Entropy methods in kinetic equations, parabolic equations and quantum models

Transcription

1 Entropy methods in kinetic equations, parabolic equations and quantum models Jean Dolbeault dolbeaul Ceremade, Université Paris-Dauphine Toulouse, April 17, 2011 in memory of Naoufel Ben Abdallah

2 Outline Vlasov-Poisson system with injection boundary conditions the electrostatic case (coll. Naoufel) Dynamical stability [Gravitational Vlasov-Poisson (Newton) system] Relative equilibria in gravitation []

3 Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria Entropy in non-dissipative kinetic theory Vlasov-Poisson system with injection boundary conditions the electrostatic case (coll. Naoufel) Dynamical stability Gravitational Vlasov-Poisson (Newton) system: Variational methods and functional inequalities Relative equilibria

4 Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria Vlasov-Poisson system with injection boundary conditions: the electrostatic case

5 Vlasov-Poisson system with injection boundary conditions Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria ω is a bounded domain in R d, ω is of class C 1 Ω = ω R d and Γ = Ω = Ω R d are the phase space and its boundary (incoming and outgoing boundary ) Σ ± (x) = {v R d : ± v ν(x) > 0} Γ ± = {(x, v) Γ : v Σ ± (x)} Vlasov-Poisson system with injection boundary conditions t f + v x f ( x ϕ + x ϕ 0 ) v f = 0 f t=0 = f 0, f Γ R +(x, v, t) = γ(1 2 v 2 + ϕ 0 (x)) ϕ = ρ = f dv, (x, t) ω R + R d and ϕ(x, t) = 0, (x, t) Ω R +

6 Assumptions Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria The initial condition f 0 is a nonnegative bounded function The external electrostatic potential is nonnegative and smooth The function γ is defined on (min x ω ϕ 0 (x), + ), bounded, smooth, strictly decreasing with values in (0, ), and + sup x ω 0 s d/2 γ(s + ϕ 0 (x)) ds < + γ is strictly decreasing: β = g 0 γ 1 (z) dz is strictly convex

7 Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria Stationary solution: the nonlinear Poisson equation Let U : L 1 (Ω) W 1,d/(d 1) 0 (ω): U[g] = u the unique solution in W 1,d/(d 1) 0 (ω) of u = g(x, v) dv R d M(x, v) = γ ( 1 2 v 2 + U[M](x) + ϕ 0 (x) ) is a stationary solution: v x M ( x ϕ 0 + x U[M]) v M = 0 It is the unique critical point in H0 1 (ω) of the strictly convex coercive functional U 1 2 U 2 dx G(U + ϕ 0 ) dx ω where G (u) = g(u) = R d γ ( 1 2 v 2 + u ) dv = 2 d/2 1 S d s d/2 1 γ(s + u) ds ω

8 Relative entropy Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria F[g h] := Ω (β(g) β(h) (g h)β (h)) dx dv where β is the real function defined by β(g) = Ω g 0 ω U[g h] 2 dx γ 1 (z) dz ( ) 1 f 2 v U[f ] + ϕ 0 dx dv [ = 1 2 (f M)U[f M] 1 2 M U[M] f β (M) ] dx dv F[f ] = Ω Ω ( ( ) 1 β(f ) + 2 v U[f ] + ϕ 0 f Ω ( β(m) + ) dx dv ( 1 2 v U[M] + ϕ 0 ) M ) dx dv

9 Relative entropy and irreversibility Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria Theorem Assume that f 0 L 1 L + is such that F[f 0 M] < + d dt F[f (t) M] = F+ [f (t) M] F + is the boundary relative entropy flux F + [g h] = (β(g) β(h) (g h)β (h)) dσ Γ + Proof. Integration by parts, like in [Darrozès, Guiraud] d β(f ) dx dv = β(f ) dσ dt Ω ± Γ d dt Ω f ( 1 2 v U[f ] + ϕ ) 0 dx dv = ± ± f ( ) 1 Γ 2 v 2 + ϕ 0 dσ f ( 1 Γ ± 2 v 2 + ϕ 0 (x) ) dσ = f β (M) dσ Γ ±

