Global existence of strong solution for the compressible - PowerPoint PPT Presentation

Presentation of the results Idea of the Proof Global existence of strong solution for the compressible Navier-Stokes equations with large initial data on the irrotational part Boris Haspot, Universit´ e Paris Dauphine 1 Presentation of the results 2 Idea of the Proof Boris Haspot

Presentation of the results Idea of the Proof Let us recall the compressible Navier-Stokes equations: Mass equation : ∂ t ρ + div ρ u = 0 , Momentum equation : ∂ t ( ρu ) + div( ρu ⊗ u ) − div( µ ( ρ ) D ( u )) − ∇ ( λ ( ρ )div u ) + ∇ P ( ρ ) = 0 , 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 2 ( ∇ u + t ∇ u ) the strain tensor. and D ( u ) = 1 We denote by λ and µ the two viscosity coefficients of the fluid, which are assumed to satisfy µ > 0 and λ + 2 µ > 0 (such a condition ensures ellipticity for the momentum equation and is satisfied in the physical cases where λ + 2 µ N > 0). In the sequel we shall only consider the shallow-water system which corresponds to: µ ( ρ ) = µρ with µ > 0 and λ ( ρ ) = 0 . We supplement the problem with initial condition ( ρ 0 , u 0 ). Boris Haspot

Presentation of the results Idea of the Proof We aim at solving the system in the case where the data ( ρ 0 , u 0 ) have critical regularity. By critical, we mean that we want to solve our system in functional spaces with invariant norm for the scaling of the equations. It is easy to see that the following transformations: ( ρ 0 ( x ) , u 0 ( x )) − → ( ρ 0 ( lx ) , lu 0 ( lx )) , ∀ l ∈ R . → ( ρ ( l 2 t, lx ) , lu ( l 2 t, lx ) , l 2 P ( l 2 t, lx )) . ( ρ ( t, x ) , u ( t, x ) , P ( t, x )) − have this property of invariance, provided that the pressure term has been N N 2 − 1 ) N . changed in l 2 P . A good candidate corresponds to the space H 2 × ( H Some results of global strong solutions A. Matsumura and T. Nishida [80s], Global strong solutions with small initial data. D. Hoff [90s,07], Global weak-strong solution with small initial data with discontinuous initial density ρ 0 ∈ L ∞ . R. Danchin [2000], Global strong solutions with small initial data in critical N N 2 − 1 space for the scaling of the equations with u 0 ∈ B , ( ρ 0 − 1) ∈ B 2 , 1 . 2 2 , 1 Z. Chen et al, F. Charve and R. Danchin, BH [2010,2011], Global strong N N N p − 1 p − 1 p solutions with small initial data with u 0 ∈ B , ( ρ 0 − 1) ∈ B ∩ B p, 1 p, 1 p, 1 with 1 ≤ p ≤ N . In particular we have large initial oscillary initial data in L N , u 0 ( x ) = φ ( x ) sin ( ε − 1 x · ω ) n, with ω and n some unit vectors and φ ∈ C ∞ 0 ( R N ). Boris Haspot

Presentation of the results Idea of the Proof How to obtain some global results for a family of large initial data? To do this we are going to consider some irrotational data. More precisely we are interested in constructing irrotational solution. It is natural to deal with such solution, indeed let us mention the case of the so-called compressible Euler system with quantum pressure which is transformed in a non linear Schr¨ dingier equations via the famous transformation of Madelung. More precisely when P ( ρ ) = 0 we can check that: ( ρ 1 , − µ ∇ ln ρ 1 ) , is a particular solution of our system (with initial data( ρ 0 , − µ ∇ ln ρ 0 )) if ρ 1 verifies an heat equation: � ∂ t ρ 1 − µ ∆ ρ 1 = 0 , ( ρ 1 ) /t =0 = ( ρ 1 ) 0 . We are going to call ( ρ 1 , − µ ∇ ln ρ 1 ) a quasi-solution . The idea in the sequel will consist in working around these profiles. Remark In order to ensure the irrotational form of the solution, we need that the term ∇ ln ρ · Du be a gradient, which implies the structure in ∇ ln ρ for the velocity. Very surprisingly we have some regular effects on the density which is a priori governed by a transport equation. Boris Haspot

