Diffusion and kinetic transport with very weak confinement

. This paper is devoted to Fokker-Planck and linear kinetic equations with very weak conﬁnement corresponding to a potential with an at most logarithmic growth and no integrable stationary state. Our goal is to understand how to measure the decay rates when the diﬀusion wins over the conﬁnement although the potential diverges at inﬁnity.


1.
Introduction. This paper addresses the large time behavior of the solutions to the macroscopic Fokker-Planck equation and to kinetic equations with Fokker-Planck or scattering collision operators.
The first part of this paper deals with the macroscopic Fokker-Planck equation where x ∈ R d , d ≥ 3, and V is a potential such that e −V ∈ L 1 (R d ), that is, e −V dx is an unbounded invariant measure. We shall investigate the two following examples V 1 (x) = γ log |x| and V 2 (x) = γ log x with γ < d and x := 1 + |x| 2 for any x ∈ R d . These two potentials share the same asymptotic behavior as |x| → ∞. The potential V 1 is invariant under scalings, whereas V 2 is smooth at the origin. In both cases, the only integrable equilibrium state is 0. Thus, if the initial datum u 0 is such that u 0 ∈ L 1 (R d ), we expect that the solution to (1) converges to 0 as t → +∞. When γ > 0, the potential V is very weakly confining in the sense that, even if it eventually slows down the decay rate, it is not strong enough to produce a stationary state of finite mass: the diffusion wins over the drift. Our goal to establish the rate of convergence in suitable norms. We shall use the notation · p := · L p (dx) in case of Lebesgue's measure and specify the measure otherwise.
Theorem 1.1. Assume that d ≥ 3, γ < (d − 2)/2 and V = V 1 or V = V 2 . Then any solution u of (1) with initial datum u 0 ∈ L 1 + ∩ L 2 (R d ) satisfies, for all t ≥ 0, u(t, ·) Here C Nash denotes the optimal constant in Nash's inequality [22,10] u Note that the rate of decay is independent of γ and we recover the classical estimate due to J. Nash when V = 0 (here γ = 0). The proof of Theorem 1.1 and further considerations on optimality are collected in Section 2.1. Theorem 1.1 does not cover the interval (d − 2)/2 < γ < d. This range is covered by employing the natural setting of L 2 e V and by requiring additional moment bounds.
The proof of Theorem 1.2 is done in Section 2.2. Although this is a side result, let us notice that the case in which the potential contributes to the decay, i.e., when γ < 0, is also covered in Theorem 1.2. The scale invariance of (1) with V = V 1 can be exploited to obtain intermediate asymptotics in self-similar variables. Let us define The following result on intermediate asymptotics allows us to identify the leading order term of the solution of (1) as t → +∞. It is the strongest of our results on (1) but initial data need to have a sufficient decay as |x| → ∞. Theorem 1.3. Let d ≥ 1, γ ∈ (0, d) and V = V 1 . If for some constant K > 1, the function u 0 is such that where c ⋆ is chosen such that u ⋆ 1 = u 0 1 then the solution u of (1) with initial datum u 0 satisfies More detailed results will be stated in Section 2.3. Let us quote some relevant papers for (1). In the case without potential, the decay rates of the heat equation is known for more than a century and goes back to [15]. Standard techniques use the Fourier transform, Green kernel estimates and integral representations: see for instance [14]. There are many other parabolic methods which provide decay rates and will not be reviewed here like, for instance, the Maximum Principle, Harnack inequalities and the parabolic regularity theory: see for instance [25].