10 The large time limit Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria F[f (t) M] + t 0 F+ [f (s) M]ds F[f 0 M] (f n (x, v, t), ϕ n (x, t)) = (f (x, v, t + t n ), ϕ(x, t + t n )) lim F + [f n (s) M] ds = 0 n + R + sup F[f n (t) M] C. t>0 + Dunford-Pettis criterion: (f n, ϕ n ) (f, ϕ ) weakly in L 1 loc(dt, L 1 (Ω)) L 1 loc (dt, H1 0 (ω)) f M on Γ

11 Is f stationary? Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria f M on Γ: f M on Ω? A partial answer for d = 1. Theorem Consider a solution (f, ϕ ) such that f M on Γ on the interval ω = (0, 1). If ϕ is analytic in x with C (in time) coefficients and if ϕ 0 is analytic with d2 ϕ 0 dx 0 on ω, then (f, ϕ) is the unique stationary 2 solution, given by: f = M, ϕ = U[M] Proof. Let ϕ 0 (0) = 0, ϕ 0 (1) 0: ϕ 0 (0) 0. Characteristics X t = V, X(s; x, v, s) = x, V t = ϕ x (X, t) dϕ 0 dx (X), V(s; x, v, s) = v are defined on (T in (s; x, v), T e (s; x, v)) : either T in (s; x, v) = or (X in, V in )(s; x, v) Γ ; either T e (s; x, v) = + or (X e, V e )(s; x, v) Γ +. Step 1: the electric field is repulsive at x = 0.

12 Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria Step 2 : Analysis of the characteristics in a neighborhood of (0, 0, t). f (X in, V in, T in ) = γ ( 1 2 V in 2) f (X e, V e, T e ) = γ ( 1 2 V e 2) The function γ is strictly decreasing: V in (t 0, x 0, 0) = V e (t 0, x 0, 0) Characteristics are parametrized by Let e ± (X) = 1 2 V 2 (X) dt ± dx = 1 V, dv dx = 1 Φ V x (X, t± ) X t ± (X) = t 0 x 0 dy 2e± (Y ) X [0, x 0 ] Rescaling: x 0 = ε 2, de ± dx = Φ x (X, t± (X)), e ± (x 0 ) = 0 X = ε 2 (1 x) and e ± (X) := ε2 2 eε ± (x)

13 Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria ( de ε ± dx = 2 Φ x ε 2 (1 x), t 0 ± ε x 0 ) dy e ε ± (y) with the condition e ε +(1) = e ε (1) for any ε > 0 small enough Lemma + e± ε = ε n e n ± n=0 With the above notations, for all n N, we have the following identities: (i) de± 2n 2(1 x)n (x) = dx n! (ii) de± 2n+1 (x) = dx n+1 ±2(1 x)n (n + 1)! x Φ(0, t 0 ) ( x t x n+1 Φ(0, t 0 ), (iii) t x n+1 Φ(0, t 0 ) = 0 0 dy e0 (y) )

14 Stability Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria [ M. J. Cáceres, J. A. Carrillo, and J. Dolbeault. Nonlinear stability in Lp for a confined system of charged particles. SIAM J. Math. Anal., 34 : , 2002 ] Theorem Let ϕ 0 be s.t. (x, s) s 3/2 1 γ(s + ϕ 0 (x)) L 1 L (R 3, L 1 (R)). If f is a weak solution of Vlasov-Poisson with f 0 in L 1 L p0, p 0 = ( )/11, s.t. β(f 0 ), ( ϕ 0 + v 2 )f 0 L 1 (R 3 ) and if for some p [1, 2], inf s (0,+ ) β (s)/s p 2 > 0, then f f 2 L p C 2 (ϕ 0 ϕ ) 2 dx R 3 + C [β(f 0 ) β(f ) β (f )(f 0 f )]dx dv R 6 where ( f (x, v) = γ( 1 2 v 2 + ϕ 0 (x) + ϕ (x)), ϕ ). Here γ 1 = β

