Interpreting systems of continuity equations in spaces of probability measures through PDE duality

Abstract

We introduce a notion of duality solution for a single or a system of transport equations in spaces of probability measures reminiscent of the viscosity solution notion for nonlinear parabolic equations. Our notion of solution by duality is, under suitable assumptions, equivalent to gradient flow solutions in case the single/system of equations has this structure. In contrast, we can deal with a quite general system of nonlinear non-local, diffusive or not, system of PDEs without any variational structure.

1 Introduction

Many PDE modelling instances of applied analysis lead to transport equations for a density function $\rho$ of the form $\partial _t \rho = \nabla \cdot (\rho \varvec{v})$, where $t> 0$, $x \in {\mathbb {R}^d}$, and $\varvec{v}$ is the velocity field. In many of these applications, there is a ubiquitous choice of convolutional-type velocities depending on the density well motivated by applications in mathematical biology, social sciences and neural networks, see for instance [4, 18, 21, 36, 43, 44, 48]. Moreover, in many of these applications, $\varvec{v}$ is given by a function of $\rho$ itself, and it may happen that the density $\rho$ is not an integrable function, but rather a measure. Since the mathematical treatment of these problems is rather tricky, and indeed many “non-physical” solutions may appear, the natural way to approach these problems is the so-called vanishing-viscosity method. With this method linear diffusion $D \Delta \rho$ is included to the right-hand side, leading to the problem

$$\begin{aligned} \partial _t \rho = \nabla \cdot (\rho \varvec{v}[\rho ]) + D \Delta \rho . \end{aligned}$$

The “physical” or entropic solution is the one obtained in the limit $D \searrow 0$. In this work, we deal with systems of such equations, where several species inhabit a domain. We will consider that the evolution of these species is coupled only by the velocity field. We are interested in systems of n aggregation–diffusion equations of the form

$$\begin{aligned} \partial _t \mu ^i_t = \nabla \cdot \left( \mu ^i_t \varvec{K}^{i}_t [\varvec{\mu }_t] \right) + D^i \Delta \mu _t^i, \qquad i = 1, \cdots , n \,\text { and }\, (t,x) \in (0,\infty ) \times {\mathbb {R}^d}\end{aligned}$$

(1.1)

where $D^i \ge 0$ and $\varvec{K}^i_t$ are velocity-fields depending non-locally on the full vector of densities $\varvec{\mu } = (\mu ^1, \cdots , \mu ^n)$ and possibly in the spatial or time variables, under some hypotheses described below. In fact, we are to work with a generalisation allowing $K_t$ to depend on the previous time states see (P). Notice that we cover the cases with and without diffusion ($D_i>0$ and $D_i =0$) in a unified framework. We will introduce a new notion of solution, which we call dual-viscosity solution, with well-posedness, and that is able to detect this vanishing-viscosity limit.

A particularly interesting case is when the problem (1.1) has a 2-Wasserstein gradient-flow structure. For example, in the scalar setting ($n = 1$) without diffusion ($D=0$) and with an interaction potential energy functional of the form

$$\begin{aligned} \varvec{K} [ \mu ] = \nabla \frac{\delta \mathcal {F}}{\delta \rho } [ \mu ], \qquad \mathcal {F} [\mu ] = \frac{1}{2} \int _{\mathbb {R}^d}\int _{\mathbb {R}^d}W(x-y) \text {d}\mu (x) \text {d}\mu (y) , \end{aligned}$$

the classical gradient flow theory developed in [2, 3] applies when W is $\lambda$-convex and smooth, and this later generalised to different settings [6, 8, 13, 14, 26, 27, 29, 32, 33, 38] for instance. This can also be applied to (1.1) with $D > 0$ by including the Boltzmann entropy in the free energy functional, see [7, 15, 20, 22, 23, 42, 49, 51]. We refer to [19] for a recent survey of results in aggregation–diffusion equations. We show in Sect. 6 that our new notion of dual viscosity solutions is equivalent, in some examples, to the notion of gradient flow solutions.

In many situations, a theory of entropy solutions in the sense of Kruzkov (see [16, 40]) is possible, when we have initial data and solutions in $L^1 ({\mathbb {R}^d}) \cap L^\infty ({\mathbb {R}^d}) \cap BV ({\mathbb {R}^d})$. For example, the work [30] deals with $d = n = 1$, $D= 0$ and $K =\theta (\mu ) W' * \mu$ where $\theta (M) = 0$ for some $M > 0$, and W very regular. Then, clearly initial data $0 \le \rho _0 \le M$ lead to solutions $0 \le \rho _t \le M$. The authors prove uniqueness of entropy solutions in the sense of Kruzkov in the space $L^\infty (0,T; L^1 ({\mathbb {R}^d}) \cap L^\infty ({\mathbb {R}^d}) \cap BV({\mathbb {R}^d}))$, by using the continuous dependence results in [39] and a fixed point argument. This requires fairly strict hypothesis on the regularity of the initial data (namely bounded variation). They also introduce an interesting constructive method based on sums of characteristics of domains evolving in time. A recent extension of [30] to related problems can be seen in [35].

However, in the diffusion-less regime $D = 0$, these models based on transport equations can describe particle systems described by Dirac deltas, which interact through an ODE. The idea of working with Dirac deltas leads to numerical methods for general integrable data by approximating the initial datum by a sum of Diracs and regularizing the entropy [17, 24, 25]. Kruzkov’s notion of entropy solutions cannot be extended to measures. Furthermore, for less regular W (or $\varvec{K} [ \mu ] = \nabla V(x)$ known), the finite-time formation of Dirac deltas even for smooth initial data is not known. The paper [32] studies uniqueness of distributional solutions for $n = 2$, when $D^i = 0$,

$$\begin{aligned} \varvec{K}^{i} [\varvec{\mu }] = \nabla \mathcal {H}_i * \mu ^i + \nabla \mathcal {K}_i * \mu ^j , \qquad \{j \} = \{1,2 \} \setminus \{i\} \end{aligned}$$

(1.2)

where $\mathcal {H}_i, \mathcal {K}_i \in \mathcal {W}^{2,\infty }$. The authors indicate that any distributional solution is a push-forward solution (pointing to [2, Chapter 8]) of the flow that solves

$$\begin{aligned} \frac{\text {d}X^i_t}{\text {d}t} = \varvec{K}^i_t [ \varvec{\mu }_t ] (X^i_t), \qquad X_0^i (y) = y. \end{aligned}$$

(1.3)

This push-forward structure is not available for problems with diffusion. The main goal of this work is to construct a notion of well-posed solutions that works for the case

$$\begin{aligned} \varvec{K}^{i} [\varvec{\mu }] = \nabla \sum _{j=1}^n W_{ij} * \mu ^j, \end{aligned}$$

(1.4)

$d, n \ge 1$, $D \ge 0$, and measure initial data. We will work in 1-Wasserstein space, and recover continuous dependence estimates by discussing the “dual” problem, for which we use viscosity solutions by studying the corresponding Lipschitz estimates. We show also how our results can be adapted to different spaces (e.g., $\dot{H}^{-1}$ and $H^1$).

