Fourier Law and Non-Isothermal Boundary in the Boltzmann Theory - PowerPoint PPT Presentation

Fourier Law and Non-Isothermal Boundary in the Boltzmann Theory Joint work with Raffaele Esposito, Yan Guo, Rossana Marra Chanwoo Kim DPMMS, University of Cambridge ICERM November 8, 2011

Steady Boltzmann Equation Steady Boltzmann Equation v · ∇ x F = Q ( F , F ) F ( x , v ) : density distribution of rarefied gas 3 D velocity space v ∈ R 3 Ω : bounded, connected domain in R d for d = 1 , 2 , 3 nonlinear Boltzmann operator Q ( F 1 , F 2 ) : quadratic, bilinear non-local in v ∈ R 3 hard potential 0 ≤ γ ≤ 1 with angular cut-off � R 3 { 1 , v , | v | 2 } Q ( F , F )( v ) dv = 0 collision invariant : Knudsen number ∼ 1 regime Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Non-Isothermal Boundary and Diffusive BC Wall temperature θ ( x ) = θ 0 + δϑ ( x ) on x ∈ ∂ Ω Diffusive boundary condition on x ∈ ∂ Ω , n ( x ) · v < 0 � F ( x , v ) = µ θ ( x , v ) F ( x , u ) { n ( x ) · u } du n ( x ) · u > 0 Wall Maxwellian � � − | v | 2 1 µ θ ( x , v ) = 2 πθ ( x ) 2 exp 2 θ ( x ) � n ( x ) · v > 0 µ θ ( x , v ) { n ( x ) · v } dv = 0 with Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Purpose of This Work Analyze the thermal conduction phenomena in the kinetic regime(Knudsen number ∼ 1) Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Purpose of This Work Analyze the thermal conduction phenomena in the kinetic regime(Knudsen number ∼ 1) when the wall temperature do not oscillate too much! ⇓ Steady Boltzmann equation with Duffuse BC | θ ( x ) − θ 0 | ≪ 1 , | ϑ ( x ) | ≤ 1 and δ ≪ 1 ⇓ F s ∼ µ Regime Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Natural Questions and Previous Works Existence, Uniqueness, Non-Negativity for Steady Solution S.-H.Yu : existence and stability, Ω is slab (length ≪ 1), ARMA 2009 L.Arkeryd, A.Nouri : Ann. Fac. Sci. Toulouse, Math. 2000 : Existence in L 1 − space, Ω is slab Regularity (Continuity and Singularity) Y.Guo : for IBVP, Ω convex, continuity away from γ 0 : ARMA 2010 C.K : for IBVP, Ω non-convex, singularity formation and propagation : CMP 2011 Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Natural Questions and Previous Works Dynamical Stability L.Desvillettes, C.Villani : polynomial decay in H k for some BCs : Invent. Math. 2005 C.Villani : polynomial decay in H k , diffusive BC, θ ≡ θ 0 : Mem. AMS 2009 Y.Guo : θ ≡ θ 0 , e − λ t − decay in L ∞ to µ : ARMA 2010 S.-H.Yu : e − λ t − decay in L ∞ to the steady solution : ARMA 2009 Hydrodynamic Limit R. Esposito, Lebowitz, R.Marra : CMP 1994, J.Stat.Phys. 1995 Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Theorem : Existence, Uniqueness and Non-Negativity Let Ω ⊂ R d , d = 1 , 2 , 3 . For all M > 0 , Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Theorem : Existence, Uniqueness and Non-Negativity Let Ω ⊂ R d , d = 1 , 2 , 3 . For all M > 0 , there exists δ 0 > 0 such that for 0 < δ < δ 0 in | θ ( x ) − θ 0 | ≤ δ, on x ∈ ∂ Ω , then there exists a non-negative solution F s = M µ + √ µ f s ≥ 0 Ω × R 3 f s √ µ = 0 to the problem �� with � F s | γ − = µ θ v · ∇ x F s = Q ( F s , F s ) , F s d γ, γ + Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Theorem : Existence, Uniqueness and Non-Negativity Let Ω ⊂ R d , d = 1 , 2 , 3 . For all M > 0 , there exists δ 0 > 0 such that for 0 < δ < δ 0 in | θ ( x ) − θ 0 | ≤ δ, on x ∈ ∂ Ω , then there exists a non-negative solution F s = M µ + √ µ f s ≥ 0 Ω × R 3 f s √ µ = 0 to the problem �� with � F s | γ − = µ θ v · ∇ x F s = Q ( F s , F s ) , F s d γ, γ + 1 such that, for all 0 ≦ ζ < 4+2 δ , β > 4, ||� v � β e ζ | v | 2 f s || ∞ + |� v � β e ζ | v | 2 f s | ∞ � δ. If M µ + √ µ g s is an another solution with Ω × R 3 g s √ µ = 0 such �� that, for β > 4 ||� v � β g s || ∞ + |� v � β g s | ∞ ≪ 1 , then f s ≡ g s . Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Theorem : Continuity and Singularity If θ ( x ) is continuous on ∂ Ω then F s is continuous away from D . In particular, if Ω is convex then D = γ 0 . On the other hand, if Ω is not convex then we can construct a continuous function θ ( x ) on ∂ Ω such that the corresponding solution F s in not continuous. Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Theorem : Dynamical Stability 1 Let 0 ≦ ζ < 4+2 δ , β > 4. There exists ε 0 > 0, depends on δ 0 , and λ > 0 such that if ||� v � β e ζ | v | 2 [ f (0) − f s ] || ∞ ≤ ε 0 then there exists a unique non-negative dynamic solution F ( t ) = µ + f s √ µ + f ( t ) √ µ ≥ 0 to the dynamical problem � F ( x , v ) = µ θ ∂ t F + v · ∇ x F = Q ( F , F ) , F n · v n ( x ) · v > 0 for x ∈ ∂ Ω and n ( x ) · v < 0 such that ||� v � β e ζ | v | 2 [ f ( t ) − f s ] || ∞ � e − λ t ||� v � β e ζ | v | 2 [ f (0) − f s ] || ∞ Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Why δ − Expansion ? Fourier Law : a relation between the temperature and the heat flux q s = − κ ( θ s ) ∂ x θ s for suitable positive smooth function κ . Let F s be the solution to the steady Boltzmann equation � 1 R 3 | v − u s | 2 F s ( x , v ) dv θ s ( x ) = 3 ρ s � 1 u s ( x ) = R 3 vF s ( x , v ) dv ρ s � ρ s ( x ) = R 3 F s ( x , v ) dv � 1 R 3 ( v − u s ( x )) | v − u ( x ) | 2 F s ( x , v ) dv . q s ( x ) = 2 Purpose : See the first order characterization of F s Chanwoo Kim Fourier Law and Non-Isothermal Boundary