15 Stability Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria Value of p 0 [Hörst and Hunze] (weak solutions). Renormalization [DiPerna and Lions, Mischler]. [Braasch, Rein and Vukadinović, 1998]. Csiszár-Kullback inequality: [AMTU, 2000] Theorem p 0 = 2 and β(s) = s log s s. There exists a convex functional F reaching its minimum at f (x, v) = e v 2 /2 (2π) 3/2 ρ (x) s.t. f (t, ) f 2 L 2 F[f 0] To be compared with the standard Csiszár-Kullback results! de la Vallée - Poussin type of approach Here p = 1, γ(s) = e s e (ϕ +ϕ0) ϕ = ρ = f 0 L 1 e (ϕ +ϕ 0) dx

16 Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria Gravitational Vlasov-Poisson (Newton) system: Variational methods and functional inequalities

17 Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria Gravitational Vlasov-Poisson (Newton) system: Consider solutions of the gravitational Vlasov-Poisson system t f + v x f x ϕ v f = 0 ϕ = 1 4π ρ, ρ = R 3 f dv By minimizing the free energy functional F[f ] = β(f ) dx dv + 1 v 2 f dx dv 1 ϕ 2 dx R 3 R 2 3 R 3 R 2 3 R 3 stationary solutions can be found, s.t. f (x, v) = γ ( λ v 2 + ϕ(x) ) where γ = (β ) 1. With g(s) := R 3 γ ( s v 2) dv, the potential is given by the nonlinear Poisson equation: ϕ = g ( λ + ϕ(x) )

18 Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria Variational methods and functional inequalities and dynamical stability Use concentration-compactness methods (for instance) + a priori estimates and minimize F[f ] under a mass constraint R 3 f dv = M Power law (polytropic cases) case: β(f ) = 1 q κq 1 q f q, q (9/7, ) Minimization of F is exactly equivalent to build explicit interpolation inequalities of Hardy-Littlewood-Sobolev type, with optimal constants explicitly related to inf F Stability is a consequence of the conservation of the free energy (resp. free energy bound) for classical solutions (resp. weak solutions) of the Vlasov-Poisson system

19 Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria Relative equilibria

20 Relative equilibria Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria [ J. Campos, M. del Pino, and J. Dolbeault. Relative equilibria in continuous stellar dynamics. Communications in Mathematical Physics, 300: , 2010 ] Consider solutions of the gravitational Vlasov-Poisson system t f + v x f x ϕ v f = 0 ϕ = 1 4π ρ, ρ = R 3 f dv which are rotating at constant angular velocity ω: x = (x, x 3 ) (e i ω t x, x 3 ) and v = (v, v 3 ) (i ω x + e i ω t v, v 3 ) t f + v x f x ϕ v f ω 2 x v f + 2 ω i v v f = 0, ϕ = 1 4π ρ, ρ = R 3 f dv, A relative equilibrium of (2) is a stationary solution of (2) and can be obtained as a critical point of the free energy functional F[f ] = β(f ) dx dv + 1 ( v 2 ω 2 x 2) f dx dv 1 ϕ 2 dx R 3 R 2 3 R 3 R 2 3 R 3

21 Critical points Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria If ω 0, F is not bounded from below The polytropic gas model: β(f ) = 1 q κq 1 q f q for some q (9/7, ) f (x, v) = γ ( λ v 2 + ϕ(x) 1 2 ω2 x 2) where γ(s) = κ 1 q ( s) 1/(q 1) +. Nonlinear Poisson equation: ϕ = g ( λ + ϕ(x) 1 2 ω2 x 2) if x supp(ρ) with g(µ) = ( µ) p + and p = 1 q With u = ϕ, u = N i=1 ρ ω i in R 3, ρ ω i = ( u λ i ω2 x 2) p + χ i under the asymptotic boundary condition lim x u(x) = 0 m i = ρ ω i dx and ξi ω = 1 x ρ ω i dx R m 3 i R 3