So far, the velocity field depends only on t and $\varvec{\mu }_t$. We can consider a generalisation of K, denoted $\mathfrak {K}$, that depends also on the past. We denote this extended notation by $\mathfrak {K} [{\overline{\mu }}]_t$. By saying that $\mathfrak {K}$ depends only on the past we mean that $0 \le t \le s \le T$ and $\varvec{\mu } \in C([0,T]; \mathcal {P}_1 ({\mathbb {R}^d})^n)$ then

$$\begin{aligned} (\varvec{{\mathfrak {K}}}^{i}[\varvec{\mu }|_{[0,s]}])_t = (\varvec{{\mathfrak {K}}}^{i}[\varvec{\mu }|_{[0,t]}])_t. \end{aligned}$$

(H)

To simplify the notation, when it will not lead to confusion, we simply write $\mathfrak {K}[\mu ]_t$. For example, we cover the case

$$\begin{aligned} \mathfrak {K}[\mu ]_t = \int _0^t \nabla W_s * \mu _s \text {d}s, \end{aligned}$$

where $W_t$ are a family of convolution kernels. Hence, we extend the (1.1) to the more general

$$\begin{aligned} \partial _t \mu ^i_t = \nabla \cdot \Big ( \mu ^i_t (\varvec{{\mathfrak {K}}}^{i}[\varvec{\mu }])_t \Big ) + D^i \Delta \mu _t^i , \qquad i = 1, \cdots , n \,\text { and }\, (t,x) \in (0,\infty ) \times {\mathbb {R}^d}. \end{aligned}$$

(P)

This covers the previous setting by defining

$$\begin{aligned} (\varvec{\mathfrak {K}}[\varvec{\mu }])^i _t = \varvec{K}_t^i[\varvec{\mu }_t]. \end{aligned}$$

The structure of the paper is as follows. Section 2 is devoted to the main results and methods. We begin in Sect. 2.1 by motivating and introducing a new notion of solution in 1-Wasserstein space, motivated by the duality with viscosity solution of the dual problem, which we call dual viscosity solution. In Sect. 2.2 we present well-posedness results for $\varvec{K}$ regular enough. In Sect. 2.3 we present a result of stability under passage to the limit. In Sect. 2.4 we discuss a limit case of the regularity assumptions of the well-posedness theory, given by the Newtonian potential. We close the presentation of the main results with Sect. 2.5 on open problems our current results do not cover, mainly non-linear diffusion and non-linear mobility. The full proofs are postponed to the next sections. First, in Sect. 3 we prove well-posedness and estimates for the dual problem of (1.1), where we assume $\varvec{K}[\mu ]$ is replaced by a known field. Then, in Sect. 4 we prove well-posedness of (1.1) when $\varvec{K}[\mu ]$ is again replaced by a known field. In Sect. 5 we prove the results stated in Sect. 2.2 by a fixed-point argument based on the previous estimates.

We conclude the paper with two sections on comments and extensions of our main results. First, in Sect. 6, we show that, where a gradient-flow structure is available, our solutions coincide with the steepest descent solutions, under some assumptions. In Sect. 7 we extend our main results from the 1-Wasserstein framework to the negative homogeneous Sobolev space $\dot{H}^{-1}$. We recall that $\dot{H}^{-1}$ is related to the 2-Wasserstein space (see below).

2 Main results and methods

2.1 A new notion of solution via duality

Considering first the case $n = 1$, let $\varvec{E}_t = - \varvec{{\mathfrak {K}}}[\mu ]_t$, so that we are looking at the problem

Furthermore, if we naively assume the equation to be satisfied pointwise, we can multiply by a test function $\varphi$ and assume we can integrate by parts integrate by parts, we recover

$$\begin{aligned} \int _{\mathbb {R}^d}\varphi _T \text {d}\mu _T+ \int _0^T \int _{\mathbb {R}^d}\left( -\frac{\partial \varphi _t}{\partial t} - \varvec{E}_t \nabla \varphi _t - D\Delta \varphi _t \right) \text {d}\mu _t \text {d}t = \int _{\mathbb {R}^d}\varphi _0 \text {d}\mu _0. \end{aligned}$$

(2.1)

To cancel the double integral, we take $\varphi _t = \psi _{T-t}$ and $\psi _s$ a solution of

and $\psi _s$ Lipschitz in x. Using $\varphi _t = \psi _{T-t}$ we reduce (2.1) to the equivalent formulation

This is an interesting formulation because it will allow us to exploit the duality properties of the spaces for $\mu$ and $\psi$.

Hence, we are exploiting the well-known duality between the continuity problem ($\hbox {P}_{\varvec{E}}$) and its dual ($\hbox {P}_{\varvec{E}}^*$).

When $D= 0$, this connects with the push-forward formulation for first order problems. Problem ($\hbox {P}_{\varvec{E}}^*$) can be solved by a generalised characteristics field $X_t$ (e.g., [34, Sect. 3.2, p.97]), and ($\hbox {D}_{\varvec{E}}$) means precisely that $\mu _T$ is the push-forward by the same field (see, e.g., [2, Sect. 5.2, p.118]).

Notice that in the general setting we have n equations, and $\psi$ depends on T. Hence, we denote our test functions $\psi ^{T,i}$. We consider the system of n dual problems

We also recover n duality conditions

Remark 2.1

Notice that we have not introduced a condition as $|x| \rightarrow \infty$ for $\psi$. In the formal derivation of the notion of solution, we have required that we can formally integrate by parts. This is also the standard argument to define distributional solutions. As with distributional solutions, we set ($\hbox {D}_i$) as our definition and prove that this is mathematically sound in terms well-posedness. We do not claim the PDE is satisfied in any stronger sense. We will make a comment on gradient flow solutions. Furthermore, we will not rigorously integrate by parts at any point in this manuscript. We will actually discuss in detail the precise admissible initial data $\psi _0$ for ($\hbox {P}_{\varvec{E}}^*$). When we study solutions in 1-Wasserstein space we will only assume that $\psi$ are Lipschitz (but not bounded). The uniqueness of solutions of ($\hbox {P}_{\varvec{E}}^*$) in that setting is discussed in Sect. 3. In Sect. 7 we extend the results to negative Sobolev spaces, and thus we will use some decay (see Remark 7.3).

Remark 2.2

Notice that the problems for the different $\psi ^i$ are de-coupled from each other.

It is well-known that classical solutions of ($\hbox {P}_{\varvec{E}}^*$) may not exist, specially in the case $D= 0$. The approach developed by Crandall, Ishii and Lions to deal with limit is the notion of viscosity solution (see, e.g., [28]), which we introduce below. Since the theory of viscosity solutions is usually well-posed for Lipschitz functions $\psi$, a natural metric for $\mu$ is the 1-Wasserstein distance (see, e.g., [50, p. 207]). We recall that the space $\mathcal {P}_1 ({\mathbb {R}^d})$ is the space of the probability measures $\mu$ such that

$$\begin{aligned} \int _{\mathbb {R}^d}|x| \text {d}\mu (x) < \infty . \end{aligned}$$

The distance $d_1$ that makes it a complete metric space is usually constructed by a Kantorovich optimal transport problem. The reason we can exploit the duality between ($\hbox {P}_{\varvec{E}}$) and ($\hbox {P}_{\varvec{E}}^*$) is the following well-known duality characterisation (see [50, Theorem 1.14]): if $\mu , {\widehat{\mu }} \in \mathcal {P}_1 ({\mathbb {R}^d})$ then

$$\begin{aligned} d_1 (\mu , {\widehat{\mu }}) = \sup \left\{ \int _{\mathbb {R}^d}\psi \text {d}(\mu - {\widehat{\mu }}) : \psi \in C ({\mathbb {R}^d}) \text { such that }{{\,\textrm{Lip}\,}}(\psi ) \le 1 \right\} , \end{aligned}$$

(2.2)

where, here and below

$$\begin{aligned} {{\,\textrm{Lip}\,}}(\psi ) = \sup _{x \ne y } \frac{ |\psi (x) - \psi (y)| }{|x-y|}. \end{aligned}$$

We recall the notion of viscosity solution.

