LARGE-TIME BEHAVIOR OF NON-SYMMETRIC FOKKER-PLANCK TYPE EQUATIONS

Transcription

1 Serials Publications LARGE-TIME BEHAVIOR OF NON-SYMMETRIC FOKKER-PLANCK TYPE EQUATIONS ANTON ARNOLD, ERIC CARLEN, AND QIANGCHANG JU Abstract. Large time asymptotics of the solutions to non-symmetric Fokker- Planck type equations are studied by extending the entropy method to this case. We present a modified Bakry-Emery criterion that yields covergence of the solution to the steady state in relative entropy with an explicit exponential rate. In parallel it also implies a logarithmic Sobolev inequality w.r.t. the steady state measure. Explicit examples illustrate that skew-symmetric perturbations in the Fokker Planck operator can help to improve the constant in such a logarithmic Sobolev inequality. 1. Introduction In this paper we consider the large-time behavior of the Cauchy problem for linear Fokker-Planck type equations advection-diffusion equations for probability densities ρx, t: ρ t = Lρ := divd ρ + ρ φ + F ], x IR n, t > 0, 1.1a ρt = 0 = ρ I L 1 +IR n, 1.1b with the confinement potential φ = φx satisfying e φ L 1 IR n, and the symmetric, locally uniformly positive definite diffusion matrix D = Dx = d ij x. Due to the divergence form we obviously have the conservation property ρx, tdx = ρ I xdx, 1. IR n IR n and without restriction of generality we shall always assume ρ I xdx = e φx dx = 1. IR n IR n Now suppose that the given vector field F x and the scalar potential φx satisfy divde φ F = 0 on IR n. 1.3 Then the unique normalized steady state of 1.1a is = e φ L 1 +IR n. Because of 1.3, L SS ρ := divdρf is the skew-symmetric part of the operator L in L IR n ; ρ 1 dx acting on ρ, and this skew-symmetric part annihilates the steady state. Hence, the steady state is independent of F. 000 Mathematics Subject Classification. Primary 35K57, 35B40, 6D10 ; Secondary 37A35, 47D07, 60J60. Key words and phrases. Entropy method, Fokker-Planck equation, large-time convergence, logarithmic Sobolev inequality, Bakry-Emery condition. 153

2 154 ANTON ARNOLD, ERIC CARLEN, AND QIANGCHANG JU In the sequel we shall assume that the data φ, D, F, and ρ I are sufficiently regular for example, φ W, loc IRn ; IR, d ij W, loc IRn ; IR, i, j = 1,..., n, and F = F i x W 1, loc IRn ; IR n such that 1.1 has a unique solution ρ C0,, L 1 +IR n, and ρ C0,, L IR n ; ρ 1 dx if ρ I L IR n ; ρ 1 dx. We remark that by a simple minimum principle ρ I x 0 implies ρx, t > 0 for all x IR n, t > 0. Simple examples of 1.1a include the symmetric Fokker-Planck equation 18, ] ρ t = div ρ + xρ, x IR n, t > 0, where φx = x / + const, D = I I being the identity matrix, and F = 0. As t its solution converges with an exponential rate towards the Gaussian steady state x = π n/ e x /. An important example for a non-symmetric equation is the quantum-kinetic Wigner-Fokker-Planck equation cf. 3, 19] with the quadratic confinement potential V y = y /: w t + v y w y v w 1.4 = D pp v w + γdiv v vw + D pq div y v w + D qq y w, y, v IR n, t > can indeed be recast in the form of 1.1a see 19] for details. It describes the evolution of the Wigner function wy, v, t with the position variable y and velocity v, and y, v plays here the role of x in 1.1a. Under simple and physically necessary assumptions on the r.h.s. of 1.4, wt also converges exponentially to the unique steady state w. Other examples of non-symmetric Fokker-Planck equations appear in the modelling of polymeric fluid flows, where ρx, t, x IR n describes the distribution of polymeric chains of length and orientation given by x. In a given homogeneous sheer flow ux = F x the scaled evolution equation reads cf. 14] for details ρ t = 1 div ρ + ρ φ F x. 1.5 In this paper we are interested in the possibly exponential decay rate of ρt towards in relative entropy, i.e. eρ := ρ ln ρ dx IR n This exponential convergence is closely related to the hypercontractivity of the semigroup generated by L and to the validity of a logarithmic Sobolev inequality LSI w.r.t. the steady state measure cf. 1, 13, 4]. In the case Dx I this inequality would read, if it holds, f ln f dx IR n f dx IR n ln f dx IR n C f dx IR n 1.7 for some fixed C < and all f L IR n ; dx. Notice that φ enters the inequality through, but that F does not. The question to be addressed here

