Semigroup factorization and relaxation rates of kinetic equations

Transcription

1 Semigroup factorization and relaxation rates of kinetic equations Clément Mouhot, University of Cambridge Analysis and Partial Differential Equations seminar University of Sussex 24th of february Joint Gualdani, Mischler arxiv (H-theorem and Boltzmann eqn) - Joint Mischler in progress (Fokker-Planck dynamics) - Baranger-CM 05, CM 06, CM-Neumann 06, CM-Strain 07

2 Plan Introduction The factorization method Relaxation rates for Fokker-Planck dynamics

3 The Boltzmann equation (Maxwell 1867, Boltzmann 1872) t f }{{} + v x f }{{} = Q(f, f ) }{{} time change space change collision operator on f (t, x, v) 0 Transport term v x : straight line along velocity v Collision operator Q(f, f ): bilinear, acting on v only, integral Q(f, f )(v) = v collisions [ f (v )f (v ) ] f (v)f (v }{{} ) B }{{} (v,v ) (v,v ) (v,v )... Balance-sheet of particles with velocity v due to collisions Here hard spheres: B = (v v, ω)

4 Structure of the Boltzmann equation (I) Q(f, f ) bilinear integral operator acting on v only (local in t and x), representing interactions between particles: Q(f, f )(v) := [f (v )f (v ) f (v)f (v v R 3 ω S 2 }{{} ) ] B(v v }{{}, ω) dω dv }{{} appearing disappearing collision kernel ( 0) Velocity collision rule ((d 1) free parameters ω): v := v (v v, ω)ω, v := v + (v v, ω)ω One has (microscopic conservation laws) v + v = v + v, v 2 + v 2 = v 2 + v 2

5 Structure of the Boltzmann equation (II) For ω S d 1 the map (v, v ) (v, v ) has Jacobian 1 We deduce for a test function ϕ(v) Q(f, f )ϕ(v) dv R d = 1 [f f 4 ff ]B(v v, ω)(ϕ+ϕ ϕ ϕ ) dω dv dv R 2d S d 1 Choosing correctly ϕ we deduce 1 Q(f, f ) v dv = 0 R d v 2 This implies formally d f dt R 2d 1 v dx dv = 0 v 2

6 Structure of the Boltzmann equation (III) Choosing ϕ = log f we obtain the H-theorem d dt H(f ) = d f log f dx dv = D(f ) 0 dt R 2d The entropy production is D(f ) = Q(f, f ) log f dx dv R 2d = [f f ff ] log f f B(v v, ω) dx dv 0 R 2d S d 1 ff Cancellation at ff f f : Maxwellian local equilibrium ρ M f = /2T (2πT ) d/2 e v u 2, ρ 0, u R d, T > 0 Time-irreversible equation and mathematical basis for studying relaxation to equilibrium (2-d law of thermodynamic)

7 Exponential H-theorem and Cercignani s conjecture (I) Quantify H-Theorem for the Boltzmann equation Old question... Truesdell and Muncaster 1980: Much effort has been spent toward proof that place-dependent solutions exist for all time. [... ] The main problem is really to discover and specify the circumstances that give rise to solutions which persist forever. Only after having done that can we expect to construct proofs that such solutions exist, are unique, and are regular. Mathematically this amounts to prove for a priori smooth solutions H(f t M) = f t log f t t + dx dv 0 T d R d M with the correct timescale (hence requires constructive proofs)

8 Exponential H-theorem and Cercignani s conjecture (II) Cercignani s conjecture 1982 The present contribution is intended as a step toward the solution of the first main problem of kinetic theory, as defined by Truesdell and Muncaster, i.e. to discover and specify the circumstances that give rise to solutions which persist forever. Linearized semigroup (Hilbert, Grad, Ukai... ) suggests exponential rate Conjecture: Linear inequality on the entropy production dh(f M) D(f ) = λh(f M), λ > 0 dt Kind of nonlinear spectral gap, cf. log-sobolev inequalities... False (Bobylev-Cercignani, Wennberg) for physical B