Definition 2.3

We say that $\psi \in C([0,T] \times {\mathbb {R}^d})$ is a viscosity sub-solution of

$$\begin{aligned} \partial _s \psi _s = \varvec{E}_{T-s} \nabla \psi _s + D\Delta \psi _s \end{aligned}$$

(2.3)

if, for every $z_0 = (s_0, x_0) \in [0,T] \times {\mathbb {R}^d}$ and U neighbourhood of $z_0$ and $\varphi \in C^2(U)$ touching $\psi$ from above (i.e., $\varphi \ge \psi$ on U and $\varphi (z_0) = \psi (z_0)$) we have that

$$\begin{aligned} \frac{\partial \varphi }{\partial s} (z_0) \le \varvec{E}_{T-s_0} (x_0) \nabla \varphi (z_0) + D\Delta \varphi (z_0). \end{aligned}$$

Conversely, we say that $\psi \in C([0,T] \times {\mathbb {R}^d})$ is a super-solution if the inequalities above are reversed for functions touching from below. We say that $\psi \in C([0,T] \times {\mathbb {R}^d})$ is a viscosity solution if it is both a viscosity sub and super solution.

Thus, we introduce the following notion of solution of (1.1):

Definition 2.4

We say that $(\varvec{\mu }, \{\varvec{\Psi }^T\}_{T\ge 0})$ is an entropy pair if:

• 1. For every $T \ge 0$,

  $$\begin{aligned}{} & {} \varvec{\Psi }^T: \{ \psi _0: \nabla \psi _0 \in L^\infty ({\mathbb {R}^d}) \}\\{} & {} \qquad \longrightarrow \left\{ (\psi ^1, \cdots , \psi ^n) \in C([0,T] \times {\mathbb {R}^d})^n: \frac{\psi ^i}{1+|x|} \in L^\infty ((0,T) \times {\mathbb {R}^d}) \right\} \end{aligned}$$

  is a linear map with the following property: for every $\psi _0$ and $i = 1, \cdots , n$ we have $\psi ^{T,i} = \Psi ^{T,i}[\psi _0]$ is a viscosity solution of ($\hbox {P}^*_i$).

• 2. For each $i = 1, \cdots , n$ and $T \ge 0$, $\mu _T^i \in \mathcal {P}_1 ({\mathbb {R}^d})$ and satisfies the duality condition ($\hbox {D}_i$).

Remark 2.5

Notice that in our definition we are not assuming that $\psi _0$ are bounded or decay at infinity, only that they are Lipschitz continuous. This means that they are bounded by $C(1+|x|)$, so they can be integrated against functions in $\mathcal {P}_1({\mathbb {R}^d})$.

For convenience, we will simply denote $\varvec{\Psi } = \{ \varvec{\Psi }^T \}_{T\ge 0}$.

Definition 2.6

We say that $\varvec{\mu }$ is a dual viscosity solution if there exists $\varvec{\Psi }$ so that $(\varvec{\mu }, \varvec{\Psi })$ is an entropy pair.

These definitions can be extended to other norms where a duality characterisation exists. The 2-Wasserstein distance, which is natural for many problems because of the gradient-flow structure, does not have any known such characterisation. However, other spaces like $\dot{H}^{-1}$ do. We present extension of our results to this setting in Sect. 7.

Remark 2.7

Notice that this duality characterisation is somewhat in the spirit of the Benamou–Brenier formula for the 2-Wasserstein distance between two measures, where the optimality conditions involve viscosity solutions of a Hamilton–Jacobi equation.

Remark 2.8

Notice that we have not requested the time continuity of $\mu$, so the notion of initial trace is not clear. However, it will follow from the continuity of $\psi ^T$. Under some assumptions on $\varvec{{\mathfrak {K}}}$ we will show that $\mu \in C([0,T]; \mathcal {P}_1 ({\mathbb {R}^d}))$. However, in more general settings, the definition guarantees that the initial trace is satisfied in the weak-$\star$ sense. Notice that if $\psi _0 \in C_c$ and $\psi _t$ is continuous in time then

$$\begin{aligned} \int _{\mathbb {R}^d}\psi _0 \text {d}\mu _t = \int _{\mathbb {R}^d}\psi _t \text {d}\mu _0 \rightarrow \int _{\mathbb {R}^d}\psi _0 \text {d}\mu _0. \end{aligned}$$

Remark 2.9

(General linear diffusions). Our definitions, methods, and results can be extended to the case where $D\Delta u$ is replaced by any other linear operator that commutes with $\partial _x$, and that satisfies the maximum principle, e.g., the fractional Laplacian $D(-\Delta )^s$.

2.2 Well-posedness when $\varvec{K}$ is regularising

So far, $\varvec{E}_t = -\varvec{K}_t [\varvec{\mu }_t]$, i.e., we assume that at every time t the convection comes from a function $\varvec{K}_t : \varvec{\mu }\rightarrow \varvec{E}$. This covers, for example, non-local operators in space. However, our arguments extend to much broader arguments, where $\varvec{K}$ could be non-local in time as well. We will consider the more general case of

$$\begin{aligned} \varvec{\mathfrak {K}} : C([0,t];X ) \rightarrow C([0,t] ; Y ), \qquad \forall t \in [0,T]. \end{aligned}$$

The results in this section correspond to $X = \mathcal {P}_1({\mathbb {R}^d})^n$ and Y the space

$$\begin{aligned} {Lip}_0 ({\mathbb {R}^d}; {\mathbb {R}^d}) = \left\{ \varvec{E} \in C_{loc} ({\mathbb {R}^d}; {\mathbb {R}^d}) : |\nabla \varvec{E}| \in L^\infty ({\mathbb {R}^d}) \right\} \end{aligned}$$

(2.4)

However, the scheme of the proof is general and can be extended to other X, Y. In Sect. 7 we work on the negative homogeneous Sobolev space.

Notice that ${Lip}_0 ({\mathbb {R}^d}; {\mathbb {R}^d})$ is very similar $\dot{W}^{1,\infty }$, but we avoid this terminology to endow it with the following special norm

$$\begin{aligned} \Vert \varvec{E} \Vert _{{Lip}_0} = \Vert \nabla \varvec{E} \Vert _{L^\infty } + \left\| \frac{\varvec{E}}{1 + |x|} \right\| _{L^\infty }. \end{aligned}$$

The reason for the ${Lip}_0$ nomenclature is that

$$\begin{aligned} |\varvec{E}(0)| + \Vert \nabla \varvec{E}\Vert _{L^\infty } \le \Vert \varvec{E} \Vert _{{Lip}_0} \le 2 (|\varvec{E}(0)| + \Vert \nabla \varvec{E}\Vert _{L^\infty }), \end{aligned}$$

so it guarantees the bound of $\varvec{E}(0)$. It is not difficult to see that this is a Banach space. In this setting, we prove the following well-posedness result

Theorem 2.10

Let $D\ge 0$, $T > 0$, and assume that for all $t > 0$

$$\begin{aligned} \varvec{\mathfrak {K}}^i : C([0,t]; \mathcal {P}_1({\mathbb {R}^d})^n) \rightarrow C([0,t]; {Lip}_0 ({\mathbb {R}^d}; {\mathbb {R}^d})), \end{aligned}$$