3 LARGE-TIME BEHAVIOR OF FOKKER-PLANCK TYPE EQUATIONS 155 is whether it is ever advantageous to consider a non-reversible evolution i.e., one with F 0 when attempting to establish the validity of 1.7 through the entropy method of 5, 7, 4]. Perhaps surprisingly, the answer is yes. To be more specific we shall now briefly outline this idea for the simplest case when D = I, following the preliminary note ]: Consider the symmetric part in L IR n ; ρ 1 dx of the Fokker-Planck operator in 1.1a, i.e. L S ρ := div e φ ρ e φ and assume that the potential φx is uniformly convex, i.e. it satisfies a Bakry- Emery condition BEC: A1 λ 1 > 0 such that φ = φx λ 1 I i j i,j=1,...,n x IR n in the sense of positive definite matrices. Then it is well known that ρ S t, the solution of ρ t = L S ρ converges to in relative entropy with an exponential rate of at least λ 1, and the LSI 1.7 holds with C = /λ 1 cf. 5, 7, 4]. Moreover, 1.3 implies that also ρt, the solution to the non-symmetric Fokker-Planck equation 1.1 converges to in relative entropy with rate of at least λ 1 cf. 4]. On the other hand, consider now L SS with F x 0 as a skew-symmetric perturbation of L S and assume that φ, F satisfy a generalized Bakry-Emery condition GBEC: A λ > 0 such that where F i,j = F i j φ 1 F + F λ I x IR n, denotes the Jacobian of F. As we shall show, the relative entropy of ρt then decays exponentially with rate at least λ, and the LSI 1.7 then holds with C = /λ. And in some cases the perturbation F gives rise to a better constant λ > λ 1, hence improving 1.7. The goal of this paper is threefold: to understand the large-time behavior of non-symmetric Fokker-Planck equations with applications like 1.4, 1.5, to analyze skew-symmetric perturbations L SS in order to possibly improve the LSI. And, finally, our analysis furnishes a proof of the entropy method for symmetric Fokker-Planck equations with full diffusion matrices D which was not included in 4]. This paper is organized as follows. We begin Section by introducing the class of entropies with which we work. We then proceed to calculate the second derivative of the entropy in the presence of the skew-symmetric term, and derive the generalized Bakry Emery condition. To get an estimate on the initial entropy in terms of the initial entropy production by the Bakry-Emery method, we need to know that the final entropy is zero. For this we need a theorem asserting that the entropy vanishes in the large time limit. We prove this in Theorem.5 for regular initial data. This part of the proof is somewhat technically involved, but once we have it, even for regular densities, the rest is straight-forward: We then use the

4 156 ANTON ARNOLD, ERIC CARLEN, AND QIANGCHANG JU results obtained up to this point to obtain a LSI for regular densities. Once this is extended by simple closure, the fact that the entropy vanishes in the large time limit, exponentially fast, then follows easily for general initial data. In Section 3 we discuss several examples in which the skew-symmetric term plays a crucial role in establishing the LSI.. Entropy Method for Non-symmetric Fokker-Planck Equations.1. Admissible relative entropies. As a generalization of the logarithmic entropy 1.6 we now introduce the relative entropies that we shall use in the sequel. Definition.1. Let ψ C0, C 4 0, satisfy the conditions ψ1 = ψ 1 = 0,.1a ψ > 0, on IR +,.1b ψ 1 ψ ψ IV on IR +..1c Let ρ 1, ρ L 1 +IR n with ρ 1 dx = ρ dx = 1 and ρ 1 /ρ IR + 0 ρ dx a.e. Then e ψ ρ 1 ρ := ψ ρ 1 ρ dx 0. IR n ρ is called an admissible relative entropy of ρ 1 with respect to ρ with generating function ψ. Our class of generating functions ψ coincides with those considered in 5] up to the normalizations.1a. The most typical examples of admissible relative entropies are the physical relative entropy 1.6 generated by ψ 1 σ := σ ln σ σ + 1, σ 0, and the p-entropies or Tsallis relative entropies 3] generated by ψ p σ := σp pσ p 1 + 1, σ 0; 1 < p..3 For p = we have e ψ ρ 1 ρ = ρ 1 ρ L IR n,ρ 1 dx. The well-known Csiszár-Kullback inequality 10, 16, 4, 4] shows that the relative entropies. are a measure for the distance between two normalized L 1 +IR n -functions ρ 1, ρ : 1 ρ 1 ρ L 1 IR n 1 η e ψ ρ 1 ρ,.4 with the notation η := ψ 1. We remark that for each admissible relative entropy e ψ, there exists a quadratic superentropy e ϕ such that 0 ψσ η σ 1 =: ϕσ, σ > 0, and hence e ψ ρ 1 ρ e ϕ ρ 1 ρ cf. Lemma.6 in 4].