In his celebrated paper [22], J. Nash was able to reduce the question of the decay rates for the heat equation to (3): see [7] for detailed comments on the optimality of such a method. Entropy methods have raised a considerable interest in the recent years, but the most classical approach based on the so-called carré du champ method applies to (1) only for potentials V with convexity properties and a sufficient growth at infinity: typically, if V (x) = |x| α , then α ≥ 1 is required for obtaining a Poincaré inequality and the rate of convergence to a unique stationary solution is then exponential, when measured in the appropriate norms; see [3] for a general overview. An interesting family of weakly confining potentials is made of functions V with an intermediate growth, such that e −V is integrable but lim |x|→∞ V (x)/|x| = 0: all solutions of (1) are attracted by a unique stationary solution, but the rate is expected to be algebraic rather than exponential. A typical example is V (x) = |x| α with α ∈ (0, 1). The underlying functional inequality is a weak Poincaré inequality: see [24,20], and [2] for related Lyapunov type methodsà la Meyn and Tweedie or [5] for recent spectral considerations. We refer to [1] and [27,28,29] for further considerations on, respectively, weighted Nash inequalities and spectral properties of the diffusion operator.
The second part of this paper is devoted to kinetic equations involving a degenerate diffusion operator acting only on the velocity variable or scattering operators, for very weak potentials like V 1 or V 2 . Various hypocoercivity methods have been developed over the years in, e.g., [16,17,21,26,12], in order to prove exponential rates in appropriate norms, in presence of a strongly confining potential. In that case, the growth of the potential at infinity has to be fast enough not only to guarantee the existence of a stationary solution but also to provide macroscopic coercivity properties which typically amount to a Poincaré inequality. A popular simplification is to assume that the position variable is limited to a compact set, for example a torus. Such results are the counterpart in kinetic theory of diffusions covered by the carré du champ method, as emphasized in [4].
Recently, hypocoercivity methods have been extended in [6] to the case without any external potential by replacing the Poincaré inequality by Nash type estimates. The sub-exponential regime or the regime with weak confinement, i.e., of a potential V such that a weak Poincaré inequality holds, has also been studied in [9,18]. What we will study next is the range of very weak potentials V , which have a growth at infinity which is below the range of weak Poincaré inequalities, but are still such that lim |x|→∞ V (x) = +∞. This regime is the counterpart at kinetic level of the results of Theorems 1.1, 1.2 and 1.3. As in the case of (1) when γ ≥ 0, the drift is opposed to the diffusion, but it is not strong enough to prevent that the solution locally vanishes.
Let us consider the kinetic equation where Lf is one of the two following collision operators: (a) a Fokker-Planck operator We consider the case of a global equilibrium of the form We shall say that the gaussian function M (v) is the local equilibrium and assume that the scattering rate σ(v, v ′ ) satisfies is defined with respect to an unbounded measure. As in the case of (1), the only integrable equilibrium state is 0. Thus, if the initial datum f 0 is such that f 0 ∈ L 1 (dx dv), we expect that the solution to (5) converges to 0 locally as t → +∞ and look for the rate of convergence in suitable norms. When V = 0, the optimal rate of convergence of a solution f of (5) with initial datum f 0 is known. In [6], it has been proved that there exists a constant C > 0 such that where dµ = M −1 dx dv and by factorization, the result is extended with same rate for an arbitrary ℓ > d to the Our main result on (5) is a decay rate in the presence of a very weak potential. It is an extension of the results of Theorem 1.2 to the framework of kinetic equations. Theorem 1.4. Let d ≥ 3, V = V 2 with γ ∈ [0, d) and k > max {2, γ/2}. We assume that (H1)-(H2) hold and consider a solution f of (5) with initial datum Then there exists C > 0 such that Standard methods of kinetic theory can be used to establish the existence of solutions of (5) when V = V 2 . We will not give details here. At formal level, similar results can be expected when V = V 1 but the singularity at x = 0 raises difficulties which are definitely outside of the scope of this paper.
The expression of the constant C is explicit. However, due to the method, we cannot claim optimality in the estimate of Theorem 1.4, but at least the asymptotic rate is expected to be optimal by consistency with the diffusion limit, as it is the case when V = 0 studied in [6]. The strategy of the proof and further relevant references will be detailed in Section 3.