22 A result Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria Theorem Let N 2 and p (3/2, 3) (3, 5). For almost any m i, i = 1,...N, and for any sufficiently small ω > 0, there exist at least [2 N 1 (N 2) + 1] (N 2)! distinct stationary solutions f ω of (2) s.t. R 3 f ω dv = N i=1 ρω i + o(1) as ω 0 + where o(1) means that the remainder term uniformly converges to 0 i = 1,...N, ρ ω i (x ξi ω ) = λ p ( (p 1)/2 i ρ λ i x ) + o(1) where ρ is non-negative, radially symmetric, non-increasing, compactly supported function. m i = λ (3 p)/2 i R ρ 3 dx + o(1) and ξi ω = ω 2/3 (ζi ω, 0) are s.t. ζi ω R 2 converges to a critical point of V(ζ 1,...ζ N ) = 1 N m i m j 8π ζ i ζ j + 1 N m i ζ i 2 2 i j=1 i=1

23 Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria [J.I. Palmore]: classification of relative equilibria for the N-body problems + find critical points of J[u] = 1 2 R 3 u 2 dx 1 p + 1 N i=1 R 3 ( u λi ω2 x 2) p+1 + χ i dx by using the solution of w = (w 1) p + =: ρ in R 3 as building brick on each of the connected components. With W ξ := N i=1 w ( (p 1)/2 i, w i (x) = λ i w λ i (x ξ i ) ) and ξ = (ξ 1,... ξ N ), we want to solve the problem ϕ + N i=1 p ( W ξ λ i ω2 x 2) p 1 + χ i ϕ = E N[ϕ] with lim ϕ(x) = 0, where E = W ξ + N ( i=1 Wξ λ i + 1 x 2 ω2 x 2) p + χ i

24 Vlasov-Poisson system with injection boundary conditions Gravitational Vlasov-Poisson system and functional inequalities Relative equilibria J[W ξ ] = N i=1 λ (5 p)/2 i e ω 2/3 V(ζ 1,...ζ N ) + O(ω 4/3 ) where e = 1 2 w 2 dx 1 R 3 p+1 (w 1) R p dx and ζ i = ω 2/3 ξ i if the points ξ i are such that, for a large, fixed µ > 0, and all small ω > 0, we have ξ i < µ ω 2/3 and ξ i ξ j > µ 1 ω 2/3. To localize each K i in a neighborhood of ξ i, we impose the orthogonality conditions ϕ xj w i χ i dx = 0 i = 1, 2...N, j = 1, 2, 3 R 3 to the price of Lagrange multipliers and solve the problem by fixed point methods Since ξ J[W ξ ] is a finite dimensional function, if ξ i = (ζ i, 0) is such that (ζ 1,... ζ N ) is in a neighborhood of a non-degenerate critical point of V, we can find a critical point ϕ for which the Lagrange multipliers are all equal to zero

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26 Some references Diffusive limits [ J. Dolbeault, P. Markowich, D. Oelz, and C. Schmeiser. Non linear diffusions as limit of kinetic equations with relaxation collision kernels. Archive for Rational Mechanics and Analysis, 186(1): , 2007 ] Fast diffusion equations [ M. Bonforte, J. Dolbeault, G. Grillo, and J. L. Vázquez. Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities. Proceedings of the National Academy of Sciences, 107 : , 2010 ] Hypocoercivity [ J. Dolbeault, C. Mouhot, and C. Schmeiser. Hypocoercivity for linear kinetic equations conserving mass. Preprint ] [ J. Dolbeault, C. Mouhot, and C. Schmeiser. Hypocoercivity for kinetic equations with linear relaxation terms. Comptes Rendus Mathématique, 347 : , 2009 ]

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28 [ J. Dolbeault, P. Felmer, M. Loss, and E. Paturel. Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems. J. Funct. Anal., 238: , 2006 ] [ J. Dolbeault, P. Felmer, and J. Mayorga-Zambrano. Compactness properties for trace-class operators and applications to quantum mechanics. Monatshefte für Mathematik, 155: 43-66, 2008 ] [ J. Dolbeault, P. Felmer, and M. Lewin. Orbitally stable states in generalized Hartree-Fock theory. Mathematical Models and Methods in Applied Sciences, 19: , 2009 ] [ G. L. Aki, J. Dolbeault, and C. Sparber. Thermal effects in gravitational Hartree systems. To appear in Annales Henri Poincaré ]