(2.5)

with the property (H), maps bounded sets into bounded sets, and satisfies the following Lipschitz condition

$$\begin{aligned} \sup _{\begin{array}{c} t \in [0,T]\\ i=1,\cdots ,n \end{array}} \left\| \frac{\varvec{\mathfrak {K}}^i[\varvec{\mu }]_t - \varvec{\mathfrak {K}}^i[ \widehat{\varvec{\mu }}]_t}{1 +|x|} \right\| _{L^\infty ({\mathbb {R}^d})} \le L \, \sup _{\begin{array}{c} t \in [0,T]\\ i=1,\cdots ,n \end{array}} d_1 (\mu _t^i, {\widehat{\mu }} _t^i ). \end{aligned}$$

(2.6)

Then, for each $\varvec{\mu }_0 \in \mathcal {P}_1 ({\mathbb {R}^d})^n$ there exists a unique dual viscosity solution of (P), $\mu \in C([0,T]; \mathcal {P}_1({\mathbb {R}^d})^n)$, and it depends continuously on the initial datum with respect to the $d_1$ distance. This solution $\mu$ also satisfies (P) in distributional sense. Furthermore, if $\varvec{\mathfrak {K}}^i[\varvec{\mu }]_t = K^i[\varvec{\mu }_t]$, then the map $S_T : \varvec{\mu }_0 \in \mathcal {P}_1 ({\mathbb {R}^d})^n \mapsto \varvec{\mu }_T \in \mathcal {P}_1({\mathbb {R}^d})^n$ is a continuous semigroup.

Going back to (2.2) we have that

$$\begin{aligned} \left| \int _{\mathbb {R}^d}\psi \text {d}(\mu - {\widehat{\mu }}) \right| \le d_1 (\mu , {\widehat{\mu }}) {{\,\textrm{Lip}\,}}(\psi ), \qquad \forall \psi \text { such that } {{\,\textrm{Lip}\,}}(\psi ) < \infty . \end{aligned}$$

Thus, we can use the duality relation ($\hbox {D}_i$) to get the following upper bound

$$\begin{aligned} d_1 (\mu _T, {\widehat{\mu }}_T)&= \sup _{\begin{array}{c} {{\,\textrm{Lip}\,}}(\psi _0) \le 1 \end{array} } \int _{\mathbb {R}^d}\psi _0 \text {d}(\mu _T - {\widehat{\mu }}_T) \nonumber \\&= \sup _{{{\,\textrm{Lip}\,}}(\psi _0) \le 1} \left( \int _{\mathbb {R}^d}\psi _T^T \text {d}\mu _0 - \int _{\mathbb {R}^d}{\widehat{\psi }}_T^T \text {d}{\widehat{\mu }}_0 \right) \nonumber \\&= \sup _{{{\,\textrm{Lip}\,}}(\psi _0) \le 1} \left( \int _{\mathbb {R}^d}\psi _T^T \text {d}(\mu _0 - {\widehat{\mu }}_0) + \int _{\mathbb {R}^d}( \psi _T^T - {\widehat{\psi }}_T^T ) \text {d}{\widehat{\mu }}_0 \right) \nonumber \\&\le d_1(\mu _0 , {\widehat{\mu }}_0) \sup _{{{\,\textrm{Lip}\,}}(\psi _0) \le 1} {{\,\textrm{Lip}\,}}(\psi _T^T) + \sup _{{{\,\textrm{Lip}\,}}(\psi _0) \le 1} \int _{\mathbb {R}^d}\bigg ( \psi _T^T - {\widehat{\psi }}_T^T \bigg ) \text {d}{\widehat{\mu }}_0. \end{aligned}$$

(2.7)

To exploit this fact we take advantage of the following, rather obvious, estimate

$$\begin{aligned} \left| \int \psi (x) \text {d}\mu (x) \right|&\le \left\| \frac{\psi }{1+|x|} \right\| _{L^\infty } \left( 1 + \int _{\mathbb {R}^d}|x| \text {d}\mu (x) \right) . \end{aligned}$$

In fact, we show that, on the second supremum in (2.7), we can focus on functions such that $\psi _0(0) = 0$. The reason for this election will be made clearer later.

Lemma 2.11

Let $\mu$ and ${\widehat{\mu }}$ two dual viscosity solutions of ($\hbox {P}_{\varvec{E}}$) such that $\varvec{E}, \widehat{\varvec{E}} \in C([0,T];{{\,\textrm{Lip}\,}}_0 ({\mathbb {R}^d}, {\mathbb {R}^d}))$. Then, letting $\psi ^T$ and ${\widehat{\psi }}^T$ be the corresponding solutions of the dual problems ($\hbox {P}_{\varvec{E}}^*$), we have that

$$\begin{aligned} d_1 (\mu _T, {\widehat{\mu }}_T)\le & {} d_1(\mu _0 , {\widehat{\mu }}_0) \sup _{\begin{array}{c} {{\,\textrm{Lip}\,}}(\psi _0) \le 1 \end{array}} {{\,\textrm{Lip}\,}}(\psi _T^T) \nonumber \\ {}{} & {} + \int _{\mathbb {R}^d}(1 + |x| ) \text {d}{\widehat{\mu }}_0(x) \sup _{ \begin{array}{c} {{\,\textrm{Lip}\,}}(\psi _0) \le 1 \\ \psi _0(0) = 0 \end{array}} \left\| \frac{\psi _T^T - {\widehat{\psi }}_T^T}{{1+|x|}} \right\| _{L^\infty } . \end{aligned}$$

(2.8)

Proof

Notice that $\psi ^T = \Psi ^T[\psi _0]$, by linearity

$$\begin{aligned} \Psi ^T[\psi _0] = \Psi ^T[\psi _0 - \psi _0(0)] + \psi _0(0) \Psi ^T [1] \end{aligned}$$

By the uniqueness in Proposition 3.1, which we will prove below, since the problem does not contain a zero-order term $\Psi ^T [1] = 1 ={\widehat{\Psi }}^T[1]$. Therefore, we can write

$$\begin{aligned} \Psi ^T[\psi _0] - {\widehat{\Psi }}^T[\psi _0] = \Psi ^T[\psi _0 - \psi _0(0)] - {\widehat{\Psi }}^T[\psi _0 - \psi _0(0)]. \end{aligned}$$

Thus, we can take the second supremum of (2.7) exclusively on functions such that $\psi _0(0) = 0$. $\square$

Remark 2.12

We recall a basic property of the 1-Wasserstein distance. It turns out that the first moment is precisely the distance to the Dirac delta, i.e.,

$$\begin{aligned} \int _{\mathbb {R}^d}|x| \text {d}\mu (x) = d_1 (\mu , \delta _0). \end{aligned}$$

Hence, our fixed point argument will be performed in the balls

$$\begin{aligned} B_{\mathcal {P}_1} (R) = \left\{ \mu \in \mathcal {P}_1 ({\mathbb {R}^d}) : \int _{\mathbb {R}^d}|x| \text {d}\mu (x) \le R \right\} . \end{aligned}$$

(2.9)

This explains that it is natural to work with the metric of $(1 + |x|) L^\infty$.

We will devote Sect. 3 to the estimates of $\psi$ that allow us to control each term of (2.8). The estimates are recovered by standard, albeit involved, arguments.