9 The three main developements (I) (I) Coercivity estimates in velocity for non-local collision operators For the linearized collision operator: Hilbert 1909: compactness collision operator Carleman 1957, Grad 1960s: spectral gap hard spheres Wang-Chang-Uhlenbeck 1950s, Bobylev 1970s: eigenpairs for Maxwell molecules Baranger-CM 2005: bounds spectral gap hard spheres CM 2006: bounds coercivity for cutoff interactions CM-Strain 2007: bounds spectral gap non-cutoff At nonlinear level (Cercignani s conjecture): Carlen-Carvhalo, Toscani-Villani: D(f ) λ ε H(f M) 1+ε with smoothness and moments... however polynomial time-scales

10 The three main developements (II) (II) Hypocoercivity Mixing of conservative (skew-symmetric) transport operator with partially dissipative (symmetric) collision operator to produce relaxation towards the global equilibrium Hypoellipticity: Kolmogorov 34, Hörmander Here focus on the long-time behavior not regularity, non-local Hérau-Nier 01, Desvillettes-Villani 02: kinetic Fokker-Planck in L 2 (M 1 ) Desvillettes-Villani 05: polynomial relaxation rates for confined nonlinear Boltzmann equation, assuming regularity For linearized Boltzmann equation with periodic conditions: t h + v x h = Bh, x T 3, v R 3, CM-Neumann 06: proof-estimate spectral gap in terms of the spectral gap of B in velocity only in L 2 (M 1 )

11 The three main developements (III) (III) Extension of the functional space in the linearized study Hence one has to turn to the study of semigroup decay properties (vs. functional inequality): importance of the Cauchy theory and the natural space for it Incompatibility of functional spaces: linearized study L 2 (M 1 ) and nonlinear evolution equation L 1 (poly) First non-constructive proof of exponential decay in physical space Arkeryd-Esposito-Pulvirenti 87, Arkeryd 88: although the proof can be filled to my opinion, never been used and remained debatted... Constructive proof CM 06 in the spatially homogeneous case with sharp exponential rate: connection of entropy production estimates with new quantitative linearized estimates Complete answer for Boltz. eqn in torus (hard spheres)...

12 The key new result Cauchy theory in physical space (Gualdani-Mischler-CM) Boltzmann equation for hard spheres in the torus x T d Locally well-posed around its global gaussian equilibrium M = M(v) = Ce v 2 /2 in the space L 1 v L x (1 + v k ) for k > 2. Well-posed for weakly inhomogeneous initial data, in the sense: f in is close in L 1 v L x (1 + v k ) to g in = g in (v), where the smallness condition depends on g in L 1 4. Constructive rate of convergence f t M in L 1 v L x (1 + v k ) given by Ce λt where λ optimal if k large enough Remarks: - Variants with derivatives Wv σ,1 W s, with 0 σ s, s > 6/p - Variants with other Lebesgue spaces... - Variants with stretched exponential weights... - Spectral decomposition of linearized semigroup in these spaces

13 Plan Introduction The factorization method Relaxation rates for Fokker-Planck dynamics

14 (Hypo)dissipative and (hypo)coercive operators (I) Λ a dissipative in X = (X,. X ) if f D(Λ), φ F (f ), Re (Λ a) f, φ 0 with φ F (f ) f, φ = f 2 X = φ 2 X Λ a hypodissipative in X if Λ a dissipative in (X, X ) for an equivalent norm X X In the case of a Hilbert space structure, this coincides with the notion of coercivity and hypocoercivity Notion of (maximal) m-(hypo)dissipativity and m-(hypo)coercivity if furthermore Range(Λ a) = X