29 Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems

30 A little bit of functional analysis [ J. Dolbeault, P. Felmer, M. Loss, and E. Paturel. Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems. J. Funct. Anal., 238 (1): , 2006 ] Let H = + V in R d, d > 1 with lowest λ 1 (V) and consider Theorem C 1 = sup V D(R d ) C GN (γ) = inf u H 1 (R d )\{0} Let d N. For any γ > max(0, 1 d 2 ) [ ] κ2(γ) C 1 = κ 1 (γ) C GN (γ), κ1 (γ) = 2γ d λ 1 (V) γ R d V γ+ d 2 dx u d 2γ+d L 2 (R d ) ( u L 2q (R d ) d 2γ+d 2γ 2γ+d u L 2 (R d ) ) 1+ d 2γ, κ 2 (γ) = 2+ d γ

31 How to relate the two inequalities? The estimate on the first eigenvalue amounts to the inequality λ 1 (V) γ C 1 V γ+ d 2 dx R d while the second inequality is a Gagliardo-Nirenberg inequality u L 2q (R d ) 1 C GN (γ) u d 2γ+d L 2 (R d ) 2γ 2γ+d u L 2 (R d ) To relate the two inequalities, consider the free energy u u 2 dx + V u 2 dx C 1 V γ+ d 2 dx R d R d R d and optimize either on u such that u 2 L 2 (R d ) = 1 or on V

32 Lieb-Thirring inequalities Given a smooth bounded nonpositive potential V on R d, if λ 1 (V) < λ 2 (V) λ 3 (V)... λ N (V) < 0 is the finite sequence of all negative eigenvalues of H = + V then we have the Lieb-Thirring inequality N i=1 λ i (V) γ C LT (γ) V γ+ d 2 dx R 3 (3.1) For γ = 1, N i=1 λ i(v) is the complete ionization energy [...], [Laptev-Weidl] for γ 3/2 the sharp constant is semiclassical Lieb-Thirring conjecture: d = 1, 1/2 < γ < 3/2, C LT (γ) = C 1

33 Interpolation inequalities for systems Let m > 1 and consider β(ν) := c m ν m, c m := (m 1)m 1 m, m = γ m γ 1 q := 2γ + d 2γ + d 2 and K 1 := q [C LT (γ)(γ + d/2)] q 1 Corollary For any m (1, + ), K[ν, ψ] + c m i N ν m i K ρ q dx R 3 ( ) θ ( K[ν, ψ] i N ν m i ) (1 θ) L R 3 ρ q dx, θ = d 2(γ 1) + d

34 A second type of Lieb-Thirring inequality Let V be a nonnegative unbounded smooth potential on R d : the eigenvalues of H V are 0 < λ 1 (V) < λ 2 (V) λ 3 (V)... λ N (V)... Theorem For any γ > d/2, for any nonnegative V C (R d ) such that V d/2 γ L 1 (R d ), N i=1 λ i (V) γ C(γ) V d 2 γ dx R 3 C(γ) = (2π) d/2 Γ(γ d/2) Γ(γ)

35 Let f be a nonnegative function on R + such that F(s) := 0 0 e t s f (t) dt t f (t) ( 1 + t d/2) dt t and G(s) := 0 < e t s (4π t) d/2 f (t) dt t Theorem Let V be in L 1 loc (Rd ) and bounded from below. If G(V) L 1 (R d ), then F(λ i (V)) = tr [F ( + V)] G(V(x))dx i N R d If F(s) = s γ, then G(s) = Γ(γ d 2 ) (4π) d 2 Γ(γ) s d 2 γ

36 The exponential case If F(s) = e s, then f (s) = δ(s 1) and G(s) = (4π) d/2 e s Corollary i N e λi(v) 1 (4π) d/2 R d e V(x) dx