What is δ − Expansion ? : µ δ − Expansion Wall Temperature θ ( x ) = θ 0 + δϑ ( x ) , | ϑ ( x ) | ≤ 1 , x ∈ ∂ Ω . Wall Maxwellian � � | v | 2 1 µ δ ( x , v ) = 2 π [ θ 0 + δϑ ( x )] 2 exp − 2[ θ 0 + δϑ ( x )] Taylor Expansion in δ ( µ δ is analytic in δ ) µ δ = µ + δµ 1 + δ 2 µ 2 + · · · + δ m µ m + · · · Chanwoo Kim Fourier Law and Non-Isothermal Boundary

What is δ − Expansion ? : f s ∼ δ f 1 + δ 2 f 2 + · · · Formal Expansion : � � F s = µ + √ µ δ f 1 + δ 2 f 2 + · · · f s = δ f 1 + δ 2 f 2 + · · · Plug in v · ∇ x F s = Q ( F s , F s ) with Diffusive Bounadary Condition to get the linear equation for f i (comparing the coefficients of power of δ ) Once we solve f i , define the Remainder f δ m such that f s = δ f 1 + δ 2 f 2 + · · · + δ m f δ m Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Theorem : δ − Expansion δ − Expansion is valid ! There exist f 1 , f 2 , · · · , f m − 1 and for all i = 1 , 2 , · · · m − 1 ||� v � β e ζ | v | 2 f i || ∞ � 1 for all 0 ≦ ζ < 1 4 , β > 4 and the remainder f δ m exits and ||� v � β e ζ | v | 2 f δ m || ∞ � 1 1 for all 0 ≦ ζ < 4+2 δ , β > 4 Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Theorem : Criterion for Fourier Law Let Ω = [0 , 1]. If the Fourier Law holds for F s = µ + √ µ f s , √ µ + O ( δ 2 ) √ µ = µ + δ f 1 F s θ 0 + δθ 1 + O ( δ 2 ) θ s = then θ 1 ( x ) is a linear function on [0 , 1] Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Prediction From an available numeric simulation (Ohwada, Aoki, Sone, 1989) θ 1 is not linear! ⇓ Fourier Law is not valid at the kinetic regime ! Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Hydrodynamic Part : P f Linearized Boltzmann operator √ µ [ Q ( µ, √ µ f ) + Q ( √ µ f , µ )] = ν ( v ) f − Kf Lf = − 1 semi-positive : � Lg , g � � ||{ I − P } g || 2 ν kernel = ‘hydrodynamic part’ � √ µ � a g ( t , x ) + v · b g ( t , x ) + | v | 2 − 3 P g ≡ c g ( t , x ) 2 Boltzmann equation = ⇒ macroscopic equation for b f ∆ x b f = ∂ 2 x { I − P } f + · · · ellipticity in H k = ⇒ Guo:VMB(Invent.Math.2003), VPL(JAMS2011); Gressman-Strain:BE without angular cut-off(JAMS2011) Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Difficulties with boundary conditions � √ µ � a f ( t , x ) + v · b f ( t , x ) + | v | 2 − 3 P f ≡ c f ( t , x ) 2 P f and { I − P } f do not make sense at the boundary no boundary condition for a f , c f , only b f · n ( x ) = 0 on ∂ Ω Chanwoo Kim Fourier Law and Non-Isothermal Boundary

Mathematical Framework : L 2 − L ∞ Frame Y.Guo : Initial Boundary Value Problem of BE, ARMA 2010 L 2 Posivity : We Need A New Method ! L ∞ Bound : We Need A New Method ! Chanwoo Kim Fourier Law and Non-Isothermal Boundary

New L 2 Positivity Estimate v · ∇ x f + Lf = g , f γ − = P γ f + r �� �� � Ω × R 3 f √ µ = 0 = Ω × R 3 g √ µ = with r n · v < 0 = ⇒ || P f || ν ≤ M {|| ( I − P ) f || ν + | (1 − P γ ) f | 2 , + } + · · · � �� � Good Terms ! weak formulation(Green’s identity) + test functions constructive estimate with an explicit M dimension of Ω = 1 , 2 , 3 Chanwoo Kim Fourier Law and Non-Isothermal Boundary

New L 2 Positivity Estimate Weak formulation (Green’s identity) � �� �� �� ψ f − Ω × R 3 v · ∇ ψ f = − Ω × R 3 ψ L ( I − P ) f + Ω × R 3 ψ g γ { a f + v · b f + | v | 2 − 3 c f }√ µ + ( I − P ) f bulk f = 2 boundary f γ = P γ f + (1 − P γ ) f 1 γ + + r 1 γ − Chanwoo Kim Fourier Law and Non-Isothermal Boundary