Global well-posedness of the quasi-linear Euler-Korteweg system for - PowerPoint PPT Presentation

Presentation of the results Idea of the Proof Perspectives Global well-posedness of the quasi-linear Euler-Korteweg system for small irrotational data Corentin Audiard (LJLL, Paris 6) and Boris Haspot (Ceremade, Paris Dauphine) 1 Presentation of the results 2 Idea of the Proof 3 Perspectives Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives Let us recall the compressible Navier-Stokes equations: Mass equation : ∂ t ρ + div ρ u = 0 , Momentum equation : ∂ t ( ρu ) + div( ρu ⊗ u ) + ∇ P ( ρ ) = div K, , Initial data : ( ρ, u ) /t =0 = ( ρ 0 , u 0 ) . Here u = u ( t, x ) ∈ R N stands for the velocity field, ρ = ρ ( t, x ) ∈ R + is the density, P ( ρ ) the pressure (in the sequel we will only consider P ( ρ ) = ρ 2 ) and the general Korteweg tensor reads as follows: �� � ′ ( ρ )) |∇ ρ | 2 � ρκ ( ρ )∆ ρ + 1 div K = div 2 ( κ ( ρ ) + ρκ Id − κ ( ρ ) ∇ ρ ⊗ ∇ ρ . (1) Here κ is the capillary coefficient and is regular far away of 0 (we can think to κ ( ρ ) = ρ α with α ∈ R ) Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives Dispersive Equations: Following an idea of F. Coquel, setting: � � κ ( ρ ) a ( ρ ) = ρ κ ( ρ ) , ω = − ∇ ρ and z = u + iω ρ we get the following extended formulation: � ∂ t ρ + div( ρu ) = 0 , ∂ t z − i ∇ ( a ( ρ )div u ) + u · z + i ∇ z · ω + ∇ f ( ρ ) = 0 . with f ′ ( ρ ) = P ′ ( ρ ) . We are in presence of a quasilinear degenerate Schr¨ odinger ρ equation which is degenerate because we have only ∇ ( a ( ρ )div z ) and not div( a ( ρ ) ∇ z ). Remark It is generally delicate to work with such equation since it requires non trapping conditions on the bicharacteristics, Kato smoothing effects. Due to the Mizohata condition we also have to work in weight space in order to ensure ”energy” estimates. and with irrotationnal velocity we have ∇ ( a ( ρ )div z ) = √ κ 1 ∆ z. When κ ( ρ ) = κ 1 ρ In particular we have in this case a semilinear Schr¨ odinger equation. This case corresponds to the so called quantum pressure. Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives The Madelung transform When the velocity u = ∇ θ is irrotational, the Madelung transform ψ = √ ρe θ i 2 √ κ 1 allows formally to rewrite the Euler Korteweg system as the Gross-Pitaevski equation (GP): � 2 i √ κ 1 ∂ t ψ = − 2 κ 1 ∆ ψ + ( | ψ | 2 − 1) ψ, (2) ψ (0 , · ) = ψ 0 . with the boundary condition lim | x |→ + ∞ ψ = 1. The Gross-Pitaevskii equation is the Hamiltonian evolution associated to the Ginzburg-Landau energy: � � √ κ 1 |∇ ψ ( t, x ) | 2 + 1 4 ( | ψ | 2 − 1) 2 � E ( ψ )= dx 2 R N (3) � � √ κ 1 |∇ ϕ ( t, x ) | 2 + 1 4 (2 Reϕ + | ϕ | 2 ) 2 � = dx. 2 R N with ψ = 1 + ϕ . Up to a change of variable we may take κ 1 = 1 and we consider the equation on ϕ = ψ − 1: � i∂ t ϕ + ∆ ϕ − 2 Reϕ = F ( ϕ ) , (4) ϕ + | ϕ | 2 ) ϕ. F ( ϕ ) = ( ϕ + 2 ¯ Remark The Gross-Pitaevskii equation is close from a defocusing cubic nonlinear Schr¨ odinger equation, one of the main difference is that there exists traveling waves. Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives Some results of ”weak-strong” solutions for Euler Korteweg S. Benzoni, R. Danchin and S. Descombes [03,04], Existence of strong N 2 +2+ ε × H N 2 +1+ ε . They use an solution in finite time for ( ρ − 1 , u 0 ) ∈ H energy method via the introduction of a suitable gauge. P. Antonelli and P. Marcati [09], Existence of global weak solution when κ ( ρ ) = 1 ρ and u 0 = ∇ θ 0 for N = 2 , 3 (Madelung transformation). C. Audiard and BH [14], Existence of global strong solution for N = 3 with small initial data u 0 = ∇ θ 0 when κ ( ρ ) = 1 ρ . The crucial part of the proof is relied to dispersive estimate in weight space. What about the Gross-Pitaevskii equation? N ≥ 2 [Gallo, G´ erard] , Existence of global strong solution in the energy space. N ≥ 2 [Bethuel, Chiron, Gravejat, Maris, Saut, Smets · · · ] Existence of traveling waves when N ≥ 2 via variational technics with √ 0 < | c | < 2 of finite energy, it means solutions of the form: u c ( x 1 − ct, x 2 , · · · , x N ) . N ≥ 3 [Gustafson,Nakanishi,Tsai] Scattering results for small initial data (Normal form, Space-time resonance). Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives Strategy to get global strong solution for the Euler Korteweg system Let us look at the simpler case of the quantum pressure κ ( ρ ) = 1 ρ . It suffices to solves the Gross-Pitaevskii equation and to verify that the solution ψ does not vanish in order to use the Madelung transform. Indeed in this case we can transfer the regularity of ψ on the density ρ and the velocity u = ∇ θ . To do this (since ψ = 1 + ϕ ) we shall prove that � ϕ � L ∞ remains small all along the time. Remark This is generally not right, indeed for the linear Schr¨ odinger equation we can choose arbitrary small L ∞ initial data which blow up in L ∞ norm for arbitrary small time. To prove such result, the natural tool is to use global dispersive Strichartz estimate (however in the case N = 3 , it is not sufficient because we have quadratic nonlinearities such that their exponent corresponds to the Strauss exponent). Why it is difficult to deal with a quadratic nonlinearity ? We have: � ϕ ( s ) � 2 � e i ( t − s )∆ ϕ 2 ( s, · ) � L 3 � L 3 is not in L 1 (0 , t ) . ( t − s ) In particular we shall use the theory of space-time resonance developed by S. Gustafson, K. Nakanishi , T-P. Tsai and also P. Germain, N. Masmoudi, J. Shatah. Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives Theorem (C. Audiard, BH) Let N = 3 and P ′ (1) > 0 , if u 0 = ∇ φ 0 is irrotational, there exists δ > 0 and N 0 large enough such that if � u 0 � H N 0 − 1 + � ρ 0 − 1 � H N 0 + � xu 0 � L 2 + � x ( ρ 0 − 1) � L 2 ≤ δ, then there exists a global strong solution for the Euler Korteweg system with � ρ − 1 � L ∞ ( R + × R d ) ≤ 1 / 2 . Remark Let us mention that our solution scatters. We are going to combine space-time resonance theory and energy estimates, these type of technics have allowed to prove for the first time the existence of global strong solution with small initial data for the water wave equation (see P. Germain, N. Masmoudi and J Shatah). Remark We have similar results when N ≥ 4 . The problem remains open when N ≥ 2 . This case is much tricky since we are in a situation where the lower exponent of the nonlinearities is below the Strauss exponent. Furthermore for κ ( ρ ) = 1 ρ , it exists traveling waves for the Gross Pitaevskii of arbitrary small energy when N = 2 . It seems more complicated to exhibit dispersion properties. Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives Sketch of the Proof: � � κ ( ρ ) κ ( ρ ) ρ ∇ ρ , L the primitive of such that L (1) = 1, L = L ( ρ ), Setting w = ρ z = u + iw the Euler-Korteweg system rewrites � ∂ t l + u · ∇ l + a (1 + l )div u = 0 , (5) ∂ t z + u · ∇ z + iw · ∇ z + i ∇ ( a (1 + l )div z ) = g ′ 0 (1 + l ) w. with l = L − 1. In the potential case u = ∇ φ , the system on φ, l then reads � � |∇ φ | 2 − |∇ l | 2 � ∂ t φ − ∆ l + 2 l = ( a (1 + l ) − 1)∆ l − 1 + (2 l − � g (1 + l )) , (6) 2 ∂ t l + ∆ φ = −∇ φ · ∇ l + (1 − a (1 + l ))∆ φ. with � g (1) = 0 since we look for integrable functions. Up to a change of variable g ′ (1) = 2 . The linear part precisely corresponds to the linear we can assume that � part of the Gross-Pitaevskii equation. Diagonalization of the system: In order to diagonalize the system we set � � − ∆ − ∆(2 − ∆) , φ 1 = Uφ, l 1 = l. U = 2 − ∆ , H = Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives The equation writes in the new variables  � � ( a (1 + l 1 ) − 1)∆ l 1 − 1 � |∇ U − 1 φ 1 | 2 − |∇ l 1 | 2 �  ∂ t φ 1 + Hl 1 = U + (2 l 1 − � g (1 + l 1 )) , 2  ∂ t l 1 − Hφ 1 = −∇ U − 1 φ 1 · ∇ l 1 − (1 − a (1 + l 1 )) Hφ. (7) More compactly, if we set ψ = φ 1 + il 1 , ψ 0 = ( Uφ + il ) | t =0 , the Duhamel formula gives � t e itH ψ 0 + e i ( t − s ) H N ( ψ ( s )) ds, ψ ( t ) = (8) 0 � ( a (1 + l 1 ) − 1)∆ l 1 − 1 � |∇ U − 1 φ 1 | 2 − |∇ l 1 | 2 � � N ( ψ ) = U + (2 l 1 − � g (1 + l 1 )) 2 � � � � − ∇ U − 1 φ 1 · ∇ l 1 − + i 1 − a (1 + l 1 ) Hφ . (9) What are the difficulties for getting solution of (8)-(16)? How to deal with the quadratic terms? How to avoid the possible loss of derivatives? Corentin Audiard and Boris Haspot