5 LARGE-TIME BEHAVIOR OF FOKKER-PLANCK TYPE EQUATIONS Generalized Bakry-Emery condition and Ricci tensor. As in A1 and A, a Bakry-Emery condition BEC on the coefficients φ, D, F of the Fokker-Planck operator L will be our main assumption for deriving exponential decay of the relative entropy. For the subsequent analysis it is convenient to rewrite 1.1. We set which satisfies the IVP µ := ρ/, µ t = Lµ := ρ 1 div D µ + F D µ = divd µ φ F D µ, x IR n, t > 0,.5 µ I = ρ I / L 1 IR n, dx. Condition A1 is a special case of the well-known Bakry-Emery condition for logarithmic Sobolev-inequalities 5, 6, 7, 4]. In order to extend the approach of Bakry and Emery to non-symmetric Fokker-Planck equations we shall now introduce a new generalized Bakry-Emery condition GBEC. For understanding the BEC in the case of general symmetric and uniformly positive definite diffusion matrices Dx we shall need some notions from basic differential geometry see, e.g. 8], 7, 8. Therefore we consider the Riemannian manifold M= IR n ; D 1, with Dx 1 =: d ij x as covariant metric tensor. The Ricci tensor of a symmetric Fokker-Planck operator was defined in 7] cf. also 4]. Here we shall extend this definition to non-symmetric Fokker-Planck operators that involve a vector field F = F 1,..., F n. The Fokker Planck operator in.5 can be decomposed as where L = D + X, D µ := det D 1 div det D 1 D µ ] is the Laplace Beltrami operator on M cf. 9], 1. And n X := X i i i=1 is a vector field or, equivalently, a directional derivative on M, with the components n X i x := d φx ij 1 ] ln det Dx F j..6 j j=1 The Christoffel symbols are defined as the elements of the 3-tensor: Γ l ij := 1 n d kl djk + d ki d ij..7 i j k k=1 The Riemann curvature tensor of M then reads l R kij := Γ l jk n Γ l ik + Γ l i imγ m jk j m=1 n Γ l jmγ m ik.8 m=1

6 158 ANTON ARNOLD, ERIC CARLEN, AND QIANGCHANG JU and the symmetric Ricci-tensor of M is cf. 0], C.3 n k ρ ij := R ikj..9 k=1 The covariant derivative of a vector field X = X 1,..., X n is given by n i X j := Xj + Γ j ik Xk..10 i We define the symmetric covariant derivative -tensor of X as S X ij := 1 n d jl i X l + d il j X l..11 l=1 We now define the Ricci tensor of a non-symmetric Fokker-Planck operator as n Ric ij x := d ik d jl ρ kl S X kl x].1 k,l=1 with the components of X defined in.6 cf. 7, 4] for the symmetric counterpart. Our GBEC for a general symmetric, positive definite diffusion matrix now reads: A3 λ 3 > 0 such that Ricx λ 3 Dx x IR n in the sense of positive definite matrices. From the explicit form of Ric see.13 below one easily sees that A3 reduces to A for Dx I. And in the case of a scalar diffusion i.e. Dx = DxI it reads: A4 λ 3 > 0 such that 1 n 1 4 D D D + 1 D D φ F ] I + D φ F + φ F D φ F + φ F D + D λ 3 I x IR n. With tedious calculations see the Appendix, the explicit form of the GBEC reads: k=1 λ 3 > 0 such that U 1 D Tr D + 1 D D D D + D 1 F D + F D U DE φ F + φ F E DU 1 4 Tr E + D E D 1 N DD 1 φ D φ F D D] U.13 λ 3 U DU

7 LARGE-TIME BEHAVIOR OF FOKKER-PLANCK TYPE EQUATIONS 159 x IR n and any vector field U : IR n IR n. Here we used the matrix E = e j i := i d jk U k i is the first index in e j i. And N := U D is a scalar differential operator, which acts elementwise when applied to a matrix. The expression D denotes the formal matrix product between the Hessian operator and the matrix D, i.e. ij d jk..3. Exponential decay of the entropy dissipation and the relative entropy. In this section, we shall first obtain the exponential decay of the entropy dissipation. As in 4], we consider the entropy dissipation and the entropy dissipation rate I ψ ρt := d dt e ψρt.14 R ψ ρt := d dt I ψρt..15 Eq..14 is referred to as entropy equation. To facilitate the computations we rewrite 1.1a in the following form: ρ t = divd U + ρf.16 with the notation U = ρ. Differentiating the relative entropy e ψ ρt w.r.t. time gives ρ I ψ ρt = ρ t dx..17 IR n ψ By using.16 we obtain after an integration by parts ρ I ψ ρt = U DU dx + T.18 IR n ψ where T := IR n ψ ρ divdf ρ dx. In terms of 1.3, we get divdf ρ = U DF,.19 from which we have ρ T = ψ U DF dx IR n ρ ρ = ψ DF dx IR n = 0 by again using 1.3. Therefore I ψ ρt = IR n ψ due to the strict convexity of ψ and the positivity of D. ρ U DU dx 0,.0

8 160 ANTON ARNOLD, ERIC CARLEN, AND QIANGCHANG JU Now, we compute.15: ρ R ψ ρt = ψ ρ t U DU dx IR n ρ ρ ψ U DU t dx, IR n = R 1 + R,.1 where ρ R 1 := ψ ρ t U DU dx IR n and ρ R := ψ U DU t dx. IR n With.16 we get ρ R 1 = ψ divdu U DU dx IR n ρ ρ divdf ρu DU dx IR n ψ = R1 + T 3,. where ρ R 1 := ψ divdu U DU dx IR n and ρ T 3 := ψ divdf ρu DU dx. IR n Using.19 and an integration by parts lead to ρ T 3 = ψ ρ DF U DU dx IR n ρ ρ = ψ DF U DU dx IR n ρ ρ = ψ divdf U DU ]dx IR n ρ ρ = U DUDF dx. Then IR n ψ From.16 and.19, it follows that U t = 1 divdu + U DF. R = IR n ψ IR n ψ ρ U D e φ divde φ U] dx ρ U D U DF dx = R + T 4,.3