2.
Decay estimates for the macroscopic Fokker-Planck equation. In this section, we establish decay rates for (1) and discuss the optimal range of the parameters.
2.1. Decay in L 2 (R d ). We prove Theorem 1.1. By testing (1) with u, we obtain For from Nash's inequality (3). Integration completes the proof of (2). For the case 0 < γ < (d − 2)/2 we use the following Hardy-Nash inequalities. with The proof of Lemma 2.1 is given in Appendix C. We use Lemma 2.1 with δ = γ (d − 2)/2 and with η = γ (for V = V 2 ), and proceed as for γ ≤ 0 to complete the proof of Theorem 1.1. Remark 1. The condition δ < (d − 2) 2 /4 in Lemma 2.1 is optimal for (6) and (7). The restriction on γ in Theorem 1.1 is also optimal. Let d ≥ 3, γ > (d − 2)/2 and In the case V = V 1 , it is indeed enough to observe that (d − 2) 2 /4 is the optimal constant in Hardy's inequality (see Appendix C). The case V = V 2 follows from the case V = V 1 by an appropriate scaling.

Decay in
In the case V = V 1 , we have e V = |x| γ and (8) takes the form 1 2 If γ ≤ 0 and a = d−γ d+2−γ , the inequality follows from the Caffarelli-Kohn-Nirenberg inequalities (see Appendix A, Ineq. (26) applied with k = 0 to v = |x| γ u). The conservation of the L 1 norm of u gives The conclusion of Theorem 1.2 follows by integration. An analogous argument based on the inhomogeneous Caffarelli-Kohn-Nirenberg inequality Without additional assumptions, it is not possible to expect a similar result for γ > 0. Let us explain why. In the case V = V 1 and with v = |x| γ u, consider the quotient . Let us consider the case γ > 0. For the proof of Theorem 1.2 in the case 0 < γ < d, V = V 1 , we start by estimating the growth of the moment which evolves according to where we have used Hölder's inequality and M 0 (t) = M 0 (0) = u 0 1 . Integration gives and, after integration, with a and b depend only on the quantities entering into the constant c of Theorem 1.2. Let θ = 2k/(d + 2k − γ) and observe that In the case V = V 2 we can adopt the same strategy, based on a moment now defined as and on the inhomogeneous Caffarelli-Kohn-Nirenberg inequality . This completes the proof of Theorem 1.2.

Decay in self-similar variables and intermediate asymptotics.
We prove Theorem 1.3. With the parabolic change of variables which preserves mass and initial data, where Φ(τ, ξ) = V (e τ ξ) + 1 2 |ξ| 2 . We investigate the long-time behavior of solutions of (1) by considering quasi- of (11) with an appropriately chosen M (τ ).
If a quasi-equilibrium of the form (12) satisfies which holds for both examples (13) and (14) if γ > 0, then v ⋆ is obviously a supersolution of (11), thus proving the following result on uniform decay estimates.
For 0 < γ < d, we obtain a pointwise decay: the attracting potential is too weak for confinement (no stationary state can exist, at least among L 1 (R d ) solutions) but it slows down the decay compared to solutions of the heat equation (that is, solutions corresponding to V = 0).
The result of Proposition 1 is also true for γ ≤ 0 if V = V 1 . In that case, a repulsive potential with γ < 0 accelerates the pointwise decay, but does not change the uniform decay rate as t → +∞ because In order to obtain an estimate in L 2 e V dx , let us state a result on a Poincaré inequality. We introduce the notations Moreover, for any γ ∈ (0, d), min σ∈[0,1] λ γ,σ > 0.
Proof. Let us consider a potential ψ on R d . We assume that ψ is a measurable function such that where B c r := x ∈ R d : |x| > r and D(B c r ) denotes the space of smooth functions on R d with compact support in B c r . According to Persson's result [23, Theorem 2.1], the lower end of the continuous spectrum of the Schrödinger operator − ∆ + ψ is ℓ.
In the special case σ = 0, it is possible to compute λ γ,0 as follows.
Lemma 2.4. Assume that d ≥ 1 and γ ∈ (0, d). With the above notations, we have Proof. A decomposition in spherical harmonics shows that the lowest eigenvalue associated with a non-radial eigenfunction (in dimension d ≥ 2) is of the form If k = 0, g ≡ 1 is optimal and the eigenvalue is k (k+d−2) with k = 1. Otherwise k = 0 and g is the lowest non-trivial Hermite polynomial with zero average on R + ∋ r in dimension n = d − γ, that is g(r) = r 2 − n and the corresponding eigenvalue is 4n. Notice that n is not necessarily an integer, but can be considered as a real parameter. All other eigenvalues are larger. We conclude by taking the minimum of the two eigenvalues. If d = 1, a similar conclusion holds with f (ξ) = ξ.
An interesting consequence of Lemma 2.4 is a result of intermediate asymptotics, which allows to identify the leading order term of the solution of (1) as t → +∞. Corollary 1. Assume that d ≥ 1, γ ∈ (0, d) and V = V 1 . With the above notations, if u solves (1) with an initial datum u 0 ∈ L 1 Proof. By definition of u ⋆ , we have Then, using the Poincaré inequality (16) and Lemma 2.4, we know that from which we deduce that This concludes the proof using the parabolic change of variables (10).
Proof of Theorem 1.3. A Cauchy-Schwarz inequality shows that The Hölder interpolation inequality combined with the results of Proposition 1 and Corollary 1 concludes the proof after taking (15) and the expression of λ γ,0 stated in Lemma 2.4 into account.
3. Decay estimate for the kinetic equation with weak confinement. In this section, we prove Theorem 1.4 by revisiting the L 2 approach of [12] in the spirit of [6].