37 Interpolation: Gagliardo-Nirenberg inequalities for systems β(ν i ) + i N i N ν i ( ψi 2 + V ψ i 2) dx + G(V(x)) dx 0 R 3 R 3 for any sequence of nonnegative occupation numbers (ν i ) i N and any sequence (ψ i ) i N of orthonormal L 2 (R d ) functions Method: For fixed ν = (ν i ) i N, ψ = (ψ i ) i N K[ν, ψ] := ν i ψ i 2 dx and ρ := ν i ψ i 2 R d i N i N H(s) := [ G (G ) 1 ( s) + s (G ) 1 ( s) ] Assume that G is invertible and optimize on V: the optimal potential V has to satisfy G (V) + ρ = 0 V ρ dx + R 3 G(V(x)) dx = R 3 H(ρ(x)) dx R 3

38 J. Dolbeault Entropy( in kinetic, parabolic and quantum ) theory Theorem with ρ = K[ν, ψ] + β(ν i ) H(ρ) dx i N R 3 ν i ψ i 2 i N Here (ν i ) i N is any nonnegative sequence of occupation numbers and (ψ i ) i N is any sequence of orthonormal L 2 (R d ) functions If β(ν) := ν log ν ν, then β (ν) = log ν = λ, F(s) = e s, G(s) = (4π) d/2 e s and H(ρ) = ρ log ρ, and we get a logarithmic Sobolev inequality for systems Corollary K[ν, ψ] + ν i log ν i ρ log ρ dx + d i N R 2 log(4π) ρ dx 3 R 3

39 Stability for the linear Schrödinger equation E[ψ] := R d ( ψ 2 + V ψ 2 )dx, eigenvalues of H := + V λ i (V) := inf F L 2 (R d ) dim(f) = i sup E[ψ] ψ F The eigenfunction ψ i form an orthonormal sequence: ( ψ i, ψ j ) L 2 (R d ) = δ ij i, j N Free energy of the mixed state (ν, ψ) = ((ν i ) i N, (ψ i ) i N ): F[ν, ψ] := β(ν i ) + ν i E[ψ i ] i N i N Assumption (H) holds if β is a strictly convex function, β(0) = 0, β( ν i ) < and ν i λ i (V) < i N i N where ν i := (β ) 1 ( λ i (V)) for any i N

40 A discrete Csiszár-Kullback inequality Lemma Under Assumption (H), if ψ = (ψ i ) i N is an orthonormal sequence, F n [ν, ψ] F n [ ν, ψ] = ( ) n i=1 β(ν i ) β( ν i ) β ( ν i )(ν i ν i ) + ( n i=1 ν i E[ψ i ] E[ ψ ) i ] Corollary Assume that inf s>0 β (s)s 2 p =: α > 0, p [1, 2]. If i N β(ν i) and i N ν i β ( ν i ) are absolutely convergent, then (ν i ν i ) i N l p and ( ) β(ν i ) β( ν i ) β ( ν i )(ν i ν i ) α ν { ν 2 l p min ν p 2 2 2/p l, p ν p 2 i N l p }

41 Stability for the Hartree-Fock system with temperature

42 Orbitally stable states in generalized Hartree-Fock theory: the repulsive case J. Dolbeault, P. Felmer, and M. Lewin. Orbitally stable states in generalized Hartree-Fock theory. Mathematical Models and Methods in Applied Sciences, 19 (3): , P.A. Markowich, G. Rein, G. Wolansky. Existence and Nonlinear Stability of Stationary States of the Schrödinger-Poisson System. Journal of Statistical Physics, Vol. 106, Nos. 5/6, March 2002

43 We consider free energy functionals of the form γ E HF (γ) T S(γ) where E HF is the Hartree-Fock energy and the entropy takes the form S(γ) := tr ( β(γ) ) for some convex function β on [0, 1]. Has the free energy a minimizer? The Hartree-Fock energy E HF (γ) is tr ( ( )γ ) ρ γ (x) Z dx + 1 R x 2 D(ρ γ, ρ γ ) 1 R3 R3 γ(x, y) 2 dx dy 2 x y 3 with D(f, g) := f (x) g(y) R 3 R 3 x y dx dy (direct term of the interaction). The last term is the exchange term Evolution is described by the von Neumann equation i dγ dt = [H γ, γ] Here H γ is a self-adjoint operator depending on γ but not on β. Orbital stability of the solution obtained by minimization?

