Regularity of the Boltzmann equation in bounded domains Daniela - PowerPoint PPT Presentation

Introduction Boundary conditions Known results Linear transport in-flow Existence Convex case Non convex case Regularity of the Boltzmann equation in bounded domains Daniela Tonon joint work with Y. Guo, C. Kim and A. Trescases CEREMADE, Université Paris Dauphine Advanced School & Workshop on Nonlocal Partial Differential Equations and Applications to Geometry, Physics and Probability ICTP May, 23rd 2017

Introduction Boundary conditions Known results Linear transport in-flow Existence Convex case Non convex case The dynamics of rarefied gases are governed by Boltzmann equation (1872) ∂ t F + v · ∇ x F = Q ( F , F ) , � �� � � �� � Free Transport Collisions where ∀ t ≥ 0 , ∀ x , v ∈ R 3 , F ( t , x , v ) denotes the particles distribution and Q ( F , F ) is the collision operator which takes the form Q ( F 1 , F 2 ) := Q gain ( F 1 , F 2 ) − Q loss ( F 1 , F 2 ) Mesoscopic description: statistic description aiming at describing particles behavior

Introduction Boundary conditions Known results Linear transport in-flow Existence Convex case Non convex case Free transport We suppose that the considered gas is made up of monoatomic identical particles In the absence of external forces, if the interactions between the particles are not considered, they move along straight lines with constant speed ∀ t ≥ 0 , ∀ x , v ∈ R 3 F ( t , x + vt , v ) = const = F ( 0 , x , v ) Hence, their distribution is given by ∀ t ≥ 0 , ∀ x , v ∈ R 3 ∂ t F ( t , x , v ) + v · ∇ x F ( t , x , v ) = 0

Introduction Boundary conditions Known results Linear transport in-flow Existence Convex case Non convex case Collision operator Binary collisions Instantaneous collisions Elastic collisions : conservation of momentum and kinetic energy v ′ + v ′ | v | 2 + | v ∗ | 2 = | v ′ | 2 + | v ′ ∗ | 2 v + v ∗ = ∗ � �� � � �� � pre-collisional post-collisional This is equivalent to the existence of a unitary vector ω ∈ S 2 such that � v ′ = v + [( v ∗ − v ) · ω ] ω, v ′ ∗ = v ∗ − [( v ∗ − v ) · ω ] ω � � � � ⇒ | v − v ∗ | = | v ′ − v ′ � v ′ − v ′ � v − v ∗ � � � � ∗ | and | v − v ∗ | · ω � = ∗ | · ω ∗ | v ′ − v ′ � Microreversible collisions Molecular chaos

Introduction Boundary conditions Known results Linear transport in-flow Existence Convex case Non convex case Under the previous hypotheses Boltzmann proved that the general equation becomes ∂ t F + v · ∇ x F = Q ( F , F ) , � � � � v − v ∗ � � Q loss ( F 1 , F 2 )( t , x , v ) = S 2 B ( | v − v ∗ | , | v − v ∗ | · ω � ) F 1 ( v ∗ ) F 2 ( v ) d ω d v ∗ � � � R 3 � � � � v − v ∗ � � Q gain ( F 1 , F 2 )( t , x , v ′ ) = S 2 B ( | v − v ∗ | , | v − v ∗ | · ω � ) F 1 ( v ∗ ) F 2 ( v ) d ω d v ∗ � � � R 3 Q ( F 1 , F 2 )( t , x , v ) = Q gain ( F 1 , F 2 ) − Q loss ( F 1 , F 2 ) � � � � v − v ∗ � � � � F 1 ( v ′ ∗ ) F 2 ( v ′ ) − F 1 ( v ∗ ) F 2 ( v ) = S 2 B ( | v − v ∗ | , | v − v ∗ | · ω � ) d ω d v ∗ � � � R 3

Introduction Boundary conditions Known results Linear transport in-flow Existence Convex case Non convex case In our case, the collision operator takes the form � � � � S 2 | v − v ∗ | κ q 0 ( θ ) F 1 ( v ′ ∗ ) F 2 ( v ′ ) − F 1 ( v ∗ ) F 2 ( v ) Q ( F 1 , F 2 ) = d ω d v ∗ , R 3 where θ is the deviation angle and the collision rule is � v ′ = v + [( v ∗ − v ) · ω ] ω, v ′ ∗ = v ∗ − [( v ∗ − v ) · ω ] ω Hard potential 0 ≤ κ ≤ 1 v ∗ − v Angular cutoff 0 ≤ q 0 ( θ ) ≤ C | cos θ | with cos θ = | v ∗− v | · ω

Introduction Boundary conditions Known results Linear transport in-flow Existence Convex case Non convex case Bounded domain Ω ⊂ R 3 The boundary of the phase space is (x,v) ∈ γ 0 Ω (x,v) γ := { ( x , v ) ∈ ∂ Ω × R 3 } , ∈ γ + n(x) (x,v) x ∈ γ where n = n ( x ) the outward - normal direction at x ∈ ∂ Ω We decompose γ as γ − = { ( x , v ) ∈ ∂ Ω × R 3 : n ( x ) · v < 0 } , the incoming set γ + = { ( x , v ) ∈ ∂ Ω × R 3 : n ( x ) · v > 0 } , the outcoming set γ 0 = { ( x , v ) ∈ ∂ Ω × R 3 : n ( x ) · v = 0 } , the grazing set