9 LARGE-TIME BEHAVIOR OF FOKKER-PLANCK TYPE EQUATIONS 161 where and ρ R := ψ U D e φ divde φ U] dx IR n ρ T 4 := ψ U D U DF dx. IR n Clearly, the computations which lead to.0 and.1 are formal. However, they can easily be justified for initial data ρ I L IR n ; ρ 1 dx L 1 IR n and for entropy generators without singularity at σ = 0 by taking into account the semigroup property of the evolution in L IR n ; ρ 1 dx, and the fact that ρ > 0 on IR n, t > 0. General admissible entropies can easily be dealt with by a local cut-off at σ = 0. Remark.. Following the approach from 4] we have to give a meaning to I ψ ρ even when ρ becomes zero which may be the case at the initial state. For positive and differentiable functions µ = µx we have ψ µ µ D µ = µ 1 µ ψ s ds D ψ s ds..4 1 Hence, we set for ρ 0 I ψ ρ := w D w dx, IR n w := h ψ ρ,.5 if w H 1 loc IRn with h ψ µ := µ 1 ψ s ds, µ > 0..6 As shown is 4], h ψ is Hölder continuous with exponent 1/ locally at µ = 0. We now return to proving the exponential decay of e ψ ρt under additional assumptions on φ, F, and D. At first we shall derive an exponential decay rate for the entropy dissipation I ψ by using the special form of the entropy dissipation rate.1. Lemma.3. Let the initial condition ρ I L IR n ; ρ 1 dx satisfy I ψ ρ I < for the admissible entropy e ψ. Assume that the coefficients φx, F x, and Dx of 1.1a satisfy the condition A3. Then the entropy dissipation converges to 0 exponentially: I ψ ρt e λ3t I ψ ρ I, t > 0..7

10 16 ANTON ARNOLD, ERIC CARLEN, AND QIANGCHANG JU Proof. An integration by parts which can be justified as mentioned above yields R 1 = ψ IV e φ ρu DU e φ dx IR n + ψ e φ ρe φ DU U IR n DUdx + ψ e φ ρe φ d U ij i U j d kl U l dx IR n k = ψ IV e φ ρu DU e φ dx IR n + ψ e φ ρe φ U D U IR n D + DU ] D U dx. Here and in the sequel we use the Einstein summation convention for double indices. Also we abbreviate i by i. Motivated by the scalar diffusion case cf. 4], we introduce S 1 := ψ e φ ρe φ DU U IR n DU + U d ij ] iu j d kl U l dx k = ψ e φ ρe φ U D U ] IR n D + D DU U dx.8 = ψ e φ ρe φ U DUDUdx IR n = ψ e φ ρdivd U DU dx IR n = ψ e φ ρdivd U DU φd U DU ]dx. IR n In a cumbersome calculation we obtain S := R S 1 = ψ e φ ρe φ U D D φ D IR n D 1 φ D D + 1 TrD D + 1 D D ] 1 4 Tr E + DED 1 N DD 1 dx + ψ e φ ρe φ Tr IR n =: T 1 + T. U D U + 1 E + 1 DED 1 1 N DD 1] dx Next we rewrite T 3 and T 4 as T 3 = ψ e φ ρe φ F D U IR n DU + F D DU ] U dx,

11 LARGE-TIME BEHAVIOR OF FOKKER-PLANCK TYPE EQUATIONS 163 and T 4 = ψ e φ ρe φ U D F IR n where we used U = U Using the fact that DU T 3 + T 4 = = IR n ψ e φ ρe φ DU + U D DU, since U is a gradient. = E + U D, we have U D F IR n ψ e φ ρe φ 1 U D ] F dx, DU + U 1 E D DEF F + F DU +U 1 F D DU U D EF + F E DU ] dx. ] dx Then 1 T 1 + T 3 + T 4 = ψ e φ ρe {U D φ Tr IR n D + 1 D D D D φ + D D 1 F D + F D 1 φ F D ] D U + 1 U DE φ F + φ F E DU 1 4 Tr E + D E D 1 N DD 1 } dx. Condition A3 or.13 leads to the estimates: All in all we have by using.8 T 1 + T 3 + T 4 λ 3 IR n ψ e φ ρe φ U DU dx. R 1 + R = R 1 + S 1 + T + T 1 + T 3 + T 4 { ψ IV e φ ρu DU + ψ e φ ρu D U D + DU D U IR n + ψ e φ ρe φ Tr D U + 1 E + 1 D E D 1 1 N DD 1] + λ 3 IR n ψ e φ ρe φ U DU dx. The first integral can be written as IR n TrXYe φ dx, } e φ dx