The first term in (2.8), which corresponds to continuous dependence respect to the initial datum, requires only the estimation of a coefficient. If $\varvec{\mathfrak {K}}$ is well-behaved, we expect to obtain an estimate of the type

$$\begin{aligned} \sup _{{{\,\textrm{Lip}\,}}(\psi _0) \le 1} {{\,\textrm{Lip}\,}}(\psi _T^T) \le C(T, \varvec{\mathfrak {K}}, \mu _0 ). \end{aligned}$$

For the well-posedness theorem ${\widehat{\mu }}_0 = \mu _0$, we can avoid the first term in (2.8) and this constant is not relevant.

The second term in (2.8), which represents continuous dependence on $\varvec{E}$ for the dual problem, is harder to estimate. We prove first that even if $\psi _0 \notin L^\infty$ but is Lipschitz and $\varvec{E}, \varvec{\widehat{E}}$ are bounded, then $\psi _s^T - {\widehat{\psi }}_s^T \in L^\infty ({\mathbb {R}^d})$. A second, more elaborate, argument will allow us to work of the case where $\varvec{E}$ grows no-more-than linearly at infinity. If ($\hbox {P}_{\varvec{E}}^*$) has good continuous dependence on $\varvec{E}$, we can expect

$$\begin{aligned} \left\| \frac{\psi _T^T - {\widehat{\psi }}_T^T}{{1+|x|}} \right\| _{L^\infty } \le C \omega (T) \Vert \varvec{\mathfrak {K}}[ \mu ] - \varvec{\mathfrak {K}} [{\widehat{\mu }} ]\Vert , \end{aligned}$$

where we leave the second norm unspecified for now. So, if $\varvec{\mathfrak {K}}$ is Lipschitz in this suitable norm, then we will be able to use Banach’s fixed point theorem.

Remark 2.13

Karlsen and Risebro [39] prove that if $\rho _0, {\widehat{\rho }}_0 \in L^1 \cap L^\infty \cap BV$ and $\rho , {\widehat{\rho }} \in L^\infty ([0,T]; L^1)$ are entropy solutions of

$$\begin{aligned} \partial _t \rho _t + \nabla \cdot (f(\rho ) \varvec{E}(x)) = \Delta A (\rho ), \qquad \partial _t {\widehat{\rho }} + \nabla \cdot ({\widehat{f}}({\widehat{\rho }}) \varvec{\widehat{E}}(x)) = \Delta A({\widehat{\rho }}) \end{aligned}$$

where $\varvec{E}, \varvec{\widehat{E}}: {\mathbb {R}^d}\rightarrow {\mathbb {R}^d}$, and we have an a priori estimates $\rho , {\widehat{\rho }}: {\mathbb {R}^d}\rightarrow I \subset \mathbb {R}$ with I bounded and

$$\begin{aligned} {[}\rho _t]_{BV({\mathbb {R}^d})} ,[{\widehat{\rho }}_t]_{BV({\mathbb {R}^d})} \le C_{\rho } , \qquad \forall t \in (0,T). \end{aligned}$$

Then,

$$\begin{aligned} \Vert \rho _t - {\widehat{\rho }}_t \Vert _{L^1} \le \Vert \rho _0 - {\widehat{\rho }}_0 \Vert _{L^1} + C t \Big ( \Vert \varvec{E} - \varvec{\widehat{E}} \Vert _{L^\infty } + [ \varvec{E} - \varvec{\widehat{E}} ]_{BV} + \Vert f - {\widehat{f}} \Vert _{W^{1,\infty }(I;\mathbb {R})} \Big ) \end{aligned}$$

where $\rho , {\widehat{\rho }} : {\mathbb {R}^d}\rightarrow I \subset \mathbb {R}$ and C depends on $C_\rho$ and the norms of $f,{\widehat{f}} \in W^{1,\infty }$, $\varvec{E}, \varvec{\widehat{E}} \in L^\infty \cap BV$. In [30] the authors use the fixed-point argument of entropy solutions only to prove uniqueness. Existence is done by proving the convergence of the particle systems, which is their main aim. They use [39] which deals directly with the conservation law in $L^1$, we obtain simple estimates for the Hamilton–Jacobi dual problem in $L^\infty$ using the maximum principle. Our duality argument is not directly extensible to $L^1 ({\mathbb {R}^d})$. Applying the duality characterisation of $L^1$ and ($\hbox {D}_i$) we can compute

$$\begin{aligned} \Vert \mu _T - {\widehat{\mu }}_T \Vert _{L^1}&= \sup _{|\psi _0| \le 1} \int _{\mathbb {R}^d}\psi _0 \text {d}(\mu - {\widehat{\mu }}) \le \sup _{|\psi _0| \le 1} \left( \int _{\mathbb {R}^d}\psi _T^T \text {d}(\mu _0 - {\widehat{\mu }}_0) \right. \\ {}&\left. \quad + \int _{\mathbb {R}^d}( \psi _T^T - {\widehat{\psi }}_T^T ) \text {d}{\widehat{\mu }}_0 \right) \end{aligned}$$

Due to the maximum principle we will show that $|\psi _s^T| \le 1$ and hence

$$\begin{aligned} \Vert \mu _T - {\widehat{\mu }}_T \Vert _{L^1} \le \Vert \mu _0 - {\widehat{\mu }}_0 \Vert _{L^1} + \Vert {\widehat{\mu }}_0 \Vert _{L^1} \sup _{|\psi _0| \le 1} |\psi _T^T - {\widehat{\psi }}_T^T|. \end{aligned}$$

There is no apparent way to bound the second supremum with a quantity related only to $|\varvec{E} - \varvec{\widehat{E}}|$. A quantity depending on $|\nabla \psi _0|$ appears, and there is no available bound of its supremum over functions such that $|\psi _0| \le 1$.

2.3 Stability theorem

We conclude the main results by stating a stability theorem. This allows to prove existence of solutions by approximation, in more general settings that the well-posedness theory. We will use it below for several applications.

Theorem 2.14

Let $(\varvec{\mu }^k, \varvec{\Psi }^k)$ be a sequence of entropy pairs in the sense of Definition 2.4 corresponding to some operators $\varvec{{\mathfrak {K}}}^k$ under the assumptions of Theorem 2.10. Assume that

• 1. $D^k \rightarrow D^\infty$.

• 2. For every $t> 0$, $\varvec{\mu }_t^k \overset{\star }{\rightharpoonup }\varvec{\mu }_t^\infty$ in $\mathcal {M} ({\mathbb {R}^d})$.

• 3. For every $\psi _0 \in C_c^\infty ({\mathbb {R}^d})$ and $T > 0$ we have $\varvec{\Psi }^{T,k} [\psi _0] \rightarrow \varvec{\Psi }^{T,\infty }[ \psi _0 ]$ uniformly over compacts of $[0,T] \times {\mathbb {R}^d}$. This allows to pass to the limit in estimate (3.3) below.

• 4. $\varvec{{\mathfrak {K}}}^k [\varvec{\mu }^k] \rightarrow \varvec{{\mathfrak {K}}}^\infty [\varvec{\mu }^\infty ]$ uniformly over compacts of $[0,T] \times {\mathbb {R}^d}$.

• 5. For every $\psi _0 \in C_c^\infty ({\mathbb {R}^d})$ we have that

  $$\begin{aligned} \int _{\mathbb {R}^d}\varvec{\Psi }^{T,k}[\psi _0]_T \text {d}\varvec{\mu }_0^k \rightarrow \int _{\mathbb {R}^d}\varvec{\Psi }^{T,\infty }[\psi _0]_T \text {d}\varvec{\mu }_0^\infty . \end{aligned}$$

Then $(\varvec{\mu }^\infty , \varvec{\Psi }^\infty )$ is an entropy pair of the limit problem.