Introduction Boundary conditions Known results Linear transport in-flow Existence Convex case Non convex case Boundary conditions on γ − In-flow boundary condition: ∀ t ≥ 0 , ∀ ( x , v ) ∈ γ − F ( t , x , v ) = g ( t , x , v ) where g precribes the density of the incoming particles. Specular reflection boundary Bounce-back reflection boundary condition: ∀ t ≥ 0 , ∀ ( x , v ) ∈ γ − condition: ∀ t ≥ 0 , ∀ ( x , v ) ∈ γ − F ( t , x , v ) = F ( t , x , R x v ) , F ( t , x , v ) = F ( t , x , − v ) where R x v := v − 2 n ( x )( n ( x ) · v )

Introduction Boundary conditions Known results Linear transport in-flow Existence Convex case Non convex case Diffuse boundary condition: ∀ t ≥ 0 , ∀ ( x , v ) ∈ γ − � F ( t , x , v ) = c µ T µ T ( v ) F ( t , x , u ) { n ( x ) · u } d u n ( x ) · u > 0 2 π T e − | v | 2 � 1 2 T is a Where c µ T n ( x ) · u > 0 µ T ( u ) { n ( x ) · u } d u = 1 and µ T = global Maxwellian distribution with constant temperature T > 0

Introduction Boundary conditions Known results Linear transport in-flow Existence Convex case Non convex case Known results in a general bounded domain Existence and uniqueness of solutions Existence of renormalized DiPerna-Lions solutions (weak regularity) : Hamdache, Arkeryd, Cercignani, Maslova, Mischler,... Perturbative framework (stronger solutions) : Domains with a particular geometry: Ukai, Asano, Guiraud, General Domains: Guo Time-decay towards an absolute Maxwellian µ = e − | v | 2 2 Desvillettes-Villani, Villani : If F ( t ) exists in H k with uniform in t bound, k >> 1 then F ( t ) → µ with some polynomial rate Guo : F ( t ) → µ in L ∞ with e − λ t rate

Introduction Boundary conditions Known results Linear transport in-flow Existence Convex case Non convex case [Guo 2010] [Kim 2011] Existence and uniqueness of a In the case of a non convex strong global solution in a domain, for diffuse, in-flow, weighted in speed L ∞ x , v space bounce-back boundary Time-decay towards an conditions, a discontinuity absolute Maxwellian with an may appear in non convexity exponential rate points and propagates inside In the case of a strictly convex the domain through a linear domain, for general boundary trajectory conditions, C 0 x , v regularity away from γ 0 for all positive time Ω Continuity Continuity Discontinuity Ω

Introduction Boundary conditions Known results Linear transport in-flow Existence Convex case Non convex case Regularity Estimates BV, Sobolev, Hölder regularity results for the Vlasov equation in a half space with various boundary conditions [Guo 1995] Hölder regularity results for the Vlasov equation in convex domains with Specular BC [Hwang-Velazquez 2010] In the case of Boltzmann equation very rare results exist when the domain is non-trivial and in the presence of boundary conditions

Introduction Boundary conditions Known results Linear transport in-flow Existence Convex case Non convex case Perturbative framework Let µ = e − | v | 2 be a global normalized Maxwellian 2 IDEA : look for solutions of the form F = √ µ f Then f satisfies ∂ t f + v · ∇ x f = Γ gain ( f , f ) − ν ( √ µ f ) f where ν ( √ µ f )( v ) = ν ( F )( v ) := √ µ f Q loss ( √ µ f , √ µ f )( v ) 1 � � � S 2 | v − v ∗ | κ q 0 ( θ ) = µ ( v ∗ ) f ( v ∗ ) d ω d v ∗ R 3 √ µ Q gain ( √ µ f 1 , √ µ f 2 )( v ) 1 Γ gain ( f 1 , f 2 )( v ) := � � � S 2 | v − v ∗ | κ q 0 ( θ ) µ ( v ∗ ) f 1 ( v ′ ∗ ) f 2 ( v ′ ) d ω d v ∗ = R 3

Introduction Boundary conditions Known results Linear transport in-flow Existence Convex case Non convex case The corresponding boundary conditions for f are followings : In-flow boundary condition : f ( t , x , v ) = g ( t , x , v ) , on γ − � µ ( v ) Diffuse boundary condition : � � � f ( t , x , v ) = c µ µ ( v ) f ( t , x , u ) µ ( u ) { n ( x ) · u } d u , on γ − n ( x ) · u > 0 Specular reflection boundary condition : f ( t , x , v ) = f ( t , x , R x v ) , on γ − Bounce-back reflection boundary condition : f ( t , x , v ) = f ( t , x , − v ) , on γ −

Introduction Boundary conditions Known results Linear transport in-flow Existence Convex case Non convex case For the initial datum f 0 , compatibility conditions are necessary In-flow boundary compatibility condition: 1 f 0 ( x , v ) = g ( 0 , x , v ) on γ − � µ ( v ) Diffuse boundary compatibility condition: � � � f 0 ( x , v ) = c µ µ ( v ) f 0 ( x , u ) µ ( u ) { n ( x ) · u } d u , on γ − n ( x ) · u > 0 Specular reflection boundary compatibility condition: f 0 ( x , v ) = f 0 ( x , R x v ) , on γ − Bounce-back reflection boundary compatibility condition: f 0 ( x , v ) = f 0 ( x , − v ) , on γ −