Generalized compressible fluid flows and solutions of the Camassa-Holm variational model
The Camassa-Holm equation on a domain , in one of its possible multi-dimensional generalizations, describes geodesics on the group of diffeomorphisms with respect to the metric. It has been recently reformulated as a geodesic equation for the metric on a subgroup of the diffeomorphism group of the cone over . We use such an interpretation to construct an analogue of Brenier’s generalized incompressible Euler flows for the Camassa-Holm equation. This involves describing the fluid motion using probability measures on the space of paths on the cone, so that particles are allowed to split and cross. Differently from Brenier’s model, however, we are also able to account for compressibility by employing an explicit probabilistic representation of the Jacobian of the flow map. We formulate the boundary value problem associated to the Camassa-Holm equation using such generalized flows. We prove existence of solutions and that, for short times, smooth solutions of the Camassa-Holm equations are the unique solutions of our model. We propose a numerical scheme to construct generalized solutions on the cone and present some numerical results illustrating the relation between the generalized Camassa-Holm and incompressible Euler solutions.
The Camassa-Holm (CH) equation is the geodesic equation for the metric on the group of diffeomorphims of the circle or the real line . This can be derived as an approximation for ideal fluid flow with a free boundary in the shallow water regime. In this context,  showed that its natural generalization to a higher dimensional domain consists in replacing the norm with the norm. In other words, the CH equation is the Euler-Lagrange equation for the Lagrangian
where is the Eulerian velocity field, are constants and is the Lebesgue measure on . This is a particular instance of a class of right-invariant Lagrangians on the diffeomorphism group of considered in , which for can be written as
Such Lagrangians give rise to several important fluid dynamics models, including the EPDiff equation for the Sobolev norm of vector fields and the Euler- model [17, 18], both of which have also been regarded as possible multi-dimensional versions of the CH equation, but also the Hunter-Saxton equation .
In one dimension, the CH equation is bi-Hamiltonian and completely integrable. It also possesses soliton solutions named peakons, i.e. non-smooth traveling wave solutions which interact and collide without changing their shapes. On the real line, these (weak) solutions have the following expression
where and determine the height and speed of the wave, respectively, and is an independent constant determining its width . Peakons always emerge from appropriate smooth initial data satisfying a certain decay property on the real line, yielding therefore a model for wave breaking [12, 27]. In other words, since the emergence of peakons corresponds to blow-up in an appropriate norm, strong solutions may have finite existence time. Furthermore, even weak solutions cannot be defined globally ; the collision of peakons, for instance, gives an explicit example of finite time breakdown (blow up) of solutions. In this case, at the collision time, the Lagrangian map ceases to be injective and after this, weak solutions are not uniquely defined.
Recently,  put forward a novel interpretation of the CH equation which emphasizes its connection with the incompressible Euler equations. In order to describe this, we consider first the incompressible Euler model. In this case, the configuration space of the system is given by a subgroup of the diffeomorphism group which consits of all diffeomeorphisms preserving the Lebesgue measure, i.e. satisfying
In fact, this can be seen as an isotropy subgroup once we interpret the push-forward as an action of the diffeomorphism group on the space of densities on . This point of view establishes a remarkable connection with optimal transport theory  (see also  for a description of the geometrical connection between the diffeomorphism group and the space of densities). Incompressible Euler flows are minimizers of the action
subject to (1.4) and with the additional constraint that , the identity map on , and , a given diffeomorphism on , which prescribes the final position of each particle in at the final time .
Shnirelman proved that the infimum of this problem is not generally attained when and that even when there exist final configurations which cannot be connected to the identity map with finite action . This motivated Brenier to introduce a relaxation whose solutions are not diffeomorphims, but rather describe the flow in a probabilistic fashion. More precisely, Brenier defined generalized incompressible flows as probability measures on , the space of continuous curves on the domain , satisfying
where is the evaluation map at time defined by . In this interpretation, the marginals are probability measures on the product and describe how particles move and spread their mass across the domain. Of course, classical deterministic solutions, i.e. curves of volume preserving diffeomorphisms , also fit in this definition and correspond to the case where the marginals are concentrated on the graph of . Then, equation (1.6) is the equivalent of the incompressibility constraint in the generalized setting; in fact, when is deterministic it coincides with (1.4). The minimization problem in terms of generalized flows consists in minimizing the action
among generalized incompressible flows, with the constraint . Brenier proved that this model is consistent with classical solutions of the incompressible Euler equations . In particular, smooth solutions correspond to the unique minimizers of the generalized problem if the pressure has bounded Hessian and for sufficiently small times. On the other hand, for any coupling there exists a unique pressure, defined as a distribution, associated to generalized solutions. This result was later improved by Ambrosio and Figalli  who showed that the pressure can be actually defined as a function and defined optimality conditions for generalized flows based on this result.