Let us use the notation u[f ] := e V ρ[f ] and observe that
where the last identity follows from To build a suitable Lyapunov functional, as in [11,12,6] we introduce the operator A defined by A := Id + (TΠ) * (TΠ) −1 (TΠ) * .
As in [12] we define the Lyapunov functional H by and obtain by a direct computation that (17) where we have used that Af , Lf = 0. For the first term in D[f ], we rely on the microscopic coercivity estimate (see [12]) The second term ATΠf , Πf is expected to control the macroscopic contribution Πf . In Section 3.2 the remaining terms will be estimated to show that for ε small enough D[f ] controls (Id − Π)f 2 + ATΠf , Πf . As in Section 2.2, estimates on moments are needed, which will be proved in Section 3.3 and used in Section 3.4 to show a Nash type estimate and to complete the proof of Theorem 1.4 by relating the entropy dissipation D[f ] to H[f ] and by solving the resulting differential inequality.

Proof of the Lyapunov functional property of H[f ]. Let us define the notations
Lemma 3.1. With the above notations, we have Proof. We already know from [12, Lemma 1] that the operator TA is bounded. Let us give a short proof for completeness. The equation Af = g is equivalent to (TΠ) * f = g + (TΠ) * (TΠ) g .
Multiplying (18) by g M −1 e V , we get that Proof. Let w be a solution of (19). Since we obtain that Using (19) and integrating on R d after multiplying by Πf = u M e −V , we obtain that On the other hand, we can also write that using the Cauchy-Schwarz inequality. As a consequence, we obtain that which concludes the proof.
and it is self-adjoint on L 2 (e V dx) so that for any w 1 and w 2 . Applied first with w 1 = w and w 2 = Lw and then with w 1 = w 2 = ∇w, this shows that Proof. Assume that u = u[f ] and w solves (19).
We conclude using a Cauchy-Schwarz inequality, Lemma 3.2 and Lemma 3.3.
In order to have unified notations, we adopt the convention that σ = 1/ √ 2 if L is the Fokker-Planck operator.
We conclude using Lemma 3.2 and an estimate on j = |j| e where e ∈ S d−1 , that goes as follows: by computing we know that • If L is the scattering operator, then Notice that for a nonnegative function f , we have the improved bounds L( Finally, we apply the results of Lemmas 3.1, 3.4, 3.5 to the right hand side of (17): Lemma 3.6. With the above notations, we have The Proof. The above mentioned Lemmas imply The Lyapunov function property is a consequence of (19) and Lemma 3.2.
3.3. Moment estimates. Let us consider the case V = V 2 and define the k th order moments in x and v by J k (t) := x k f (t, ·, ·) 1 and K k (t) := |v| k f (t, ·, ·) 1 .
Our goal is to prove estimates on J k and K k . Notice that is constant if f solves (5).
There exist constants C 2 , . . . , C k such that Proof. We present the proof for a Fokker-Planck operator, the case of a scattering operator follows the same steps. A direct computation shows that A bound C ℓ for K ℓ , ℓ ∈ N, follows after observing that using Hölder's inequality twice. Next, let us compute Note that, again by Hölder's inequality, |L ℓ | ≤ J . . , k. We prove the bound on J ℓ (t) by induction. If ℓ = 2, (21) implies L 2 (t) ≤ max L 2 (0), C 2 and, thus, J 2 (t) ≤ C 2 (1 + t), up to a redefinition of C 2 . Now let ℓ > 2 and assume that We use Hölder's inequality once more for the right hand side of (21):

which implies
L ℓ ≤ C (1 + t) ℓ 2 −1 , and one more integration with respect to t establishes the estimate for J ℓ in (20), up to an eventual redefinition of C ℓ . Lemma 3.8. Let γ ∈ (0, d), k ∈ N with k > 2, V = V 2 and assume that f ∈ C R + , L 2 (M −1 dx dv) is a nonnegative solution of (5) with initial datum f 0 such Proof. The solution w of (19) is positive by the maximum principle. In what follows we use the definition of M ℓ for arbitrary integers ℓ and note that for ℓ ≤ 0, Multiplication of (19) by x ℓ−γ and integration over R d gives where J ℓ has been estimated in Lemma 3.7. Then, with ℓ = 2 and (22), we obtain M 2 (t) ≤ C 2 (1 + t). This implies by the Hölder inequality that M 1 (t) ≤ M 0 M 2 (t) ≤ C 1 (1 + t) 1/2 . For 2 < ℓ ≤ k the estimate M ℓ (t) ≤ C ℓ (1 + t) ℓ/2 follows recursively from (23).