The result follows directly from the definition and the following classical result of stability of viscosity solutions (see, e.g., [28]).

Theorem 2.15

Let $D^k \ge 0$, $\psi ^k \in C([0,T] \times {\mathbb {R}^d})$, $\varvec{E}^k \in C([0,T]\times {\mathbb {R}^d})^d$ be viscosity solutions of

$$\begin{aligned} \partial _s \psi _s^k = \varvec{E}_{T-s}^k \nabla \psi _s^k + D^k \Delta \psi _s^k \end{aligned}$$

and assume that $D^k \rightarrow D^\infty$ and $\psi ^k \rightarrow \psi ^\infty$, $\varvec{E}^k \rightarrow \varvec{E}^\infty$ uniformly over compacts of $[0,T] \times {\mathbb {R}^d}$. Then, $\psi ^\infty$ is a viscosity solution of

$$\begin{aligned} \partial _s \psi _s^\infty = \varvec{E}_{T-s}^\infty \nabla \psi _s^\infty + D^\infty \Delta \psi _s^k. \end{aligned}$$

2.4 The limits of the theory: the diffusive Newtonian potential in $d=1$

Let us consider the case $n = d = 1$ with K coming from the gradient-flow structure (1.4) and the Newtonian potential $W(x) = -|x|$. Then $\nabla W(x) = {{\,\textrm{sign}\,}}(x)$. This W falls outside the theory developed in Theorem 2.10, but can be approximated by solutions in this framework. It was proved in [11] that for $\mu _0 = \delta _0$ then the gradient flow solution is not given by particles but rather

$$\begin{aligned} \mu _t = \frac{1}{2t} \chi _{[-t,t]}. \end{aligned}$$

Naturally, any smooth approximation of W (for example $W_k = -(x^2 + \frac{1}{k})^{\frac{1}{2}}$) yields $\mu ^k_t = \delta _0$. However, solving with $\mu _0^m = \frac{1}{2m} \chi _{[-m,m]}$ gives the expected reasonable solution. We therefore have the following diagram

Clearly, $\mu _t^k \rightarrow \delta _0$ in any conceivable topology. Checking whether $\delta _0$ is or is not a distributional solution for W is not totally trivial. Notice that ${{\,\textrm{sign}\,}}$ (as a distributional derivative of |x|) is not pointwise defined at 0, so the meaning of $\int \varphi {{\,\textrm{sign}\,}}* \delta _0 \text {d}\delta _0$ is not completely clear. With the choice ${{\,\textrm{sign}\,}}(0) = 0$ one could be convinced that $\mu _t = \delta _0$ is a distributional solution. But this seems arbitrary.

In fact, if one takes the approximating entropy pairs $(\mu ^k, \Psi ^k)$, it is not difficult to show that a discontinuity appears in $\psi _s^{k,T}$ as $k \rightarrow \infty$. Hence, we do not have uniform convergence of $\psi ^{k,T}$. To show this fact, let $E^k = -\partial _x (W_k * \mu ^k)$. Since $\mu ^k = \delta _0$, we have that

$$\begin{aligned} E^k _t (x) = -\partial _x W_k = \frac{x}{\left( x^2 + \frac{1}{k}\right) ^{\frac{1}{2}}}. \end{aligned}$$

Since $E^k$ does not depend on t, $\psi ^{k,T}$ does not depend on T. Therefore, we drop the T to simplify the notation. We have to solve the equation

$$\begin{aligned} \partial _t \psi ^k_t (y) = \frac{y}{\left( y^2 + \frac{1}{k}\right) ^{\frac{1}{2}}} \partial _y \psi ^k_t. \end{aligned}$$

This equation can be solved by characteristics $\psi ^k_t (Y^k_t(y_0)) = \psi _0 (y_0)$ given by

$$\begin{aligned} \partial _t Y^k_t = - \frac{Y^k_t}{\left( Y_t^2 + \frac{1}{k}\right) ^{\frac{1}{2}}} \end{aligned}$$

As $k \rightarrow \infty$ we recover $Y^{k} _t (y_0) \rightarrow Y_t(y_0) = y_0 - {{\,\textrm{sign}\,}}(y_0) t$ or, equivalently, that

$$\begin{aligned} \psi ^k_t (y) \rightarrow \psi _t (y) = \psi _0 ( y + {{\,\textrm{sign}\,}}(y) t ). \end{aligned}$$

Therefore, $\psi _t(0^-) = \psi _0 (-t)$ and $\psi _t(0^+) = \psi _0 (t)$. Thus, $\psi _t$ is not continuous in general. So we cannot pass to the limit $\mu ^k \rightarrow \mu$ in terms of entropy pairs.

The gradient flow solution is recovered by the entropy pairs if one approximates $\delta _0$ by $\mu _0^m = \frac{1}{2m} \chi _{[-m,m]}$ and passes to the limit. It is also not difficult to show that the entropy pairs of twice regularised problem, $(\mu ^{k,m}, \Psi ^{k,m})$ do converge in the sense of entropy pairs as $k \rightarrow \infty$ to $(\mu ^m, \Psi ^m)$. And then we can also pass to the limit in the sense of entropy pairs as $m \rightarrow \infty$.

Thus, the stable semigroup solutions are entropy pairs, and this notion of solution is powerful enough to discard the wrong order of the limits.

2.5 Open problems

We finally briefly discuss the key difficulties for some related problems that our duality theory cannot cover.

2.5.1 Nonlinear diffusion

We could deal, in general, with problems of the form

$$\begin{aligned} \partial _t \mu _t^i = \nabla \cdot (\mu ^i \varvec{{\mathfrak {K}}}^i [\varvec{\mu }] ) + \nabla \cdot (\varvec{M}^i_t[ \mu ^i _t] \nabla \mu _t^i ). \end{aligned}$$

(2.10)

Going back to (2.10), the dual problem can be written in the form

$$\begin{aligned} \frac{\partial \psi _s}{\partial s} = \varvec{E}_{T-s} \nabla \psi _s + \nabla \cdot ( \varvec{A}_s \nabla \psi _s ) . \end{aligned}$$

We have not been able to estimates of $\Vert \nabla ( \psi - {\overline{\psi }}) \Vert _{L^\infty }$ of $\Vert \psi - {\overline{\psi }} \Vert _{H^1}$ with respect to $E - {\overline{E}}$ and $\varvec{A} - \overline{\varvec{A}}$ using the techniques below. The main difficulty is that letting $v = \frac{\partial \psi }{\partial x_i}$ and ${\overline{v}} = \frac{\partial {\overline{\psi }}}{\partial x_i}$, the reasonable extension of (3.11) contains a term $-\nabla \cdot ( (\varvec{A} - \overline{\varvec{A}}) \nabla {\overline{v}})$. Controlling these terms requires estimates of $D^2 \psi$, which are not in principle present for the initial data we discuss. However, smarter estimates and choices of space for $\mu$ and $\psi$ may lead to a successful extension of our results.

