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Long Time Estimates of Solutions to Hamiltonian Nonlinear Pdes

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Long Time Estimates of Solutions to Hamiltonian Nonlinear Pdes Long time estimates of solutions to Hamiltonian nonlinear PDEs Patrick Gérard Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France [email protected] UNC PDE mini-school February 2-4, 2016 Abstract This minicourse is devoted to long time behavior of solutions to nonlinear PDEs such as nonlinear Schrödinger equation or nonlinear wave equations. More precisely, we would like to provide an introduction to the following general question, closely connected to wave turbulence: assume that such a nonlinear PDE is globally well-posed on high regularity Sobolev spaces; how big can the high Sobolev norms of generic solutions be as time goes to infinity? The second part of the course will be focused on the special case of the cubic Szegő equation, which is a model of a nonlinear wave evolution and enjoys some integrable structure allowing to study its solutions in detail. Introduction Since the eighties, the study of Hamiltonian evolution partial differential equations, such as nonlinear Schrödinger equations of nonlinear wave equations, has become a topic of growing importance in mathematical analysis. Currently, a fairly general theory of initial value problems is now available, and the main effort of the experts is now mainly directed towards the long term description of solutions, starting with the case of small initial data. Though this program is well advanced if the physical space is the whole Euclidean space, where the dispersive effects are maximal and generally lead to scattering for small data solutions, it is still widely unexplored if the physical space is a bounded domain or a closed manifold, where the dispersive effects may be much weaker. In particular, a specific important question in the latter case concerns the possibility of strong oscillations of the solution as time becomes large, or equivalently the growth of Sobolev norms of high regularity. The goal of these notes is to provide an elementary introduction to this topic, as well as a focus onto one of the few Hamiltonian evolutions for which such a description is available, thanks to a special integrability structure. The content of these notes is based on a series of three lectures given in the University of North Carolina during the Winter 2016, addressed to an audience of graduate students and postdocs in PDEs coming from various universities in the United States. I am grateful to the Mathematics Department of Chapel Hill, who made this minicourse possible, and to my student Joseph Thirouin, who wrote the first version of these notes. Orsay, April 2016 Patrick Gérard 1 Lecture 1 Introduction to long time estimates 1.1 What is a Hamiltonian PDE? 1.1.1 Hamiltonian systems in finite dimension At the beginning of the nineteenth century, physicists were looking for an abstract setting to reformulate Newton’s laws of mechanics. The ideas of the French mathematician Joseph-Louis Lagrange (1736-1813) led to the so-called Euler-Lagrange equations and to the calculus of vari- ations, whereas the Irish scientist William Rowan Hamilton (1805-1865) introduced another formalism, which later inspired the theory of quantum mechanics, and gave rise to a whole field in the study of PDEs. Here is the finite-dimensional setting Hamilton proposed. Let n 2 Nnf0g, and consider n n the phase space R × R . An element (q; p) in this space corresponds to a set of “positions” n n q = (q1; : : : ; qn) and of “momenta” p = (p1; : : : ; pn). Given a smooth function H : R × R ! R, the Hamilton equations associated to H are 8 @H > q_j = (q; p); < @pj > @H :> p_j = − (q; p); @qj for all j 2 f1; : : : ; ng, where the dot stands for the time derivative. The function H is then called the energy of this system of ordinary differential equations, since it is conserved along the flow lines. n An example. Let V : R ! R be a smooth function (a potential), and consider the Hamilto- nian given by 1 H(q; p) := jpj2 + V (q); 2m 2 Pn 2 where jpj := j=1 pj and m > 0 is a real number. Then the Hamilton equations read mq_j = pj, which exactly means that pj is the momentum in a physical sense, and p_j = −@qj V (q), i.e. mq¨j = −@qj V (q). The Hamilton equations associated to H thus describe the motion of n bodies of same mass m according to Newton’s second law of Mechanics. 