L p + s f L p (3.) Proof. We have that P k f L q k( d p d q ) P k f L p k( d p d q ) f L p, as a consequence P k f L p sk s P k f L p sk s f L p P k f L q min{ k( d p d q ), k( d p d q s) }( f L p + s f L p) using the triangle inequality and the completeness property: f = k P kf, we obtain (3.). We can even get sharper result for Sobolev embedding: Proposition 3.. Assume d = 3, then ( f L 6 and nonlinear estimates sup k Pk f L k Proposition 3.3. For 0 s < d/, there holds that ) 3 f 3 L Recall that Ḣs = s L. uv Ḣs u L v Ḣs + u Ḣs v L Proof of Proposition 3.3. The estimate is trivial if s = 0. Note that, by orthogonality, f Ḣ s k sk P k f L This proof contain one important idea: looking at the support of convolution P k (uv) = P k (P k up k v) + P k (P k up k v) + P k (P k up k v). k k k k k k +0 k k k +0

7 RANDOM PROPERTIES 7 Once the support have been clarified, the proof is easy. The first sum is treated using Cauchy- Schwartz sk P k (P k up k v) L s(k k) P k u Ḣs P k v L k k k k k k s(k k) P k u v Ḣ s L k k and this is summable in k. the second and third sum are similar. We only treat the second: sk P k P k up k v L sk P k up k 0 v L k k k +0 k k sk P k u L P k 0 v L k k v L sk P k u L k k and this is square summable in k. This finishes the proof. Finally, we note one last estimate about bilinear operators as we will often consider F {B m [f, g]} (ξ) = m(ξ, η) f(ξ η)ĝ(η)dη R n Lemma 3.4. Assume that p, q, r and that p + q = r then B m [f, g] L r F m L (R d ) f L p g L q. This estimate, together with LP-decomposition will make the bulk of the analysis estimates we need. However, although we will not use it, we note the (harder) LP square-function theorem: f L p Sf L p, Sf(x) = ( k References P k f(x) ), < p <. [] E. Stein and R. Shakarchi, Princeton Lectures in analysis IV, Functional analysis, Princeton university press, ISBN Brown University address: benoit.pausader@math.brown.edu