12 164 ANTON ARNOLD, ERIC CARLEN, AND QIANGCHANG JU where X and Y are the -matrices ψ X = e φ ρ ψ e φ ρ ψ e φ ρ ψ IV e φ ρ and, resp., Y = α 1 U D U D + DU D U 1 U D U D + DU D U U DU with α = Tr D U + 1 E + 1 D E D 1 1 N DD 1]. X is non-negative definite since ψ generates an admissible entropy cf. Definition.1. Next, we will show that Y is also non-negative definite. To this end, we introduce the symmetric matrices Z and W as follows: and Z := D U 1 1 D + D E D D E D D N D D W := D U U D. Using the cyclicity of the trace, we prove and TrW Z = TrZ W Then, it follows that α = TrZ, TrW = Tr DU UDU U D] = TrDU UDU U] = U DU, D U D + 1 E D + 1 D E 1 N D = Tr = U D U D + 1 E D + 1 D E 1 ] N D U = U D U D + 1 ] E D U = 1 U D U D + DU ] D U. Y = TrZ TrZ W TrW Z TrW., ] U U

13 LARGE-TIME BEHAVIOR OF FOKKER-PLANCK TYPE EQUATIONS 165 Since Z Z W Z W W 0 by the positivity of partial traces we include a proof in Lemma.4 below for completeness, Y is nonnegative. Thus IR n TrXYe φ dx 0 and we have for the entropy dissipation rate.1: R ψ ρt λ 3 IR n ψ e φ ρe φ U DU dx = λ 3 I ψ ρt. The assertion now follows from d dt I ψρt λ 3 I ψ ρt..9 Lemma.4. Let P = P 0, and P11 P P = 1 P 1 P, where P ij, i, j = 1, are n n matrices. Then TrP11 TrP Q := 1 0. TrP 1 TrP Proof. Let I j := I kl n, j = 1,..., n, where I 1j = I,n+j = 1; the other elements are 0. Then we have I j PI pjj p j = j,n+j 0. p n+j,j p n+j,n+j Hence, Q = n j=1 I jpi j 0. Next, we shall derive the exponential decay of the relative entropy. For this purpose, we first show the convergence of ρt to in relative entropy without a rate, for the moment. We remark that the analogous result for the symmetric Fokker-Planck equation was obtained in 4],.1 using spectral theory. Specifically, σl S IR 0 when considering L S in L IR n ; ρ 1 dx. Hence, ρt L IR n ;ρ 1 dx is monotonically decaying also for the non-symmetric Fokker- Planck equation 1.1. And we have the apriori estimate ρt L IR n ;ρ 1 dx ρ I L IR n ;ρ 1 dx, t In contrast, we shall derive it here from the decay of the entropy dissipation: Theorem.5. Let ρ I / L IR n, I ψ1 ρ I <, and let the coefficients φx, F x, and Dx satisfy condition A3. Then a e ψp ρt 0 as t for 1 p <. b If, additionally, e ψ+ε ρ I < for some ε > 0, then e ψ ρt 0.

14 166 ANTON ARNOLD, ERIC CARLEN, AND QIANGCHANG JU c Let e ψ be any admissible relative entropy, and e ϕ its quadratic superentropy with I ϕ ρ I <. Then e ψ ρt 0. Proof. First we establish this result for the logarithmic physical relative entropy eρ. Its entropy dissipation satisfies ρ Iρt = ρ D ρ dx IR n ρ ρ ρ = 4 D dx. IR n Since Dx is locally uniformly strictly positive definite, > 0 and L loc IRn, Lemma.3 implies that ρt t 0 in L Ω for any bounded domain Ω IR n. By a well-known result on Beppo-Levi spaces cf. 11], 1] p. 49, or Lemma III. of 1] we have for bounded Lipschitz domains Ω and an arbitrary sequence t k : ρk c k Ω k 0 in L Ω,.31 with the notation ρ k := ρt k and c k Ω := For any Ω fixed, we have ρk L Ω = Ω Ω ρk / dx/volω. ρ k dx CΩ ρ k dx = CΩ. IR n Thus, c k Ω is uniformly bounded with respect to k for Ω fixed. Since L Ω,.31 implies that ρk c k Ω k 0 in L Ω. Because we have ρ k c k Ω L 1 Ω ρ k c k Ω L Ω ρ k L Ω + c k Ω L Ω, ρ k c kω k 0 in L 1 Ω..3 Due to the uniform boundedness of c k Ω, there exists a subsequence still denoted by {c k Ω} such that c k Ω k cω. Now we choose the domain sequence Ω N := B N 0 IR n. And take the diagonal subsequence of all {c k Ω N } such that for any N fixed, we have c k Ω N k c N Ω N..33

