Nonlinear mobility continuity equations and generalized displacement convexity
We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a non-rigorous argument indicating that they are not displacement semiconvex.
Keywords: gradient flows, displacement convexity, nonlinear diffusion equations, parabolic equations, Wasserstein distance, nonlinear mobility.
Displacement convexity and Wasserstein distance.
In [McC97], McCann introduced the notion of displacement convexity for integral functionals of the form
defined on the set of the Borel probability measures in a convex open domain , which are absolutely continuous with respect to the Lebesgue measure . Displacement convexity of means convexity along a particular class of curves, given by displacement interpolation between two given measures. These curves turned out to be the geodesics of the space endowed with the euclidean Wasserstein distance.
We recall that the Wasserstein distance between two Borel probability measures and on is defined by the following optimal transportation problem (Kantorovitch relaxed version)
where is the set of admissible plans/couplings between and , that is the set of all Borel probability measures on with first marginal and second marginal .
We introduce the “pressure” function , defined by
The main result of [McC97] states that under the assumption
the functional is convex along the constant speed geodesics induced by , i.e. for every curve satisfying
the map is convex in . This class of curves can be, equivalently, defined by displacement interpolation, using the Brenier’s optimal transportation map pushing onto (see [Vil03], for example). For power-like functions
The link with a nonlinear diffusion equation.
Among the various applications of this property, a remarkable one concerns a wide class of nonlinear diffusion equations. The seminal work of Otto [Ot01] contributed the key idea that a solution of the nonlinear diffusion equation
with homogeneous Neumann boundary condition on can be interpreted as the trajectory of the gradient flow of with respect to the Wasserstein distance. This means that the equation is formally the gradient flow of with respect to the local metric which for a tangent vector has the form
In particular, the heat equation, for , is the gradient flow of the logarithmic entropy . Let us also note that the metric above satisfies
The key property of this metric is that the length of the minimal geodesic between given two measures is nothing but the Wasserstein distance. More precisely
As for the classical gradient flows of convex functions in euclidean spaces, the flow associated with (1.5) is a contraction with respect to the Wasserstein distance. In [AGS05] the authors showed that one of the possible ways to rigorously express the link between the functional , the distance , and the solution of the diffusion equation (1.5) is given by the evolution variational inequality satisfied by the measures associated with (1.5):
A new class of “dynamical” distances.
In a number of problems from mathematical biology [H03, BFD06, BD09, DR09], mathematical physics [K93, K95, F04, F05, SSC06, CLR08, CRS08], studies of phase segregation [GL1, Sl08], and studies of thin liquid films [Ber98], the mobility of “particles” depends on the density itself. More precisely the local metric in the configuration space is formally given as follows: For a tangent vector (euclidean variation)
where is the mobility function. The global distance generated by the local metric is given by
This distance was recently introduced and studied in [DNS09] in the case when is concave and nondecreasing. Similarly to the case , it is easy to check formally that the trajectory of the gradient flow of with respect to the modified distance solves
with homogeneous Neumann boundary conditions on . Assuming that and are locally integrable, we can define in this case the function and the auxiliary function by
By means of a formal computation, detailed in Section 2, the second derivative of the internal energy functional along a geodesic curve satisfying as in (1.3)
is nonnegative, i.e. , if the following generalization of McCann condition (1.2a,b) holds
|It can also be expressed by requiring that|
As in the case of the Wasserstein distance, in dimension the condition (1.10a) reduces to the usual convexity of . In dimension , still considering the relevant example of power-like functions as in (1.4), we get
and condition (1.10a) is equivalent to
In this case the heat equation corresponds to and it is therefore the gradient flow of the functional
with respect to the distance induced by the mobility function .
Another interesting example, still leading to the heat equation, is represented by the functional
and the distance induced by , . Notice that in this case the positivity domain of the mobility is the finite interval , a case that has not been explicitly considered in [DNS09], but that can be still covered by a careful analysis (see [LM]).
Geodesic convexity and contraction properties.
Our aim is to prove rigorously the geodesic convexity of the integral functional under conditions (1.10a,b) and the metric characterization of the nonlinear diffusion equation (1.9) as the gradient flow of with respect to the distance (1.8). If one tries to follow the same strategy which has been developed in the more familiar Wasserstein framework, one immediately finds a serious technical difficulty, due to the lackness of an “explicit” representation of the geodesics for . In fact, the McCann’s proof of the displacement convexity of the functionals is strictly related to the canonical representation of the Wasserstein geodesics in terms of optimal transport maps.