2.5.2 Coupling in the dual problem

Another possible extension is the analysis of more general systems of the form

$$\begin{aligned} \frac{\partial \mu ^i}{\partial t} = \sum _{j=1}^n \nabla \cdot \left( \mu ^j \varvec{{\mathfrak {K}}}^{ij} [\varvec{\mu }]\right) = \sum _{j,k=1}^n \frac{\partial }{\partial x_k} \left( \mu ^j \varvec{{\mathfrak {K}}}^{ijk}\right) , \qquad i = 1, \cdots , n. \end{aligned}$$

(2.11)

The dual problem is now not decoupled, and we have to solve it a system

$$\begin{aligned} \frac{\partial }{\partial t} \varvec{\psi }_s^T = \nabla \varvec{\psi }_s^T \cdot \varvec{{\mathfrak {K}}}^* [\varvec{\mu }_{T-s}] . \end{aligned}$$

This kind of problems does not have a comparison principle in general, for example

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t u = \partial _x v, \\ \partial _t v = \partial _x u \end{array}\right. \end{aligned}$$

leads to the wave equation $\partial _{tt} u =\partial _{t} (\partial _x v) = \partial _x \partial _t v = \partial _{xx} u$. Some examples of equations resembling this structure appear in mathematical biology models, e.g., [47].

2.5.3 Problems with saturation

Some authors have studied the case $\varvec{{\mathfrak {K}}}_i [\varvec{\mu }] = \frac{\theta (\mu )}{\mu } H_i [\varvec{\mu }]$ where $\theta (0) = \theta (M) = 0$. This problem was studied for $n = 1$ and $D = 0$ by [30] using a fixed point argument with the estimates by [39]. In this setting the solutions with initial datum $0 \le \rho _0 \le M$ remain bounded. Therefore, we expect bounded (even continuous) solutions $\rho$. So the natural duality would be $\psi$ integrable. We point the reader to [1], where the authors study the duality between $L^\infty$ estimates for entropy solutions of conservation laws and $L^1$ bounds for viscosity solutions of Hamilton–Jacobi equations.

3 Viscosity solutions of ($\hbox {P}_{\varvec{E}}^*$)

The aim of this section is to prove the following existence, uniqueness, regularity and continuous dependence result for ($\hbox {P}_{\varvec{E}}^*$) which is one of the key tools in this paper. The usefulness of the estimates obtained is clear going back to Lemma 2.11.

Proposition 3.1

Let $D \ge 0$, $\varvec{E} \in C([0,T]; {{\,\textrm{Lip}\,}}_0 ({\mathbb {R}^d}))$ and $\psi _0$ be such that $\nabla \psi _0 \in L^\infty ({\mathbb {R}^d})$. Then, there exists a unique viscosity solution of ($\hbox {P}_{\varvec{E}}^*$) such that $\frac{ \psi }{ 1 + |x| } \in L^\infty ((0,T)\times {\mathbb {R}^d})$. Furthermore, it satisfies the following estimates

$$\begin{aligned} \Vert \psi _s \Vert _{L^\infty }&\le \Vert \psi _0 \Vert _{L^\infty }, \end{aligned}$$

(3.1)

$$\begin{aligned} \left\| \!\left| \nabla \right\| \!\right| \psi _s \left\| \!\left| _{L^\infty }\right\| \!\right|&\le \left\| \!\left| \nabla \right\| \!\right| \psi _0 \left\| \!\left| _{L^\infty }\right\| \!\right| \exp \left( C { \int _0^T \left\| \nabla \varvec{E}_{T-s }\right\| _{L^\infty } \text {d}\sigma } \right) \end{aligned}$$

(3.2)

$$\begin{aligned} \left\| \frac{\psi _s}{1 + |x|} \right\| _{L^\infty }&\le C \left\| \frac{\psi _0}{1 + |x|} \right\| _{L^\infty } \exp \left( DT + \int _0^T \left\| \frac{\varvec{E}_{T-\sigma } }{1 + |x|} \right\| _{L^\infty } \text {d}\sigma \right) , \end{aligned}$$

(3.3)

where $C = C(d)$ depends only on the dimension, and $\left\| \!\left| \nabla \right\| \!\right| \psi _s \left\| \!\left| _{L^\infty }\right\| \!\right| = \sup _i \Vert \tfrac{\partial \psi }{\partial x_i}\Vert _{L^\infty }$. If $\psi _0 \ge 0$ then $\psi \ge 0$. Moreover, we obtain the following time-regularity estimate

$$\begin{aligned} \left\| \frac{\psi _{s+h} - \psi _s}{1 + |x|} \right\| _{L^\infty } \le C \Bigg ( h + D^{\frac{1}{2}} \left( h^{\frac{1}{2}} + h^{\frac{3}{2}}\right) + \sup _{\sigma \in [0,s]} \left\| \frac{\varvec{E}_{T-(\sigma +h)} - \varvec{E}_{T-\sigma }}{1 + |x|} \right\| _{L^\infty } \Bigg ) , \nonumber \\ \end{aligned}$$

(3.4)

where $C = C(T,D, \left\| \psi _0 \right\| _{Lip_0} , \sup _\sigma \Vert \varvec{E}_\sigma \Vert _{Lip_0} )$. Lastly, given ${\widehat{\psi }}_0 = \psi _0$ and $\varvec{\widehat{E}}$ in the same hypotheses above, there exists a corresponding solution of (2.3), denoted by ${\widehat{\psi }}$, and we have the continuous dependence estimate

$$\begin{aligned} \left\| \frac{\psi _s - {\widehat{\psi }}_s}{1 + |x|} \right\| _{L^\infty } \le C \int _0^T \left\| \frac{E_{T-\sigma } - \varvec{\widehat{E}}_{T-\sigma }}{1+|x|} \right\| _{L^\infty } \text {d}\sigma \end{aligned}$$

(3.5)

where $C = C\left( d, D, \Vert \psi _0 \Vert _{Lip_0}, \int _0^T \left\| E_{T-\sigma } \right\| _{Lip_0} \right)$ is monotone non-decreasing in each entry.

We will prove this result in Sect. 3.2. The uniqueness of viscosity solutions leads to the following

Corollary 3.2

Assume that $\varvec{{\mathfrak {K}}}: C([0,T]; \mathcal {P}_1 ({\mathbb {R}^d})) \rightarrow \mathcal {C}([0,T]; {Lip}_0({\mathbb {R}^d}; {\mathbb {R}^d}))$, and let $(\mu , \Psi )$ and $(\mu , {\widehat{\Psi }})$ be entropy pairs of (1.1), then $\Psi = {\widehat{\Psi }}$.

There are some immediate consequences that come from the linearity of equation.

Remark 3.3

Notice that the properties above imply a comparison principle. Given two solutions, we have that $\psi _0 \le {\widehat{\psi }}_0$, then ${\widetilde{\psi }}_0 = {\widehat{\psi }}_0 - \psi _0 \ge 0$. Then, by linearity, uniqueness, and preservation of positivity ${\widehat{\psi }} - \psi = {\widetilde{\psi }} \ge 0$.

Notice that the generic initial datum ${{\,\textrm{Lip}\,}}(\psi _0) \le 1$, need not be bounded. This can initially seem like a problem for uniqueness, since we cannot prescribe conditions at infinity. However, following Aronson [5, Theorem 2 and 3], existence and uniqueness are obtained under the assumption that $e^{-\lambda |x|^2} u_t \in L^2 ((0,T) \times {\mathbb {R}^d})$ if the coefficients are bounded. We tackle this issue by studying weighted versions of our solutions $v_s(x) = \psi _s(x) / \eta (x)$, which solve the following problem

$$\begin{aligned} \partial _s v = v \frac{ E_{T-s} \cdot \nabla \eta + D\Delta \eta }{\eta } + \nabla v \cdot \frac{E_{T-s} +2 D\nabla \eta }{\eta } + D\Delta v. \end{aligned}$$

(3.6)

Notice that, if $\eta (x) = (1 + |x|^2)^{k/2}$ with $k \ge 1$, then $\nabla \eta / \eta \sim |x|^{-1}$ at infinity. If E is Lipschitz, then all the coefficients of the equation above are bounded. And, if $k > 1$, then $|v_0(x)|\le C(1+|x|)^{1-k} \rightarrow 0$ as $|x|\rightarrow \infty$.