Decay estimate for the kinetic equation (proof of Theorem 1.4).
Lemma 3.9. Let γ ∈ (0, d), k ≥ max{2, γ/2}, V = V 2 and assume that f ∈ C R + , L 2 (M −1 dx dv) is a nonnegative solution of (5) with initial datum f 0 such that R d ×R d x k f 0 dx dv < +∞ and R d ×R d |v| k f 0 dx dv < +∞. Assume the above notations, in particular with M k defined as in Lemma 3.8, with the constant K from (30), and with a = d+2k−γ d+2+2k−γ . Then Proof. If u = u[f ] and w solves (19), we recall that (19), we also deduce that By inequality (30), we have that Combining these inequalities gives As a consequence of Lemmas 3.1, 3.6, 3.9 and of the properties of Φ we have implying, with Lemma 3.8, The decay of H[f ] can be estimated by the solution z of the corresponding ODE problem By the properties of Φ it is obvious that z(t) → 0 monotonically as t → +∞, which implies that the same is true for dz dt . Therefore, there exists t 0 > 0 such that, in the rewritten ODE the first term is smaller than the second for t ≥ t 0 , implying the differential inequality dz dt with an appropriately defined positive constant κ. Integration and estimation as in Section 2.2 gives thus completing the proof of Theorem 1.4.
A.1. The general Caffarelli-Kohn-Nirenberg inequalities. The main result of [8] goes as follows. Assume that p ≥ 1, q ≥ 1, r > 0, 0 ≤ a ≤ 1 and Assume moreover that Then there exists a positive constant C such that the inequality holds for any v ∈ C ∞ 0 (R d ). These interpolation inequalities are known in the literature as the Caffarelli-Kohn-Nirenberg inequalities according to [8] but were introduced earlier by V.P. Il'in in [19]. Next we specialize Ineq. (24) to various cases of Nash type corresponding to q = 1.
A.2. Weighted Nash type inequalities. We consider special cases corresponding to r = p = 2 and q = 1.
Without loss of generality, we can assume that the function g is nonnegative and radial, by spherically non-increasing rearrangements. From now on, we will only consider nonnegative, radial, non-increasing functions g and the corresponding functions v(x) = |x| γ/2 g(x). For any R > 0, let On the other hand, let us define v R := R d vR |x| k−γ dx B R |x| 2k−γ dx and observe that Let us consider the weighted inequality The existence of a positive, finite constant λ R 1 can be deduced from elementary variational techniques as in [13]. We infer from the definition of v R that this inequality is equivalent to With λ 1 := λ 1 1 , a simple scaling shows that λ R 1 = λ 1 R −2 . Let us come back to the estimation of R d v 2 |x| −γ dx. By definition of v R , we know that After summing (27) and (29), we arrive at and notice that using k > 0 and v R ≤ v, for some numerical constant c which depends only on d and γ. Collecting terms, we have found that We can summarize our observations as follows.
The numerical value of κ can be deduced from the expression of c and from the coefficients that arise from the optimization with respect to R > 0.
Appendix B. Inhomogeneous Caffarelli-Kohn-Nirenberg inequalities of Nash type. Our goal is to establish an extension of (26) adapted to the inhomogeneous case.
Proof. Again we rely on the method of E. Carlen and M. Loss in [10]. The computations are similar to the ones of Proposition 2 except that |x| has to be replaced by x . With g = x −γ/2 v, (30) is equivalent to .
Without loss of generality, we assume that the function g is nonnegative, radial by spherically non-increasing rearrangements, and nonnegative. Let v(x) = x γ/2 g(x) and g R := g ½ BR and v R (x) = x γ/2 g R (x) for any R > 0. We observe that g − g R is supported in R d \ B R and On the other hand, using we deduce from the weighted Poincaré inequality and from the definition of v R that By definition of v R , we also know that After summing (31) and (32), we arrive at and such that lim R→0+ R d b(R) ∈ (0, +∞), lim R→+∞ R d+2k−γ b(R) ∈ (0, +∞), lim R→+∞ R −2 a(R) = 1/λ 1 where λ 1 is the optimal constant in Proposition 2 while lim R→0+ R −2 a(R) = 1/λ is related with Nash's inequality as in [10] and such that B1 In order to prove (30), we can use the homogeneity of the inequality and assume that R d x −γ v 2 dx = 1. What we shown so far is that With the choice R = X −(1−a)/2 , we get that there exists a constant K > 0 such that a(R) X + b(R) < K X a . This proves (30) with K ≤ K.
C.1. Proof of Lemma 2.1. We start with the proof of (7) by first showing a Hardy type inequality. For some α ∈ R to be fixed later we compute We deduce that so that, by writing |x| 2 = x 2 − 1, we obtain  (1 + |x| 2 ) 2 dx ≥ 0 . (33)