Presentation of the results Idea of the Proof Littlewood-Paley decomposition corresponds to a dyadic decomposition of the space in Fourier variables. Let ( ϕ, ψ ) a couple of regular functions such that ϕ has support in C = { ξ ∈ R N / 3 4 ≤ | ξ | ≤ 8 3 } , and ψ in B (0 , 4 3 ) with: � ϕ (2 − l ξ ) = 1 . ψ ( ξ ) + l ∈ N Let us note h = F − 1 ϕ , we can define the dyadic blocks by: � ∆ l u = ϕ (2 − l D ) u = 2 lN R N h (2 l y ) u ( x − y ) dy if l ∈ Z � S l u = ∆ k u . k ≤ l − 1 ′ ( R N ): We have then modulo the polynoms in S � u = ∆ k u. k ∈ Z Definition ′ The Besov space B s p,r corresponds to the set of temperated distributions u ∈ S h such that : � � � 1 2 lsr � ∆ l u � r r < + ∞ . � u � B s p,r = L p l ∈ Z Boris Haspot

Presentation of the results Idea of the Proof Definition Let ρ ∈ [1 , + ∞ ] , T ∈ [1 , + ∞ ] and s 1 ∈ R . We define the space � L ρ T ( B s 1 p,r ) as L ρ T ( B s 1 follows, u is in � p,r ) if: � � � 1 2 lrs 1 � ∆ l u ( t ) � r r . � u � � p,r ) = T ( B s 1 L ρ L ρ T ( L p ) l ∈ Z Proposition Let s ∈ R , ( p, r ) ∈ [1 , + ∞ ] 2 . Assume that u 0 ∈ B s p,r and f ∈ � L 1 T ( B s p,r ) . Let u be a solution of: � ∂ t u − µ ∆ u = f u t =0 = u 0 . Then there exists C > 0 depending only on N, µ, ρ 1 and ρ 2 such that: � � � u � � p,r ) ≤ C � u 0 � B s p,r + � f � � . T ( B s +2 L 1 L 1 T ( B s p,r ) If in addition r is finite then u belongs to C ([0 , T ] , B s p,r ) . Boris Haspot

Presentation of the results Idea of the Proof We now search some solutions written under the following form ln ρ = ln ρ 1 + h 2 ( ρ = ρ 1 e h 2 , with ρ 1 = 1 + q 1 ) and u = − µ ∇ ln ρ 1 + u 2 , we have then modulo that the density does not admit vacuum:  ∂ t ln ρ + u · ∇ ln ρ + div u = 0 ,   ∂ t u + u · ∇ − µ ∆ u − µ ∇ ln ρ · Du + ∇ F ( ρ ) = 0 , (1)   (ln ρ, u ) /t =0 = (ln ρ 0 , u 0 ) . By using the fact that ( ρ 1 , u 1 ) = ( ρ 1 , − µ ∇ ln ρ 1 ) is a quasi solution with: � ∂ t ρ 1 − µ ∆ ρ 1 = 0 . ρ 1 (0 , · ) = ρ 1 0 . we can rewrite the system under the following form:  ∂ t h 2 + u · ∇ h 2 + div u 2 = − u 2 · ∇ ln ρ 1 ,     ∂ t u 2 + u · ∇ u 2 − µ ∆ u 2 + a ∇ h 2 = − a ∇ ln ρ 1 − u 2 · ∇ u 1 + µ ∇ ln ρ 1 · Du 2  + µ ∇ h 2 · Du 1 + µ ∇ h 2 · Du 2 ,      ( h 2 , u 2 ) /t =0 = ( h 2 0 , u 2 0 ) . (2) where we have assumed to simplify that P ( ρ ) = aρ . We are going to solve the system (2) with small initial data ( h 2 0 , u 2 0 ) and with large ρ 1 . Boris Haspot