In analogy to the incompressible Euler case, the CH equation can be reformulated as a geodesic equation for the cone metric on a certain isotropy subgroup of the diffeomorphism group of . More precisely, CH flows are represented by time dependent maps in the form
where and . This set of maps is a group under composition and is known as the automorphism group of . The isotropy subgroup is given by
Differently from (1.4), this condition does not enforce incompressibility but it relates and by requiring . Therefore, automorphisms satisfying (1.9) provide us with an alternative way to represent diffeomorphisms of . Importantly, in this picture we cannot capture the blow up of solutions as induced by peakon collisions, as in this case the Jacobian would locally vanish. In addition, the metric space equipped with the cone metric is not complete, which complicates the construction of generalized solutions following Brenier’s approach. We are then led to work with the cone , which allows us to represent solutions with vanishing Jacobian by paths on the cone reaching the apex.
In this paper, we construct a framework to solve the boundary value problem associated to the CH equation using generalized flows interpreted as probability measures on the space of continuous paths on the cone . We will show that in this setting, the isotropy subgroup condition in (1.9) is then replaced by
where is the projection from the cone onto and now is the evaluation map for continuous paths on the cone. The generalized minimization problem for CH consists in minimizing the action
among generalized flows satisfying (1.10) and an appropriate coupling constraint. The issue of choosing the correct coupling constraint will occupy an important part in the paper. It will be evident that enforcing the coupling of points on the cone in the same way Brenier did for Euler is inappropriate for our case. The same holds also when enforcing the constraint on the base space only. We will define a weaker form of coupling which allows us to prove existence of solutions but it is still compatible with the original model. In particular, we will prove three main results on this problem: existence of solutions of generalized problem; existence and uniqueness of the pressure as a distribution; and correspondence with smooth solutions of the CH equation.
The main difficulty in carrying out this program lies on the necessity to work on an unbounded cone domain and on the impossibility to “cut it” without limiting the class of functions that can be represented by the model. This issue is directly linked to the choice of the correct coupling constraint. In fact, we will introduce a sufficiently weak coupling constraint in order to be able to represent a sufficiently large class of generalized flows on the cone and consequently prove existence of solutions. In principle, this allows us to represent solutions that charge paths reaching the apex of the cone and therefore are characterized, in some sense, by a vanishing Jacobian locally in space. It is natural to ask whether these solutions are actually realized by appropriate couplings. In this paper, we will not answer this question but we will address its complementary side, that is, we will show that smooth solutions of the CH equation (that do not reach the apex) are the unique minimizers for our model for sufficiently short times and upon some regularity conditions on the pressure similar to the Euler case. Interestingly, the decoupling between the Lagrangian flow map and its Jacobian we use to define generalized solutions has also been used in  to construct global weak solutions of the CH equation. However, in their case, one continues solutions after the blowup by allowing the square root of the Jacobian to become negative, which does not occur in our construction.
It should be noted that the cone construction has been developed and used extensively in  in order to characterize the metric side of the Wasserstein-Fisher-Rao (WFR) distance (which is also called Hellinger-Kantorovich distance) on the space of positive Radon measures. In fact, as noted in  this has the same relation to the CH equation as the Wasserstein distance does to the incompressible Euler equations. In the geodesic problem associated the WFR distance the isotropy subgroup relation in (1.10) is used to prescribe the initial and final density. The resulting problem can then be expressed without recurring to the cone construction, yielding an optimal entropy-transport problem, a widespread form of unbalanced optimal transport based on the Kullback-Leibler divergence [11, 10, 24]. Unfortunately, we cannot easily relate such a formulation to our generalized CH problem and in fact this latter does not coincide with a multi-marginal entropy-transport problem. This means that we will develop our construction on the cone without looking at the possibility of reducing the problem using objects defined on the base space only.