15 LARGE-TIME BEHAVIOR OF FOKKER-PLANCK TYPE EQUATIONS 167 In view of.3 and.33, we obtain ρ k k c NΩ N in L 1 Ω N..34 Since > 0 in IR n, we conclude that c N Ω N = c for all N. Using.30 and the Hölder inequality we have 1/ ρ k dx ρ k L IR n ;ρ 1 dx e φ N dx 0, uniformly in k IN. Ω c N Thus Ω c N ρ k k c in L 1 IR n. Due to 1., we deduce that c = 1 and hence.35 ρ k k in L 1 IR n..36 Therefore µ k := ρ k 1 in measure in the measure space IR n, dx. The three assertions of the Lemma will now be discussed separately. Part a: In order to apply Vitali s convergence theorem we rewrite e ψp ρ k = 1 p 1 Proceeding as for.35 we obtain Ω IR n : µ p k dx ρ k p L IR n ;ρ 1 Ω µ k pl pir n;ρ dx 1 ], 1 < p <. dx 1 p/ dx..37 Ω And this yields both the uniform integrability of {µ p k } and the uniform decay of its tails. Thus, Vitali s theorem yields µ k k 1 in L p IR n ; dx, and hence e ψp ρ k 0. For the logarithmic entropy the result follows from ψ 1 σ ψ p σ, σ 0. Part b: From.0 we obtain the apriori estimate for the + ε entropy: ε µt +ε L +ε IR n ;dx 1 ] = e ψ+ε ρt e ψ+ε ρ I, t 0. Now, estimating Ω µ k dx analogously to.37 proves the assertion. Part c: Here we consider the decay of the quadratic superentropy e ϕ that satisfies 0 e ψ ρt e ϕ ρt := η ρt L IR n ;ρ 1 dx..38

16 168 ANTON ARNOLD, ERIC CARLEN, AND QIANGCHANG JU From.0 its entropy dissipation satisfies I ϕ ρt = η ρt D ρt dx. IR n A similar analysis as before yields that for any bounded Lipschitz domain Ω IR n and an arbitrary sequence t k, it holds: ρ k d k Ω k 0 in L Ω ρ k with d k Ω := Ω dx/volω. Since ρk ] dx CΩ Ω because of.30, d k Ω is also uniformly bounded with respect to k for Ω fixed. Now we can take the diagonal subsequence of all d k Ω N such that for any N fixed, we have d k Ω N k d N.39 and ρ k k d N Ω N in L Ω N..40 From the previous analysis we know that d N Ω N = c N Ω N and d N = 1 for all N. Since > 0,.40 implies ρ k 1 dx k Ω N The monotone decay of ρt L IR n ;ρ 1 dx and.30 imply that there exists a subsequence still denoted by {ρ k } with ρ k ρ in L IR n ; ρ 1 dx and ρ k L IR n ;ρ 1 dx τ then implies ρ = and the weak lower semicontinuity of the norm yields 1/ τ L IR n ;ρ 1 dx = dx = 1. Indeed, we have τ = 1, since ε > 0 : The strong convergence.41 then implies N = Nε with L Ω N ;ρ 1 dx 1 ε. ρ k L Ω N ;ρ 1 dx L Ω N ;ρ 1 dx 1 ε, and hence τ = lim ρ k k L Ω N ;ρ 1 dx 1 ε. Weak convergence of ρ k and convergence of its norms then imply 0 e ψ ρt e ϕ ρt = η ρt k 0. L IR n ;ρ 1 dx This proves the assertion.

17 LARGE-TIME BEHAVIOR OF FOKKER-PLANCK TYPE EQUATIONS 169 In the above theorem, the assumptions ρ I / L IR n and I ψ1 ρ I < are perhaps unnaturally restrictive. However, once we have proved a logarithmic Sobolev inequality for ρ I smooth with compact support, simple closure yields us the inequality in full generality, and then the conclusion of the Theorem follows immediately without this assumption. Thus, nothing is lost in making this assumption. We then obtain: Theorem.6. Let e ψ be an admissible relative entropy and e ψ ρ I <. Let the coefficients φx, F x, and Dx satisfy condition A3. Then the relative entropy converges to 0 exponentially: e ψ ρt e λ3t e ψ ρ I, t > 0..4 Moreover, the convex Sobolev inequality LSI for ψ = ψ 1 ρ ψ dx 1 ρ ρ h ψ D h ψ dx.43 IR n λ 3 IR n with h ψ from.6 holds ρ L 1 +IR n with ρdx = IR n dx. IR n.44 This inequality, of course, does not require our usual normalization ρxdx = 1. Note that L 1 +IR n in.44 can be replaced by L 1 IR n if ψ is quadratic. Proof. We proceed in two steps and first derive.4 for ρ I S := {ρ L +IR n, ρ 1 dx I ψ1 ρ + I ϕ ρ < }. From the Theorem.5c and Lemma.3 we then know that e ψ ρt 0 and I ψ ρt 0 as t. Hence, integrating.9 which also holds under condition A3 over t, gives I ψ t = d dt e ψt λ 3 e ψ t, t 0,.45 which proves the exponential entropy decay for sufficiently regular initial data. In explicit terms.45 just is the convex Sobolev inequality.43 for all sufficiently regular ρ I. Then, by simple closure cf. the proof of Corollary.18 in 4], we obtain this inequality in full generality. Once we have this, we no longer need Theroem.5 to prove e ψ ρt 0, and we obtain the full result. The desired L 1 convergence of ρt to is now a direct consequence of Theorem.6 and the Csiszár-Kullback inequality.4: Corollary.7. Let e ψ be an admissible relative entropy and e ψ ρ I <. Let the coefficients φx, F x, and Dx satisfy condition A3. Then the solution of 1.1 satisfies with the notation η = ψ 1. ρt L 1 IR n e λ3t η e ψ ρ I, t > 0,.46