15 (Hypo)dissipative and (hypo)coercive operators (II) Lumer-Phillips Theorem The m-dissipativity implies e tλ B(X ) e at Hille-Yosida Theorem The m-hypodissipativity implies e tλ B(X ) C a e at, C a 1 Add to these definition finite number of discrete eigenvalues e tλ (1 Π Λ,a ) B(X ) C a e at, C a 1 or { Λ a is m-hypodissipative on invariant set X0 X 0 closed, codimx 0 < Stronger notions of self-adjoint and of sectorial operators: R(x + iy) C x + iy a 1 for y = ±µ(x a), x a, for some µ (0, + )

16 Spectral mapping theorem General problem in semigroup theory Prove that Σ(e tl ) = e Σ(tL) When true, spectral localization implies semigroup decay In general hard problem for non-self-adjoint operators Hörmander operators type I: X 0 X 0 + X 1 X 1 self-adjoint, already hard problem for regularity and semigroup decay Hörmander operators type II: X 0 + X 1 X 1 non self-adjoint but still symmetry structure, semigroup decay solved recently Hérau-Nier, Villani 2000s Here non-self-adjoint and non-symmetric situations, in Banach spaces, with possibly non-diffusive operators Idea: use theories in a small space E in order to get results in a larger Banach space E

17 Assumptions in the small space E Abstract method that can be used for different equations Notation: half complex plane a := {z C; Re z > a} 1. Localization of the spectrum Σ(L) a = {ξ 1,..., ξ k } ξ 1,..., ξ k := discrete eigenvalues Π L,ξj := eigenspace projector (finite dimensional eigenspace) 2. Growth estimate on the semigroup e tl Π L,a := Π L,ξ1 + + Π L,ξk e tl (1 Π L,a ) B(E) C a e a t

18 Idea of the result E E Banach spaces, L generator of C 0 -semigroup s.t. L E = L If L decomposes as L = A + B with A : E E bounded ( regularizing term) B a is dissipative (coercive term spectral localization) Then L inherits the spectral properties of L Σ(L) a = {ξ 1,..., ξ k }, with ξ j discrete eigenvalue Π L,ξj E = Π L,ξj = spectral projector And e t L inherits the growth estimate of e tl t 0, a > a, e tl e tl Π L,a E E C a e a t Remark: Quantitative partial spectral mapping theorem in E Σ(e L t ) e a t = e Σ(L) t e a t

19 Main difficulties and strategy (I) Difficulties L and L may be non symmetric E, E may not have a Hilbert space structure we want constructive estimates Factorization of resolvents for decomposing the semigroups and relating their decays Semigroup S L (t) Resolvent (L ξ) 1 factorization Semigroup S L (t) Resolvent (L ξ) 1

20 Main difficulties and strategy (II) The horizontal arrows: Quantitative dictionary between semigroup decay estimates and resolvent estimates homogeneous case: complex integration for sectorial operators inhomogeneous case: inversion of Laplace transforms = complex integration on vertical lines in C The vertical arrow (factorization): L = A + B smooth + well known RL = R B R L A R B or more generally R L = R B R B (A R B ) + + ( 1) n R L (A R B ) n Additional difficulty of the discrete spectrum...

21 A model statement E E Banach spaces, L, L generators s.t. L E = L with for a < 0: (H1) L is coercive: known (i) Σ(L) a = Σ d (L) a = {0} (localization of the spectrum) (ii) L a is dissipative on Range(I Π L,0 ) (H2) Decomposition of L: A, B s.t. L = A + B and (i) B a is hypodissipative : e Bt (t) B(E) C a e a t (ii) A B(E) (iii) T n := (A S B ) ( n) satisfies T n (t) B(E,E) C a e a t for n N Theorem (i) Σ(L) a = Σ d (L) a = {0}, Π L,0 E = Π L,0 (ii) a > a, C a > 0 s.t. t 0, e tl Π L,0 B(E) C a e a t