The framework we develop here for the CH equation opens several new directions in terms of the numerical treatment of these equations. This can be useful both to study the nature of generalized CH flows and as a basis to develop novel numerical tools to simulate classical solutions to this problem. In the case of the incompressible Euler equations, recent advances in numerical methods for optimal transport have inspired several methods which can also be reformulated for the CH problem. The introduction of the Sinkhorn’s algorithm for the entropic regularization of optimal transport problems  has paved the way for the development of a number of efficient algorithms for several applications . This methodology has been used in  to develop a numerical scheme to compute generalized incopressible Euler flows. In this paper, we construct a similar numerical scheme for the generalized CH problem, which however cannot represent the “blow up”, i.e. solutions with vanishing Jacobian. It should be noted that different methodologies based on semi-discrete optimal transport  could provide a better description of generalized CH flows. Semi-discrete schemes for the incompressible Euler problem have been developed in  for the boundary value problem and in  for the initial value problem, and indicate a promising direction for the development of numerical schemes for the CH model as well.
Besides numerical applications, our approach to solve the variational CH model also suggests several new research directions. First of all, a natural question is whether our construction can provide any insight on the continuation of CH solutions after blow up, for instance in relation to the analytical approach in . In addition, it is also natural to ask whether different models arising from the right-invariant Lagrangian (1.2) on the group of either compressible or incompressible diffeomorphisms can be treated in the same way as the CH model. A unified approach to treat this Lagrangian could shed light on the relation between several important fluid dynamics models and provide a deeper understanding of their solutions.
The rest of the paper is structured as follows. In section 2 we describe the notation and provide some background measure theoretical notions. In section 3 we describe the variational interpretation of the generalized CH equation as geodesic equation on the group of automorphisms of the cone. In section 4 we build on such an interpretation proposing a definition for compressible generalized flows which allows us to define solutions of the boundary value problem associated to the CH equation. In section 5 we prove that for any given final configuration of the flow defining the boundary conditions of the generalized CH problem there exists a unique pressure defined as a distribution; this mimics Brenier’s analogue result for the incompressible Euler equations . In section 6 we prove that smooth solutions of the CH equation are the unique minimizers of our generalized model for sufficiently short times. In section 7 we construct a numerical algorithm based on entropic regularization and Sinkhorn’s algorithm to compute generalized CH flows and provide some numerical results. Conclusions and open questions are collected in section 8.
2. Notation and preliminaries
In this section, we describe the notation and some basic results used throughout the paper. Because of the similarities between our setting and the one of , we will adopt a similar notation for the cone construction and the measure theory objects we will employ.
2.1. Function spaces
Given two metric spaces and , we denote by the space of continuous functions , and with the space of real-valued continuous functions . If is compact is a Banach space with respect to the sup norm . The set of Lipschitz continuous function on is denoted by and the associated norm is given by
where denotes the distance function on . If is a manifold, we will denote by the group of smooth diffeomorphisms of .
2.2. The cone and metric structures
Let be a compact domain. We will denote by the Euclidean metric tensor on , with the Euclidean distance on and with the Euclidean norm. We denote by the cone over . A point on the cone is an equivalence class , with equivalence relation given by
The distinguished point of the cone is the apex of and it is denoted by . Every point on the cone different from the apex can be identified with a couple where and . Moreover, we fix a point and we introduce the projections and defined by
We endow the cone with the metric tensor defined on . We denote the associated norm by . We use the superscripts and for differential operators, e.g., , and so on, to indicate that they are computed with respect to either one of these metrics. The distance on the cone is given by
The closed subset of the cone composed of points below a given radius is denoted by , or more precisely
Given an interval , we denote by and the spaces of, respectively, continuous and absolutely continuous curves . We will generally use the notation
so that and . Note that if is continuous (resp. absolutely continuous), then so is the path but not . However, is continuous (resp. locally absolutely continuous) when restricted to the open set . Then, if we define by
we have that coincides for a.e. with the metric derivative of with respect to the distance . We denote by the space of absolutely continuous curves such that . Then, the following variational formula for the distance function holds
We will extensively use the class of homogeneous functions on the cone defined as follows. A function is -homogeneous (in the radial direction) if for any constant and for all -tuples ,
In particular, a -homogeneous function satisfies . Similarly, a functional is -homogeneous if for any constant and for any path ,
2.3. Measure theoretic background
Let be a Polish space, i.e. a complete and separable metric space. We denote by the set of non-negative and finite Borel measures on . The set of probability measures on is denoted by . Let be another Polish space and a Borel map. Given a measure we denote by the push-forward measure defined by for any Borel set . Given a Borel set we let the restriction of to defined by for any Borel set . Note that we will generally use bold symbols to denote measures on product spaces, e.g., .