18 170 ANTON ARNOLD, ERIC CARLEN, AND QIANGCHANG JU 3. Examples In this section we shall construct examples to illustrate how the non-symmetric perturbation divdρf can help to improve the constant in the LSI 1.7. For simplicity of the presentation we confine ourselves here to the case Dx I. Assume that φx is smooth on IR n and satisfies φ i φ0 = 0; x > 0, x 0 ; φ ii λi > 0 on IRn \B δ 0 for some small δ > 0 ; φ iii 0 = , φ 0 n where φ n 0 > 0. Clearly, this confinement potential φx satisfies the BEC A1 only with the convexity constant λ 1 = 0. Let = e φx be normalized on IR n. Our goal is to find a vector field F = F 1 x,..., F n x with div F = 0 such that the generalized Bakry-Emery condition GBEC holds, i.e. A λ > 0 such that φ 1 F + F λ I x IR n. More precisely, we shall construct F LipIR n with suppf L, L] n and L > 0 sufficiently small, such that F 0 = , 3.1 n 1 where the derivatives j F n 0 are yet unspecified we use the abbreviation j := j. The first principal minors of G := φ 0 ε F + F 0 are ε,..., ε n 1, and its determinant is of the form φ 0 ε n 1 + Oε n. n Then, for some ε > 0 sufficiently small, it holds: det G > 0 and φ, εf clearly satisfies the GBEC A. We remark that F could be chosen as smooth as desired, by using easy modifications of the strategy below. Now we shall construct two vector fields F and J = J 1 x,..., J n 1 x that satisfy

19 LARGE-TIME BEHAVIOR OF FOKKER-PLANCK TYPE EQUATIONS 171 F = n J 1 n J n J n 1 1 J 1 J n 1 J n 1 and hence, div F = 0. For j = 1,..., n 1 we put 0, L x n L/; n 1 F j x 1,..., x n := fx j cos xn L π cos x k π L, L/ x n L/; k=1 k j g j x 1,..., x n 1 sin x n L π, L/ x n L, with L > 0 to be chosen later. fs is a smooth function on IR with support in L, L] and it satisfies Further, which implies L L g j x 1,..., x n 1 n 1 := fx j i f±l = f ±L = f ±L = 0; ii f 0, f 0 = 1, f 0 = 0. k=1 k j cos x k π L L/ L/ cos x n L π x 1,..., x n dx n L L/ sin xn L πρ, x 1,..., x n dx n F j dx n = 0, x 1,..., x n 1 IR n 1, j = 1,..., n Next we define for j = 1,..., n 1: J j x 1,..., x n { xn := L F jx 1,..., x n 1, x n x 1,..., x n 1, x n d x n, x n L; 0, x n > L. Due to 3. we have J j LipIR n. Finally we put n 1 F n := ρ 1 k J k. k=1 In order to verify 3.1, one easily finds for j = 1,..., n 1: j F j 0 = f 0 = 1; k F j 0 = 0, k j.

20 17 ANTON ARNOLD, ERIC CARLEN, AND QIANGCHANG JU In order to analyze n F n we use φ0 = 0 in div F = 0 and obtain n 0 j F j 0 = 0. Hence j=1 n 1 n F n 0 = j F j 0 = n 1. j=1 Thus, F 0 is of form 3.1 and φ, εf satisfies the GBEC A for some small ε > 0. Appendix: Calculation of the Ricci Tensor The definition.1 of the Ricci tensor gives using the Einstein summation convention U RicxU = U i Ric ij U j = U i d ik d jl ρ kl U j U i d ik d jl S X kl U j =: W 1 + W. Using the definitions.7-.9, after a long computation, we have W 1 = U i d ik d jl R p kpl U j = U i d ik d jl p Γ p lk lγ p pk + Γp pmγ m lk Γ p lm Γm pk = 1 U id qp pq d ji U j 1 U i + 1 U i q d ji p d pq U j 1 4 U i +d qp d lk p d jl q d ik + m d jp p d im U j d ik kp d pj + d jl lp d pi U j d ik d ql p d jl k d qp d jl d kq p d ik l d qp U j + 1 U i d ik d mr d sp d jl l d sm k d rp + d qp d lk p d jl q d ik d ik d jl l d qp k d qp dik d rp m d rp k d jm djl m d rp l d im 1 4 dsm d rp m d rp s d ji U j 1 = U Tr D D + 1 ] D D D D U 1 4 Tr E + D E D 1 N DD D lndet D 4 U D 1 4 U ] lndet D D + DU, D lndet D U