Existence of a minimal geodesic connecting two measures at a finite distance has been proved by [DNS09]. However, an explicit representation is no longer available. On the other hand in [DS08], following the eulerian approach introduced in [OW05], the authors presented a new proof of McCann’s convexity result for integral functionals defined on a compact manifold without the use of the representation of geodesics. Here, following the same approach of [DS08], we reverse the usual strategy which derives the existence and the contraction property of the gradient flow of a functional from its geodesic convexity. On the contrary, we show that under the assumption (1.10a) smooth solutions of (1.9) satisfy the following Evolution Variational Inequality analogous to (1.7)
This is sufficient to construct a nice gradient flow generated by and metrically characterized by (1.11), as showed in [AGS05] and [AS06]. The remarkable fact proved by [DS08] is that whenever a functional admits a flow, defined at least in a dense subset of , satisfying (1.11), the functional itself is convex along the geodesics induced by the distance . As a by-product we obtain stability, uniqueness, and regularization results for the solutions of the problem (1.9) in a suitable subspace of metrized by .
Concerning the assumptions on , its concavity is a necessary and sufficient condition to write the definition of with a jointly convex integrand [DNS09], which is crucial in many properties of the distance, in particular for its lower semicontinuity with respect to the usual weak convergence of measures. Since on the concavity implies that the mobility must be nondecreasing. This is the case considered in [DNS09]. However we are also able to treat the case when the mobility is defined on an interval where it is nonnegative and concave. It that case the configuration space is restricted to absolutely continuous measures with densities bounded from above by . Such mobilities are of particular interest in applications as mentioned before.
Plan of the paper.
In next section, we show the heuristic computations for the convexity of functionals with respect to . Section 3 is devoted to introduce the notation and to review the needed concepts on from [DNS09]. Moreover, we prove a key technical regularization lemma: Lemma 3.5. Subsection 3.4 addresses the question of finiteness of , providing new sufficient conditions on and in order to ensure that . After a brief review of some basic properties of the diffusion equation (1.6), in Section 4 we try to get some insight on the features of the generalized McCann condition (1.10a,b), we recall some basic facts on the metric characterization of contracting gradient flows and their relationships with geodesic convexity borrowed from [AGS05, DS08], and we state our main results Theorems 4.11 and 4.13. The core of our argument in smooth settings is collected in Section 5, whereas the last Section concludes the proofs of the main results. At the end of the paper we collect some final remarks and open problems.
We first discuss, in a formal way, the conditions for the displacement convexity of the internal, the potential and the interaction energy, with respect to the geodesics corresponding to the distance (1.8). For simplicity, we assume that and that densities are smooth and decaying fast enough at infinity so that all computations are justified.
We first obtain the optimality condition for the geodesic equations in the fluid dynamical formulation of the the new distance (1.8). As in [B03], we insert the nonlinear mobility continuity equation (1.8)
inside the minimization problem as a Lagrange multiplier. As a result, we get the unconstrained minimization problem
Applying a formal minimax principle and thus taking first an infimum with respect to we obtain the optimality condition and the following formal characterization of the distance
which provides the further optimality condition
We thus end up with a coupled system of differential equations in [DNS09, Rem. 5.19]
2.2 Internal energy
We use the formal equations (2.3) for the geodesics associated to the distance (1.8) to compute the conditions under which the internal energy functional is displacement convex. If therefore is a smooth solution of (2.3), which decays sufficiently at infinity, we proceed as usual [CMV03, Vil03, OW05] to obtain the following formulas:
As usual, the Bochner formula
and the fact that , allow us to estimate it as
Therefore, under conditions of concavity of the mobility and the generalized displacement McCann’s condition (1.10a), the functional is convex along the geodesics of the distance .
2.3 Potential energy
with a given smooth potential. As before, it is easy to check that the second derivative of along a geodesic satisfying (2.3) is
This formula allows us to show that this functional cannot be convex along geodesics if is not linear. Technically, the reason is the presence of the terms linearly depending on . We present a simple example:
Example. Let us first construct the example in one dimension. The expression for the second derivative of the functional above reduces to
Consider the case that is nontrivial. Then on some interval. For notational simplicity, we assume that
Since the mobility we are considering is not a linear function of there exists such that . Again for notational simplicity, let us assume that
The fact that we chose to be positive and negative is irrelevant because the sign of term can be controlled by the sign of . Let be a piecewise linear function on :
The fact that the function is Lipschitz, but not smooth is irrelevant; smooth approximations of the given , can also be used in the construction. Let . Let , supported in , such that on and . Let . Note that . A typical profile of is given in Figure 1.