We endow with the topology induced by narrow convergence, which is the convergence in duality with the space of real-valued continuous bounded functions . Then, can be identified with a subset of with the weak-* topology (see Remark 5.1.2 in ). Moreover, given a lower semi-continuous function , the functional defined by
is also lower-semicontinuous (see Lemma 1.6 in ) .
As usual in this setting, we will use Prokhorov’s theorem for a characterization of compact subsets of endowed with the narrow topology.
A set is relatively sequentially compact in if and only if it is tight, i.e. for any there exists a compact set such that for any .
We also need a criterion to pass to the limit when computing integrals of unbounded functions: for this will use the concept of uniform integrability. Given a set , we say that a Borel function is uniformly integrable with respect to if for any and any there exists a such that
Lemma 2.2 (Lemma 5.1.17 in ).
Let be a sequence in narrowly convergent to and let . If is uniformly integrable with respect to the set then
For a fixed , we will denote by the space of continuous paths on . This is a Polish space so that we can use the tools introduced in this section also for probability measures . We call such probability measures generalized flows or also dynamic plans. When we will often use to denote .
Since we will work with homogeneous functions on the cone, we also introduce the space of probability measures , defined by
Then, if , where is the cone over the compact domain , it is easy to verify that any -homogeneous function on is -integrable.
Finally, we will denote by the Lebesgue measure on normalized so that .
3. The variational formulation on the cone
In this section we describe the geometric structure of the CH equation using the group of automorphisms of the cone. Such a formulation was introduced in  and it was used to interpret the CH equation as an incompressible Euler equations on the cone. In fact, in itself it is similar to that of the incompressible Euler equations originally considered by Arnold . In this section we will only focus on smooth solutions, but we will later use the variational interpretation presented here to guide the construction of generalized solutions of the CH equation. We will keep the discussion formal at this stage and we will use some standard geometric tools and notation commonly adopted in similar contexts.
Consider a compact smooth domain . For any and , we let be the map defined by . The automorphism group is the collection of such maps, i.e.
The group composition law is given by
the identity element is , where is the identity map on , and the inverse is given by
The tangent space of at is denoted by . This is the set of tangent vectors
where is a curve on defined on an open interval around 0 and satisfying . The tangent space can be identified with the space of vector fields . The collection all the tangent spaces is the tangent bundle .
We endow with the metric inherited from . This is defined as follows: given ,
where is the norm on associated to and is the Lebesgue measure on .
In  the authors found that the CH equation on coincides with the geodesic equation on the subgroup defined as follows:
In other words, the group can be regarded as the configuration space for the CH equation in the same way as the is the configuration space for the incompressible Euler equations, with
In order to see this, we first observe that the metric is right invariant when restricted to , meaning that it does not change when moving on this subgroup by right translations. In particular, for any , consider the right translation map defined by . Its tangent map at is given by
Geodesics on correspond to stationary paths on for the action functional
for a given , where the Lagrangian . The invariance of the metric implies that the geodesic equation can be expressed in terms of right trivialized (Eulerian) velocities only, or in other words in terms of the variables
Now, the constraint can be rewritten as . Moreover, we have that for any ,
Hence the constraint becomes in terms of Eulerian variables. Moreover, by right invariance,
which is the Lagrangian for the CH equation. Note that the coefficient is directly related to the choice of as cone metric. Using different coefficients in we can obtain the general form of the Lagrangian in equation (1.1).
In order to compute the geodesic equation we consider the following augmented Lagrangian
where is the Lagrange multiplier enforcing the constraint. Taking variations we obtain
Hence the Euler-Lagrange equations associated with read as follows
which can be expressed in terms of via right trivialization, yielding
Using the relation , finally gives us the CH equation for .