Presentation of the results Idea of the Proof Theorem Let ρ 0 = ( ρ 1 ) 0 e h 2 0 and u 0 = − µ ∇ ln( ρ 1 ) 0 + u 2 0 . Moreover we assume that N N N N 2 − 2 2 − 1 2 , 1 , h 2 2 , 1 and u 2 ( ρ 1 ) 0 ≥ c > 0 , ( ρ 1 − 1) 0 ∈ B ∩ B 2 0 ∈ B 2 0 ∈ B . Then it 2 , 1 2 , 1 exists ε > 0 , C > 0 and l > 0 large enough (depending on � ( ρ 1 − 1) 0 � such N 2 B 2 , 1 that if: � ≤ ε and � h 2 + � u 2 � ∆ k ( ρ 1 − 1) 0 � 0 � 0 � ≤ ε, (3) N N N 2 − 2 2 − 1 2 B B B k ≥ l 2 , 1 2 , 1 2 , 1 then it exists a global strong solution ( ρ, u ) of the shallow-water system under the following form: ρ = ρ 1 e h 2 and u = − µ ∇ ln ρ 1 + u 2 with: � ∂ t ρ 1 − µ ∆ ρ 1 = 0 , (4) ( ρ 1 ) t =0 = ( ρ 1 ) 0 . and such that: N N h 2 ∈ � C ( R + , B 2 , 1 ) ∩ L 1 ( R + , B 2 2 , 1 ) 2 N N 2 − 1 2 +1 and u 2 ∈ � C ( R + ; B ) ∩ L 1 ( R + , B ) . 2 , 1 2 , 1 Boris Haspot

Presentation of the results Idea of the Proof Remark We have a super-critical condition of smallness for the initial density, more precisely we can write this condition under the following form: � ( ρ 1 − 1) 0 � ≤ Cexp ( − C � ( ρ 1 − 1) 0 � ) . N N 2 − 2 2 B B 2 , 1 2 , 1 We can choose initial data with α and β well-chosen of the form: 0 ( x ) = 1 ε f ( x 1 , x 2 ε β e i x 3 q 1 ε γ , x 3 ) , 1 ε β + q 1 0 and f ∈ S ( R N ) . with ( ρ 1 ) 0 = In particular we allows initial density with large L ∞ norm. Remark In particular we can obtain global strong solution in dimension N = 2 with large initial data in the energy spaces. We also refer to a remarkable work of Kazhikhov and Waigant for other viscosity coefficients. Remark In a very surprising way the density is decomposed in a small part h 2 and in a regular part ρ 1 . This regularizing effect remains quite surprising. Boris Haspot

Presentation of the results Idea of the Proof In the sequel we are going to show the global existence for ( h 2 , u 2 ). First step: Construction of approximate solutions We regularize the initial data: ( h 2 ) n 0 = S n h 2 0 , ( u 2 ) n 0 = S n u 2 0 . We obtain then a solution ( h n 2 , u n 2 ) on the interval (0 , T n ) such that: N N N 2 − 1 2 − 1 2 +1 h n 2 ∈ � C ([0 , T n ] , B N ) u n 2 ∈ � ) ∩ � L 1 ([0 , T n ] , B 2 , 1 ∩ B C ([0 , T n ] , B ) . 2 , 1 2 , 1 2 , 1 Second step: Uniform estimates We are going to obtain uniform estimates on ( h n 2 , u n 2 ) in order to show that T n = + ∞ . To do this, we need to distinguish the behavior in low and high frequencies. We need to study the following system:  ∂ t h n + u n · ∇ h n + div u n 2 = F n 1 ,   2 + a ∇ h n = G n ∂ t u n 2 − µ ∆ u n 1 , (5)   h n 0 = h 0 , ( u n 2 ) /t =0 = ( u 2 ) n 0 . Let us mention that it is necessary to include in the study the convection term in order to not lose derivative on the density. Boris Haspot