The test velocity (tangent vector at ) we consider also needs to be localized near zero. A simple choice is . Let be the corresponding geodesics given by (2.1) and (2.2). Let us observe how, at , the terms and scale with :
Thus, for small enough, . Furthermore note that the square of the length of the tangent vector is
Thus for any there exists such that
which implies that is not -convex for any .
Let us conclude the example by remarking that it can be extended to multidimensional domains. In particular it suffices to extend the 1-D profile to d-D to be constant in every other direction and then use a cut-off. We only sketch the elements of the construction.
We can assume that . Let . Let . To cut-off in the directions perpendicular to we use the length scales . Let be smooth cut-off function equal to 1 on and equal to outside of ; with and . Let . Let . Checking the scaling of appropriate terms is straightforward.
2.4 Interaction energy
Consider the interaction energy functional
with a given smooth potential. As before, it is easy to check that
It can be demonstrated that if is non-linear then the interaction energy is not geodesically convex. As for the potential energy, the reason lies in the presence of derivatives of in the expression above. More precisely, in one dimension the second derivative of reduces to
It turns out that the example for the lack of (semi-)convexity provided for the potential energy is also an example (with replaced by ) for the interaction energy. The estimates of the terms are similar, so we leave the details to the reader.
3 Notation and preliminaries
In this section, following [DNS09], we shall recall the main properties of the distance introduced in (1.8). For the sake of simplicity, we only consider here the case of a bounded open domain , so that it is not be restrictive to assume that all the measures (Radon, i.e. locally finite, in the general approach of [DNS09]) involved in the various definitions have finite total variation. Since we deal with arbitrary mobility functions , these distances do not exhibit nice homogeneity properties as in the Wasserstein case; therefore we deal with finite Borel measures without assuming that their total mass is .
3.1 Measures and continuity equation
We denote by (resp. ) the space of finite positive Borel measures on (resp. with compact support) and by the space of -valued Borel measures on with finite total variation. By Riesz representation theorem, the space can be identified with the dual space of and it is endowed with the corresponding weak topology. We denote by the total variation of the vector measure . admits the polar decomposition with . If is a Borel subset of (typically an open or closed set) we denote by (resp. ) the subset of (resp. ) whose measure are concentrated on , i.e. (resp. ). Notice that if is a compact subset of then the convex set in of measures with a fixed total mass is compact with respect to the weak topology. If , is the convex subset of whose measures have fixed total mass .
Let be a bounded open subset of . Given we denote by the collection of time dependent measures and such that
is weakly continuous in with and ;
is a Borel family with ;
is a distributional solution of
If then it is immediate to check that the total mass is a constant, independent of . In particular, .
3.2 Mobility and action functional
We fix a right threshold and a concave mobility function strictly positive in . We denote by the left limit of as . We can also introduce the maximal left interval of monotonicity of whose right extreme is
We distinguish two situation:
- Case A
so that is nondecreasing and ; typically and the main example is provided by , . This is the case considered in [DNS09]. When we are in the sublinear growth case. A linear growth of corresponds to .
- Case B
, so that and is nonincreasing in the right interval (but we also allow to be constant or even decreasing in with ). Typically (in this case ) and the main example is , or, more generally, , .
Many properties proved in the case A can be extended to the case B, but there are important exceptions: we refer to [LM] for further details. Using the conventions
the corresponding action density function is defined by
It is not difficult to check that, under the convention (3.1), the function is (jointly) convex and lower semi-continuous.
Given that is concave and is convex, when we can define the recession function (recall (3.1))
We introduce now the action functional
defined on couples of measures , . In order to define it we consider the usual Lebesgue decomposition , and distinguish the following cases:
If the support of or is not contained in then ;
When (Case B), we set
notice that if then with -a.e. in and .
When and (Case A, sublinear growth) then
Finally, when and (Case A, linear growth) then we set
3.3 The modified Wasserstein distance
Let be a bounded open set. Given