44 Assumptions on the entropy term (A1) β is a strictly convex C 1 function on (0, 1) (A2) β(0) = 0 and β 0 on [0, 1] Fermions!... we introduce a modified Legendre transform of β g(λ) := argmin(λν + β(ν)) 0 ν 1 { g(λ) = sup inf { (β ) 1 ( λ), 1 } }, 0 Notice that g is a nonincreasing function with 0 g 1. Also define β (λ) := λg(λ) + (β g)(λ) (A3) β is a nonnegative C 1 function on [0, 1) and β (0) = 0 (no loss of generality) (A4) j 2 β ( Z 2 /(4 T j 2 ) ) < j 1 The ground state free energy is finite (the eigenvalues of Z/ x are Z 2 /(4 j 2 ), with multiplicity j 2 )

45 Minimization of the free energy Theorem (Minimization for the generalized HF model) Assume that β satisfies (A1) (A4) for some T > 0. 1 For every q 0, the following statements are equivalent: (i) all minimizing sequences (γ n ) n N for I β Z (q) are precompact in K (ii) I β Z (q) < I β Z (q ) for all q, q such that 0 q < q 2 Any minimizer γ of I β Z (q) satisfies the self-consistent equation γ = g ( (H γ µ)/t ), H γ = Z x + ρ γ 1 γ(x, y) x y for some µ 0 3 The minimization problem I β Z (q) has no minimizer if q 2 Z Problem I β Z always has a minimizer γ. It satisfies the self-consistent equation γ = g(h γ /T)

46 The Hartree model of boson stars with temperature

47 Non-zero temperature solutions of the gravitational Hartree system [ Gonca L. Aki, Jean Dolbeault and Christof Sparber Thermal effects in gravitational Hartree systems To appear in Annales Henri Poincaré ]

48 The free energy is well-defined and bounded from below Potential energy term: By the Hardy-Littlewood-Sobolev inequality and Sobolev s embedding E pot [ρ] = 1 n ρ (x)n ρ (y) dx dy C n ρ 3/2 2 R 3 R x y L tr( ρ) 1/2 1 3 This yields E H [ρ] tr( ρ) CM 3/2 tr( ρ) 1/2 1 4 C2 M 3 Entropy term: S[ρ] = trβ(ρ) β is convex and β(0) = 0 = 0 β(ρ) β(m)ρ for all ρ H Then β(ρ) S 1, provided ρ S 1 The free energy F T [ρ] := F T [ρ] = tr( ρ) 1 2 tr(v ρρ) + T tr β(ρ) is well defined

49 Sub-additivity of i M,T w.r.t. M (i) As a function of M, i M,T is sub-additive. In addition, for any M > 0, m (0, M) and T > 0, we have i M,T i M m,t + i m,t Consider two states as almost minimizers ρ H M m and σ H m ρ = JX λ j ϕ j ϕ j, σ = j=1 JX λ j ϕ j ϕ j with smooth eigenfunctions (ϕ j) J j=1 having compact support in a ball B(0, R). Next translate one of the states so that they have disjoint supports, observe that ρ + σ H M (ii) The function i M,T is a decreasing function of M and an increasing function of T j=1

50 Maximal temperature T Let T (M) := sup{t > 0 : i M,T < 0}. (iii) For any M > 0, T (M) > 0 is positive, maybe even infinite. As a function of M it is increasing and satisfies T (M) If T < T, we have i M,T < 0 max 0 m M m 3 β(m) i 1,0 i M,0 = M 3 i 1,0 by the homogeneity of zero-temp. minimal energy By sub-additivity i M,T n i M/n,T nβ ( ) M n T M 3 n i 2 1,0 (iv) As a consequence, T (M) = + for any M > 0, if β(s) lim s 0 + s 3 = 0

