A dispersive property of the Euler-Korteweg system Corentin Audiard Laboratoire Jacques-Louis Lions (UMR 7598) Universit´ e Pierre et Marie Curie 28 june 2012 Corentin Audiard A dispersive property of the Euler-Korteweg system
The Euler-Korteweg system consists of the following equations � ∂ t ρ + div( ρ u ) = 0 , ∂ t u + ( u · ∇ ) u + ∇ g 0 ( ρ ) = ∇ ( K ( ρ )∆ ρ + 1 2 K ′ ( ρ ) |∇ ρ | 2 ) , (EK) It is a perturbation of the classical Euler equations, that takes into account the capillarity effects. The quantities involved are the density ρ and the velocity u , K is the so called Korteweg stress tensor. On the opposite of the Navier-Stokes equations, the perturbation is dispersive . Corentin Audiard A dispersive property of the Euler-Korteweg system
The linearized system near a constant state ( u , ρ ) admits the following dispersion relation ( τ + iu · ξ ) 2 + ρ ( g ′ | ξ | 2 + K | ξ | 4 ) = 0 . At high frequencies it amounts to τ ∼ ± i � K ρ | ξ | 2 and thus bears some similarity with the usual Schr¨ odinger equation i ∂ t u + a ∆ u = 0 , whose dispersion relation is τ + ia | ξ | 2 = 0. Corentin Audiard A dispersive property of the Euler-Korteweg system
Local well-posedness of the Euler-Korteweg system in any dimension was obtained in 2007. Theorem (Benzoni-Danchin-Descombes ′ 07) Given s > d / 2 + 1 , u 0 ∈ H s ( R d ) , ρ 0 ∈ C 0 b ( R d ) such that ∇ ρ ∈ H s , there exists T > 0 and a unique solution ( ρ, u ) ∈ C T C b × C T H s , ∇ ρ ∈ C T H s of the Cauchy problem ∂ t ρ + div ( ρ u ) = 0 , ∂ t u + ( u · ∇ ) u + ∇ g 0 ( ρ ) = ∇ ( K ( ρ )∆ ρ + 1 2 K ′ ( ρ ) |∇ ρ | 2 ) , ( ρ, u ) | t =0 = ( ρ 0 , u 0 ) (1) Corentin Audiard A dispersive property of the Euler-Korteweg system
The proof relied on rather involved a priori estimates, for an extended (formally equivalent) system displaying a better structure : � ∂ t ζ + u · ∇ ζ + a ( ζ )div u = 0 , (2) ∂ t z + ( u · ∇ ) z + i ( ∇ z ) · w + i ∇ ( a div z ) = − g ′ ( ζ )Re( z ) . � where ζ = R ( ρ ), R is a primitive of the application ρ → K ( ρ ) /ρ , � R − 1 ( ζ ) K ( R − 1 ( ζ )). w = ∇ ζ , z = u + iw , a ( ζ ) = The second equation actually looks like a quasi-linear degenerate Schr¨ odinger equation . Corentin Audiard A dispersive property of the Euler-Korteweg system
Opposedly to the Schr¨ odinger equation, no dispersive estimate was proved yet for the Euler-Korteweg system. Our main result is a local smoothing property. Theorem Under the assumptions of the local well-posedness theorem, if moreover u 0 is irrotational, the curves associated to the hamiltonian a ( x , 0) | ξ | 2 are unbounded and ∇ x , t a (0 , x ) ≤ C / (1 + | x | 2 ) , then any solution ( u , ∇ ρ ) ∈ ( C T H s ) 2 additionally satisfies for some � T ( u , ∇ ρ ) / (1 + | x | ) ∈ L 2 ([0 , � T ]; H s +1 / 2 ( R d )) . Corentin Audiard A dispersive property of the Euler-Korteweg system
A few comments on the assumptions : the irrotationality seems natural since ∂ t z + ia ∇ div z = 0 admits trivial stationary solution, which precisely correspond to “purely rotational” initial data, div( z 0 ) = 0. The second assumption means that the solutions of the differential equation � X ( t ) ′ ∇ ξ ( a ( X ( t ) , 0) | Ξ( t ) | 2 ) = 2 a ( X ( t ) , 0)Ξ( t ) , = Ξ( t ) ′ −∇ x ( a ( X ( t ) , 0) | Ξ | 2 ) , = satisfy lim t →∞ � X ( t ) � = + ∞ . It is standard in the frame of linear Schr¨ odinger equations with variable coefficients. Corentin Audiard A dispersive property of the Euler-Korteweg system