21 LARGE-TIME BEHAVIOR OF FOKKER-PLANCK TYPE EQUATIONS 173 where we have used formulas such as d qp k d qp = k lndet D, d ql k d lm = d lm k d ql. Next we compute W, which involves φx and F x. We use.10 and.11 to obtain W = 1 U id ik d lm k X m + d km l X m d jl U j = 1 U id ik d lm k X m + Γ m kpx p + d km l X m + Γ m lpx p] d jl U j = 1 U id ik d lm k X m + d km l X m d jl U j 1 U id ik d lm Γ m kpx p + d km Γ m lpx p d jl U j =: V 1 + V. From.6, we have V 1 = 1 U id ik d lm k d mq q φ 1 lndet D F q] = 1 +d km l d mq q φ 1 ] lndet D F q] d jl U j U D E φ F + φ F E DU +U φ D 1 F + F DU 1 D lndet D 4 U D + From.6 and.7, we have V = 1 U id pq p d ij q φ 1 lndet D F q D lndet D U. U j = 1 φ 1 ] lndet D F DEU = 1 U ] φ F D D U U lndet D D DU. Hence, the GBEC A3 can be written as which is exactly.13. W 1 + W = W 1 + V 1 + V λ 3 U DU, Acknowledgment. A. Arnold was partially supported by the ESF in the project Global and geometrical aspects of nonlinear partial differential equations, the Wissenschaftskolleg Differentialgleichungen of the FWF, and by the DFG under Grant no. AR 77/3-3. E. Carlen was partially supported by U.S. National Science

22 174 ANTON ARNOLD, ERIC CARLEN, AND QIANGCHANG JU Foundation grant DMS Q. Ju was supported by the NNSFC grant no , the Wittgenstein 000 Award of Peter Markowich funded by the Austrian FWF, and by the Austrian-Chinese Technical-Scientific Collaboration Agreement funded by the ÖAD. The authors acknowledges fruitful discussions with Peter Markowich and Cedric Villani. References 1. Arnold, A., Bonilla, L., Markowich, P.: Liapunov functionals and large-time asymptotics of mean-field Fokker-Planck equations, Transp. Theo. Stat. Phys. 5, No Arnold, A., Carlen, E.: A generalized Bakry-Emery condition for non-symmetric diffusions: in EQUADIFF 99 Proceedings of the International Conference on Differential Equations, Berlin 1999, World Scientific Publishing, , Arnold, A., Lopez, J. L., Markowich, P.A., Soler, J.: An Analysis of Quantum Fokker-Planck Models: A Wigner Function Approach, Rev. Mat. Iberoam, 0, No Arnold, A., Markowich, P., Toscani, G., Unterreiter, A.: On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. PDE, 6 No Bakry, D., Emery, M.: Hypercontractivité de semi-groupes de diffusion, C.R. Acad. Sc. Paris, t. 99, Série I Bakry, D.: Inégalité de Sobolev faibles: un critère Γ, Lecture Notes in Mathematics 1485, J. Azéma, P.A. Meyer, M. Yor Eds. Séminaire de Probabilités XXV, Springer, Berlin, Bakry, D.: L hypercontractivité et son utilisation en théorie des semigroupes, Lecture Notes in Mathematics 1581, D. Bakry, R.D. Gill, S.A. Molchanov Eds. Lectures on Probability Theory, Springer, Berlin-Heidelberg, Boothby, W. M.: An introduction to differentiable manifolds and Riemannian geometry, Academic Press, New York, Chavel, I., Eigenvalues in Riemannian Geometry, Academic Press, Orlando, Csiszár, I.: Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis von Markoffschen Ketten, Magyar Tud. Akad. Mat. Kutató Int. Közl., Deny, J., Lions, J. L.: Les espaces du type de Beppo Levi, Ann. Inst. Fourier Grenoble Gross, L.: Logarithmic Sobolev inequalities, Amer. J. of Math., Gross, L.: Logarithmic Sobolev Inequalities and Contractivity Properties of Semigroups, Lecture Notes in Mathematics 1563, E. Fabes et al. Eds. Dirichlet Forms, Springer Berlin-Heidelberg, Jourdain, B., Le Bris, C., Lelièvre, T., Otto, F.: Long-Time Asymptotics of a Multiscale Model for Polymeric Fluid Flows, Arch. Rational Mech. Anal., Kato, T.: Perturbation theory for linear operators, Springer Verlag, Kullback, S.: Information Theory and Statistics, John Wiley, Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Risken, H.: The Fokker-Planck Equation, Springer Verlag, Sparber, C., Carrillo, J. A., Dolbeault, J., Markowich, P.: On the long time behavior of the quantum Fokker-Planck equation, Monatshefte für Mathematik 141 No Taylor, M. E.: Partial Differential Equations II, Springer Verlag, New York, Temam, R.: Infinite dimensional dynamical systems in mechanics and physics, Springer Verlag, New York, Toscani, G.: entropy dissipation and the rate of convergence to equilibrium for the Fokker- Planck equation, Quart. Appl. Math., Vol. LVII Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys., 5 No.1/

23 LARGE-TIME BEHAVIOR OF FOKKER-PLANCK TYPE EQUATIONS Unterreiter, A., Arnold, A., Markowich, P., Toscani, G.: On generalized Csiszar-Kullback inequalities, Monatshefte für Mathematik, Anton ARNOLD: Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8, A-1040 Wien, Austria address: Eric CARLEN: Department of Mathematics, Hill Center, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA address: Qiangchang JU: Institute of Applied Physics and Computational Mathematics, P.O. Box , Beijing, , China address: qiangchang