51 Euler-Lagrange equations and Lagrange multiplier Theorem (Euler-Lagrange equations) Let M > 0, T (0, T (M)] and assume that (β1) (β2) hold. Consider a density matrix operator ρ H M which minimizes F T. Then ρ satisfies the self-consistent equation ρ = (β ) 1( 1 T (µ H ρ) ) where µ 0 denotes the Lagrange multiplier associated to the mass constraint. Explicitly, it is given by µ = 1 M (tr(h ρ + T β (ρ))ρ)

52 Existence of minimizers below T Theorem (Existence of minimizers) Assume that (β1) (β3) hold. Let M > 0 and consider T = T (M) as the maximal temperature. For all T < T, there exists an operator ρ in H M such that F T [ρ] = i M,T Moreover, every minimizing sequence (ρ n ) n N for i M,T is relatively compact in H up to translations The proof relies on the concentration-compactness method once it is known that i M,T < 0: Vanishing: can be ruled out by the fact that n ρ L 7/5 Dichotomy: splitting behaviour: i M,T = i M (1),T + i M M (1),T contradicts the binding inequality i M (1) +M (2),T < i M (1),T + i M (2),T Compactness

53 Orbital stability A direct consequence of this variational approach is orbital stability: Consider the set of minimizers M M H M and denote dist MM (ρ(t), ρ) = inf ρ M M ρ(t) ρ Here ρ(t) solves the corresponding time-dependent system i d dt ρ(t) = [H ρ(t), ρ(t)], ρ(0) = ρ in where H ρ := n ρ 1 Corollary (Orbital stability) For given M > 0 let T < T (M). Then for any ε > 0, there exists δ > 0 such that, for all ρ in H M and ρ M M with dist MM (ρ in, ρ) δ it holds: sup t R + dist MM (ρ(t), ρ) ε

54 Pure states, mixed states and critical temperature Let ρ 0 = M ψ 0 ψ 0 be the (appropriately scaled) minimizer for T = 0. Then the corresponding Hamiltonian operator H 0 := ψ admits countably many (negative) eigenvalues 1 Claim: (µ 0 j ) j N with µ 0 j ր 0 A critical temperature T c (0, T ) exists, and depends on the entropy function β such that, for T < T c minimizers ρ M M are only pure states

55 Positivity of the critical temperature for all M > 0 T c (M) := max{t > 0 : i M,T = i M,0 + τβ(m) τ (0, T]} Assume that (β1) (β3) hold. Then T c (M) is positive for any M > 0 To see this, take T n 0 and consider a sequence of minimizers ρ (n) Since ρ (n) is also a minimizing sequence for F T=0, we know µ (n) j n µ 0 j 0 We assume by contradiction that lim inf n λ (n) 1 = ǫ > 0 Then the Euler-Lagrange equation implies µ (n) > µ (n) 1, yields a contradiction: M = λ (0) 0 lim n λ(n) 0 lim n (β ) 1 ( (µ 0 1 µ (n) 0 )/T n ) = + Hence [0, T c ] with T c > 0 s.t. µ (n) < µ (n) 1 for any T n [0, T c ]. Thus ρ (n) is of rank one

56 Characterization of the critical temperature T c Corollary Assume that (β1) (β3) hold. There is a pure state minimizer of mass M if and only if T [0, T c ] For any M > 0 the critical temperature satisfies T c = µ0 1 µ0 0 β (M) where µ 0 0 < µ0 1 are the two lowest eigenvalues of H 0 Step 1: Prove T c (µ 0 1 µ0 0 )/β (M) by using µ(t) = µ Tβ (M) for pure states (T T c ) Step 2: (T > T c ) Prove the equality (approaching to T c from above) by using Mµ (n) = ( ) λ (n) j µ (n) j + T (n) β (λ (n) j ) j N

57 Remarks on the maximal temperature A case in which T = : T (M) = + for any M > 0, if β(s) lim s 0 + s 3 = 0 A case in which T is finite: If p (1, 7/5) given in the entropy generating function β(s) = s p, then the maximal temperature, T (M) is finite Limit case: Assume T < +. Then, lim T T i M,T = 0 and lim T T µ(t) = 0 Open: If p (7/5, 3) then T is finite?

58 Thank you for your attention!