2 Complex formulation. We can reformulate the equations in an even shorter way, introducing n n n C = R ⊕ iR and a new variable z := q + ip. The evolution of z under the Hamilton equations is given by @H @ 1 @ @ iz_j = 2 (z); where := + i : (1.1) @z¯j @z¯j 2 @qj @pj @ := 1 @ − i @ Similarly, we define @zj 2 @qj @pj . n n Note that for any smooth function F : C ! R and any smooth curve γ : R ! C , γ(s) = (γ1(s); : : : ; γn(s)), we have n d X dγj @F dγ¯j @F F (γ(s)) = (s) (γ(s)) + (s) (γ(s)): ds ds @z ds @z¯ j=1 j j n @F @F Now denote by (·|·) the usual Hermitian product on C , and by @z (resp. @z¯ ) the n-vector @F @F j 2 f1; : : : ; ng F @F = @F formed by the collection of @zj (resp. @z¯j ), . Since is real-valued, @z¯ @z , so that we can write d @F F (γ(s)) = 2 Re γ_ (s) (z) = Im (γ _ (s)jXF (γ(s))) ; ds @z¯ @F setting XF (z) := −2i @z¯ (z). The Hamilton equations (1.1) are thus equivalent to z_ = XH (z); (1.2) and this is how the Hamiltonian formalism can be extended to the case of infinite dimensional phase spaces. 1.1.2 Extension to infinite dimensional systems 1 As a phase space, consider now E := C (M; C). In general, M could be any compact bound- aryless manifold. For instance, if dim M = 0, M is a set formed by n points, and E is then n d d isomorphic to C . In the sequel, we will restrict ourselves to M = T = (R=2πZ) , with d ≥ 1. 1 d In that case, E is isomorphic to the set of C functions on R which are 2π-periodic with respect to each coordinate. Definition. Let H : E ! R. Assume that there exists a mapping XH : E ! E such that for all smooth path γ : R ! E, d H(γ(s)) = Im (γ _ (s)jXH (γ(s))) ; (1.3) ds where Z (fjg) = f(x)g(x)dx; Td d d and dx is the Lebesgue measure renormalized by (2π) , so that T is of measure 1. 3 Then we say that XH is the Hamiltonian vector field of H, and the corresponding equation u_ = XH (u) (1.4) is called the Hamiltonian system associated to H. Remark 1. A crucial remark is that, as in the finite dimensional case, H is a conservation law for the system. Indeed, by definition, d H(u(t)) = Im(u _(t)jX (u(t))) = Im(X (u(t))jX (u(t))) = 0: dt H H H Remark 2. Note that there is no theorem insuring that solutions of (1.4) exist, even locally in time. 1.2 Some examples 1.2.1 The nonlinear Schrödinger equation 1 d Let λ 2 R. For u 2 E = C (T ; C), we set Z 1 2 λ 4 H(u) = 2 jru(x)j + 4 ju(x)j dx: Td Let us show that the hypothesis of the above definition is satisfied. Let γ : R ! E be a d smooth path (in the sequel, we write γ as a function of s 2 R and x 2 T ), and compute d Z H(γ(s)) = Re rγ_ (s; x)rγ(s; x) + λjγ(s; x)j2γ_ (s; x)γ(s; x) dx ds Td Z = Re γ_ (s; x)[−∆γ(s; x) + λjγ(s; x)j2γ(s; x)] dx Td Z = Im γ_ (s; x)[i∆γ(s; x) − iλjγ(s; x)j2γ(s; x)]dx Td 2 Thus, XH : E ! E, u 7! i(∆u − λjuj u) is the Hamiltonian vector field of H, and (1.4) reads ( iu_ + ∆u = λjuj2u (1.5) ujt=0 = u0: >From now on, we assume that λ ≥ 0 : this is called the defocusing case. The existence of solutions of (1.5) has been extensively studied, and we have the following result : 1 d Theorem 1. Let d 2 f1; 2; 3g. For any u0 2 E, there exists a unique u 2 C (R × T ; C) satisfying (1.5). The case d = 2 dates back to Brezis and Gallouët [4], and d = 3 is a consequence of the works of Bourgain [1]. In fact, more general results are available, involving less regular data. 4 1 d Definition. • Let f 2 L (T ). We define the Fourier coefficients of f by Z −ik·x d f^(k) = f(x)e dx; 8k 2 Z : Td s d 1 d • For s ≥ 1, we define the space H (T ) to be the space of all f 2 L (T ) such that X (1 + jkj2)sjf^(k)j2 < +1: k Hs is a Hilbert space, and the norm is defined as follows : 1 ! 2 X 2 s ^ 2 kfkHs := (1 + jkj ) jf(k)j : k s α 2 d • Observe that, whenever s 2 N, belonging to H is the same as requiring that @ f 2 L (T ) for all jαj ≤ s. s d Theorem 2. The nonlinear Schrödinger equation (1.5) is globally well-posed on H (T ) for any s ≥ 1 and d 2 f1; 2; 3g.
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