Some elements of proof : Since the local gain of regularity is only of 1 / 2 derivative, it can hardly be obtained by basic multiplier techniques. It is necessary to use slightly more sophisticated tools. Nonlinearities appear even in the highest order derivatives, thus the pseudo-differential calculus is not well-suited as it usually requires a lot of smoothness from the coefficients. A more fitted tool would be Bony’s paradifferential calculus. Para-differential calculus allows to replace a product uv by T u v + R ( u , v ), where R is (hopefully) smooth, and T u acts H s → H s . It is even possible for a class of symbol s ( x , ξ ) that satisfy minimal regularity assumptions to define the paradifferential operators T s . Corentin Audiard A dispersive property of the Euler-Korteweg system
A very basic sketch of proof : fix some symbol p ( x , ξ ) and consider the derivative d dt � T p z , z � = � T p ( − i ∇ a div z ) , z � + � T p z , − i ∇ a div z � + l.o.t. = � [ T p , − i ∇ a div] z , z � + l.o.t. use then the rules of para-differential calculus and the irrotationality · · · = � [ T p , T − i | ξ | 2 a ] z , z � + l.o.t. = � T { ip , | ξ | 2 a } z , z � + l.o.t. where { ip , | ξ | 2 a } is the Poisson bracket d � ∂ ξ j p ∂ x j ( a | ξ | 2 ) − ∂ x j p ∂ ξ j ( a | ξ | 2 ) . j =1 Corentin Audiard A dispersive property of the Euler-Korteweg system
We have obtained (roughly) d dt � T p z , z � ≃ � T { ip , | ξ | 2 a } z , z � . If p is a zeroth order operator, and { ip , | ξ | 2 a } ≥ c | ξ | / (1 + | x | 2 ), it is then possible to use a G¨ arding-like inequality to deduce � d 1 + | x | 2 � 2 dt � T p z , z � � c � z / H 1 / 2 ( R d ) � � � T 1 + | x | 2 � 2 ⇒ � z ( t ) / H 1 / 2 dt � � z � L ∞ ([0 , � T ] , L 2 ( R d )) , 0 which is the expected smoothing effect. Corentin Audiard A dispersive property of the Euler-Korteweg system
Why it is not that simple : The construction of p is complicated (but similar to the one of Doi for Schr¨ odinger like equations), The G¨ arding inequality is not standard, The lower order terms are actually not neglectible, Instead of a gain L 2 → H 1 / 2 we want a gain H s → H s +1 / 2 , thus instead of working on z one has to study the quantity T | ξ | s z , it raises new commutators and more “bad” lower order terms. Corentin Audiard A dispersive property of the Euler-Korteweg system
Some more details on the gauge method : set Z s = T ϕ | ξ | s z , the equation satisfied by Z s is ∂ t Z s + T u ·∇ Z s + i ( ∇ Z s ) · w + i [div a ∇ , T ϕ | ξ | s ] z − i div T a ∇ Z s = l . o . t . We have i ( ∇ Z s ) · w ≃ T − w · ξ | ξ | s ϕ z , and i [div a ∇ , T ϕ | ξ | s ] z ≃ T { a | ξ 2 , | ξ | s ϕ } z . To suppress the bad terms, it is sufficient to have { a | ξ | 2 , ϕ | ξ | s } = ϕ | ξ | s ξ · w , and the “miraculous” function ϕ = √ ρ a s / 2 works (as in Benzoni-Danchin-Descombes) ! Corentin Audiard A dispersive property of the Euler-Korteweg system
The Euler-Korteweg system admits traveling waves ( ρ, u ) that only depend on x 1 − ct , the assumptions for the smoothing effect are usually not satisfied even for the linearized equations near such waves : ∂ t z + u ∇ z + i ∇ z · w + i ∇ a div z = l . o . t . Nevertheless, it is still possible to prove local smoothing in special cases. Proposition Assume that a 0 ( x 1 ) − a 0 ′ ( x 1 ) � 2 � a 0 ( x 1 ) ≥ α > 0 , then the same local smoothing property is still true. Corentin Audiard A dispersive property of the Euler-Korteweg system
Principle of proof : Doi’s construction of p does not work, but a simpler one is actually available. Essentially, local smoothing is reduced again to the positivity of { a | ξ | 2 , p } , now the choice p = f ( x 1 − ct ) x · ξ gives | ξ | � � { a | ξ | 2 , p } = | ξ | (2 af − x 1 a ′ f ) + ξ 1 x · ξ a ′ f + 2 f ′ a . | ξ | For f = c / √ a the bad term cancels, and the positivity condition becomes � � a ′ 2 √ a − x 1 { a | ξ | 2 , p } = | ξ | √ a > 0 , which is precisely the assumption. Corentin Audiard A dispersive property of the Euler-Korteweg system
Thank you for you attention ! Corentin Audiard A dispersive property of the Euler-Korteweg system
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