Note that in the literature for the CH equation the “pressure field” is sometimes defined in a different way so that, when is one-dimensional, the first equation in (3.17) can be written as
for an appropriate function (see, e.g., ). Throughout the paper we will instead intend by pressure the Lagrange multiplier considered above.
4. The generalized CH formulation
In view of the interpretation of the CH equation as geodesic flow on , we now turn our attention to the following minimization problem:
Problem 4.1 (Deterministic CH flow problem).
Given a diffemorphism , find a curve satisfying
and minimizing the action in equation (3.10).
There is a remarkable analogy between this problem and the equivalent version for the incompressible Euler equations. This raises the question of whether we can define generalized solutions for this problem in the same way Brenier did for the Euler case. In this section we address this question by formulating the generalized CH flow problem, proving existence of solutions and discussing their nature. In the following the Lebesgue measure on the base space is renormalized in such a way that .
By generalized flow or dynamic plan we mean a probability measure on the space of continuous paths of the cone . This is a generalization for curves on the automorphism group since for any smooth curve , we can associate the generalized flow defined by
More explicitly, for any Borel functional ,
The condition is equivalent to requiring . We want to generalize this condition for arbitrary . Let be the evaluation map at time . Then, if is defined as in (4.2), we have
In fact, for any ,
where for any path and any time , and . By similar calculations, we also obtain
In other words, enforcing the boundary conditions in the generalized setting boils down to constraining a certain marginal of to coincide with a given coupling plan on the cone, i.e. a probability measure in .
Consider now the energy functional defined by
Setting in (4.3) we obtain the action for the CH equation expressed in Lagrangian coordinates. This motivates the following definition for the generalized CH flow problem.
Problem 4.2 (Generalized CH flow problem).
Given a coupling plan on the cone , find the dynamic plan satisfying: the homogeneous coupling constraint
for all 2-homogeneous continuous functions ; the homogeneous marginal constraint
and minimizing the action
We remark three basic facts on this formulation:
we allow for general coupling plans in so that the integral on the right-hand side of equation (4.8) is finite. However, we will mostly be interested in the case where the coupling is deterministic.
The first of the points above is made explicit in the following lemma, whose proof is postponed to the appendix.
The main result of this section is contained in the following proposition, which states that generalized CH flows are well-defined as solutions of problem (4.2).
Proposition 4.4 (Existence of minimizers).
Provided that there exists a dynamic plan such that , the minimum of the action in problem 4.2 is attained.
Before providing the proof of proposition 4.4, we introduce a useful rescaling operation which will allow us to preserve the homogenous constraint when passing to the limit using sequences of narrowly convergent dynamic plans. Such an operation was introduced in  in order to deal with the analogous problem arising from the formulation of entropic optimal transport on the cone. Adapting the notation in  to our setting, we define for a functional ,
Then, given a dynamic plan , if for -almost any path , we can define the dilation map
Since the constraint in equation (4.9) is 2-homogeneous in the radial coordinate , it is invariant under the dilation map, meaning that if satisfies (4.9), also does. For the same reason, we also have
The map performs a rescaling on the measure in the sense specified by the following lemma.
Given a measure and a 1-homogeneous functional such that for -almost every path , suppose that
if then and
We prove this by direct calculation. Let . By 1-homogeneity of , for -almost every path
By similar calculations we also have . ∎
The set of measures with uniformly bounded action and satisfying the homogeneous constraint in equation (4.9) is relatively sequentially compact for the narrow topology.
Due to Therorem 2.1, it is sufficient to prove that sequences of admissible measures are tight. For a given path with , for all ,
which implies that level sets of are equicontinuous. Consider now the set
For any , the set is also equicontinuous; moreover, since paths in this set are bounded at any time, it is contained in a compact subset of , by the Ascoli-Arzelà theorem.
In order to use such sets to prove tightness we need to be able to control the measure of . In particular, we now show that there exists a constant such that
Let us fix a , and an interval . Moreover, consider the following set of paths
Then integrating the constraint in equation (4.9) over such a set with being any continuous function such that for , for and , we obtain
Since the estimate is uniform in this means that
Now, consider the set because of equation (4.19) we have
This implies that if is sufficiently small
More precisely this holds for
and hence for . Therefore, if ,