22 Proof of the theorem - Step 1: right inverse of L ξ Define Ω := a \{0} (simpler example) n 1 U(ξ) := ( 1) k R B (ξ) (A R }{{} B (ξ)) k + ( 1) n R L (ξ) (A R }{{} B (ξ)) k=0 (B ξ) 1 (L ξ) 1 For ξ Ω: - (AR B (ξ)) n bounded from E to E - R B (ξ) and A bounded in E - R L (ξ) bounded in E hence U(ξ) bounded in E and right-inverse of (L ξ) (L ξ) U(ξ) = n 1 ( 1) k (A + (B ξ)) R B (ξ) (A R B (ξ)) k k=0 +( 1) n (L ξ) R L (ξ) (A R B (ξ)) n = Id E (telescopic cancellations) n

23 Step 2: L ξ is one-to-one on Ω L generates a semigroup ξ 0 a s.t. L ξ 0 is invertible R L (ξ 0 ) exists and bounded by some C R R L (ξ) exists on B(ξ 0, 1/C R ) R L (ξ) = U(ξ) on B(ξ 0, 1/C R ) Then a priori bound on U(ξ) U(ξ) B(E) R B (ξ) B(E) + R L (ξ) B(E) A B(E,E) R B (ξ) B(E) C R on a \B(0, r) Conclusion by a continuation argument: Existence of R L (ξ) = U(ξ) on Ω with the bound C R

24 Step 3: the discrete spectrum (I) On a spectrum of L = poles of R L = poles of R L = {0} First inclusion clear on the algebraic eigenspaces Range(Π L,0 ) Range(Π L,0 ) Eigenspaces and eigenprojectors: write the Laurent series R L (ξ) = l 0 ξ l R l + ξ l R l, A R B (ξ) = ξ j C j l=1 l=0 j=0 with R 1 = Π L,0 and Range(R l0 ),..., Range(R 2 ) Range(R 1 )

25 Step 3: the discrete spectrum (II) Then we have the following formula for the spectral projector Π L,0 := i R L (z) dz 2π z =r 1 = R L (z) A R B (z) dz 2iπ z =r = R 1 C 0 + R 2 C R l0 C l0 1 R(Π L,0 ) algebraic eigenspace of L Remark Another proof can be done by assuming additionnally some invertibility of B ξ in E for ξ a, however it is more convenient not to have to check this in the applications

26 Step 4: The representation formula (I) Simpler case R L = R B R L A R B : We want to establish and estimate the growth of (for a > a) f 0 D(L), t 0, e tl f 0 = Π L,0 f 0 +(Id Π L,0 ) e t B f 0 +T 1 (t) f 0 1 a +im T 1 (t) := lim e zt (Id Π L,0 ) R L (z) A R B (z) dz M 2iπ a im = [(Id Π L,0 ) S L (t)] (A S B (t)) f 0 with time convolution (S 1 S 2 )(t) := t 0 S 1 (s) S 2 (t s) ds Remark: S 1 S 2 not a semigroup but has the good decay, and convolution behaves well w.r.t. the Laplace transform

27 Step 4: The representation formula (II) Factorization of the resolvent ( ) n 1 R L = R B ( 1) k (A R B ) k + ( 1) n R L (A R B ) n k=0 Higher-order factorization of the semigroup S L (t) f 0 = Π L,0 f 0 + (Id Π L,0 ) S L (t) f 0 n 1 S L (t) f 0 = Π L,0 f 0 + ( 1) k (Id Π L,0 ) S B (AS B ) k (t) f 0 k=0 +( 1) n [(Id Π L,0 )S L ] (AS B ) n (t) f 0 The RHS is bounded thanks to assumption (iii) Related to Dyson-Phillips expansion for semigroups: quantitative version, with (non-local) general decomposition L = A + B

28 Shrinkage of the functional spaces It is also possible to go from E to E with a similar method by using the left-inverse decomposition of the resolvent n 1 ( ) R L = ( 1) k (A R B ) k R B + ( 1) n (A R B ) n R L k=0 and corresponding factorization formula on the semigroups, by changing the assumption (iii) into estimates now on (iii ) T n := (S B A) ( n) satisfies T n(t) B(E,E) C a e a t for n N Useful to increase regularity or Lebesgue integrability

29 Plan Introduction The factorization method Relaxation rates for Fokker-Planck dynamics

30 The Fokker-Planck equation in L 1 t f = Lf := v ( v f + v φ f ), v R d φ C 2 φ : R d R satisfies e φ(v) dv satisfies Poincaré s inequality φ(v) v γ, γ 2 Stationary measure M(v) dv = e φ(v) dv in R d Then f t M f in L 1 ( v k ) C e λ γ,k t f 0 M f in L 1 ( v k ) for any k > 0 with λ γ,k := λ P when γ > 2 while it is given by λ γ,k := min {λ P ; 2k + 0} for the critical case γ = 2, which degenerates to zero as k goes to 0 Optimality of these L 1 spectral gaps?

31 The kinetic Fokker-Planck equation in L 1 t f = Lf := v ( v f + v φ f ) v x f, x T d, v R d φ C 2 φ : R d R satisfies e φ(v) dv satisfies Poincaré s inequality φ(v) v γ, γ 2 Stationary measure M(v) dx dv = e φ(v) dx dv in T d R d Then f t M f in L 1 ( v k ) C e λ γ,k t f 0 M f in L 1 ( v k ) for any k > 0 with λ γ,k := λ KFP when γ > 2 while it is given by λ γ,k := min {λ KFP ; 2k + 0} for the critical case γ = 2, which degenerates to zero as k goes to 0 Optimality of these L 1 spectral gaps?

32 Idea of the proof First ingredient: Poincaré s inequality in L 2 (M 1 ) in R d or hypocoercivity estimate CM-Neumann 06 in T d R d Second ingredient: decomposition L = A + B with Bf = Lf Mχ R f, A = Mχ R f, χ R (v) = 1 v R, and dissipativity estimate d Bf v k diffusive term dt } {{ } 0 + f v k (ϕ γ,k Mχ R ) using that ϕ γ,k λ γ,k < 0 at v >> 1 Third ingredient: Regularization estimates on AS B (t): ultracontractivity in R d, hypoellipticity in T d R d (following Hérau and Villani)

33 The (kinetic) Fokker-Planck equation in W 1 t f = Lf := v ( v f + v f ) v R d t f = Lf := v ( v f + v f ) v x f, x T d, v R d with φ(v) = v 2 /2 for simplicity (in general γ 2 required) Stationary measure M(v) = e φ(v) in R d or T d R d Here f probability measure: f = 1 Then W 1 (f t, M) C e λ t W 1 (f 0, M) for some rate λ > 0 and constant C > 0 explicit

34 Idea of the proof (I) Introduce the following norm ψ F := max { ψ v 1 L v, ε v ψ L v } in the case v R d with ε chosen in the proof, or } ψ F := max { ψ v 1 L, x,v x ψ L, ε x,v v ψ L x,v in the case (x, v) T d R d with ε chosen in the proof, and check that W 1 (f, g) f g F ε 1 W 1 (f, g)

35 Idea of the proof (II) Then prove dissipativity of B for the norm F by using: duality: estimate B for F use energy estimate in L p and then pass to the limit p in a uniform way use mollification of the truncation χ R (coercive) energy estimates for ψ v 1 and x ψ the estimate for the v-derivative produces zero-order terms and x-derivatives terms, controlled by the previous estimates by adjusting ε small enough Finally use regularization estimates on AS B (t) and S B (t)a

36 Remarks and open problems Used in other contexts: - Haff law and self-similar cooling process for granular gases: Mischler-CM 09 - Wigner-Fokker-Planck equation: AGGMMS 12, Stürzer-Arnold 13 - Smoluchowski equation: Cañizo-Lods Landau equation: Carrapatoso 14 Possible to study singularities and pointwise estimates on the Green s function (fluid-kinetic) for the Boltzmann equation: First step (revisit Liu-Yu s results) K.-C. Wu Useful framework for the clustering problem in granular gases: Tristani 14, work in progress w/ Mischler & Rey Whole space with potential confinement w/ Mischler Situation with continuous spectrum under study...