Remark 3.4

A similar argument can be adapted to initial data which are weighted with respect to $1 + |x|^p$ for any $p \ge 1$, if this is satisfied by the initial datum. However, this escapes the interest of this work.

3.1 A priori estimates in $L^\infty$

3.1.1 General linear problem

For the dual problem we are interested in the existence and uniqueness of the linear parabolic problem in non-divergence form

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial u_s}{\partial s} = f_s + a_s u_s + \varvec{b}_s \cdot \nabla u_s + D\Delta u_s &{} \quad \text {for all } s> 0, x \in {\mathbb {R}^d}\\ u_s \rightarrow 0 &{} \quad \text {as } |x| \rightarrow \infty \text { for all } s > 0. \end{array}\right. \end{aligned}$$

(3.7)

The theory of existence and uniqueness of classical solutions dates back to [41]. When the coefficients are smooth, the linear problem can be rewritten in divergence form as

$$\begin{aligned} \partial _s u_s = f_s + (a_s - \nabla \cdot \varvec{b}_s) u_s + \nabla \cdot ( u_s \varvec{b}_s + D\nabla u_s ) . \end{aligned}$$

(3.8)

We focus on obtaining suitable a priori estimates assuming that the data and solutions are smooth enough, and these estimates pass to the limit to the unique viscosity solution by approximation of the coefficients.

By studying (3.7) at the point of maximum/minimum, we formally have that

$$\begin{aligned} \frac{d}{ds} \Vert u_s\Vert _{L^\infty } \le \Vert f_s \Vert _{L^\infty } + \Vert a_s\Vert _{L^\infty } \Vert u_s \Vert _{L^\infty } . \end{aligned}$$

This intuition can be made precise by the following result

Lemma 3.5

If $a,f \in C({\mathbb {R}^d})$ and bounded, $\varvec{b} \in C({\mathbb {R}^d})^d$, and u is a classical solution of (3.7) such that $\sup _{(0,T) \times \partial B_R} |u| \rightarrow 0$ as $R \rightarrow \infty$ for each T , then

$$\begin{aligned} \Vert u_s\Vert _{L^\infty } \le \Vert u_0 \Vert _{L^\infty } \exp \left( \int _0^s \Vert a_\sigma \Vert _{L^\infty } \text {d}\sigma \right) + \int _0^s \Vert f_\sigma \Vert _{L^\infty } \exp \left( \int _\sigma ^s \Vert a_\kappa \Vert _{L^\infty } \text {d}\kappa \right) \text {d}\sigma . \nonumber \\ \end{aligned}$$

(3.9)

Proof

Define

$$\begin{aligned} {\overline{u}}_s (x) = \Vert u_0 \Vert _{L^\infty } \exp \left( \int _0^s \Vert a_\sigma \Vert _{L^\infty } \text {d}\sigma \right) + \int _0^s \Vert f_\sigma \Vert _{L^\infty } \exp \left( \int _\sigma ^s \Vert a_\kappa \Vert _{L^\infty } \text {d}\kappa \right) \text {d}\sigma . \end{aligned}$$

This is a classical super-solution of (3.7) and $u_0 \le {\overline{u}}_0$. Using the weak maximum principle on a ball $B_R$ and the initial condition we get

$$\begin{aligned} \min _{(0,T) \times B_R} ({\overline{u}} - u) = \min _{\begin{array}{c} \{0\} \times B_R \\ \cup (0,T) \times \partial B_R \end{array}} ({\overline{u}} - u) \ge \min \left\{ 0, \min _{(0,T) \times \partial B_R} ( {\overline{u}} - u) \right\} . \end{aligned}$$

As $R \rightarrow \infty$ this tends to 0 by hypothesis. We conclude that $u \le {\overline{u}}$. Similarly, we prove that $-u \le {\overline{u}}$. $\square$

Now we compute estimates on the solution formally assuming that $\psi \in C ([0,T]; C_0^1( {\mathbb {R}^d}) )$, and that $\varvec{E}$ is smooth and satisfies all necessary bounds. Later, we will justify this formal computations.

3.1.2 $L^\infty$ estimates of $\psi ^T$, $\nabla \psi ^T$ and $\psi ^T/(1+|x|)$

Going to ($\hbox {P}_{\varvec{E}}^*$) when we have suitable decay of $\psi$ we deduce (3.1) by Lemma 3.5. When we define $U^i_s = \frac{\partial \psi _s^T}{\partial x_i },$ we recover that

$$\begin{aligned} \frac{\partial U^i_s}{\partial s}&= \nabla \psi _s \cdot \frac{\partial \varvec{E}_{T-s}}{\partial x_i} + \nabla U_s^i \cdot \varvec{E}_{T-s} + D\Delta U_s^i \\&= \sum _{j=1}^d U^j_s \frac{\partial E^j_{T-s}}{\partial x_i} + \nabla U_s^i \cdot \varvec{E}_{T-s} + D\Delta U_s^i . \end{aligned}$$

Since this is a system, we cannot directly apply Lemma 3.5. Nevertheless, we compute

$$\begin{aligned} \frac{1}{p} \frac{\text {d}}{\text {d}s} \int _{\mathbb {R}^d}|U_s^i|^p&= \int _{\mathbb {R}^d}|U_s^i|^{p-2} U_s^i \left( \sum _{j=1}^d U^j_s \frac{\partial E^j_{T-s}}{\partial x_i} + \nabla U_s^i \cdot \varvec{E}_{T-s} + D\Delta U \right) \\&= \sum _{j=1}^d \int |U_s^i|^{p-2} U_s^i U^j_s \frac{\partial E^j_{T-s}}{\partial x_i} + \frac{1}{p} \int \nabla |U_s^i|^p \cdot \varvec{E}_{T-s} \\ {}&\quad + D\int |U_s^i|^{p-2} U_s^i \Delta U_s^i \\&= \Vert U_s^i \Vert _{L^p}^{p-1} \sum _{j=1}^d \Vert U_s^j \Vert _{L^p} \left\| \frac{\partial E^j_{T-s}}{\partial x_i} \right\| _{L^\infty } - \frac{1}{p} \int _{\mathbb {R}^d}|U_s^i|^p \nabla \cdot \varvec{E}_{T-s}\\&\quad - (p-1) D\int _{\mathbb {R}^d}|U_s^i|^{p-1} |\nabla U_s^i|^2 \\&\le \Vert U_s^i \Vert _{L^p}^{p-1} \sum _{j=1}^d \Vert U_s^j \Vert _{L^p} \left\| \frac{\partial E^j_{T-s}}{\partial x_i} \right\| _{L^\infty } + \frac{1}{p} \Vert U_s^i \Vert _{L^p}^p \left\| \nabla \cdot \varvec{E}_{T-s} \right\| _{L^\infty } \end{aligned}$$

Using the norm equivalence of norms of ${\mathbb {R}^d}$, we have

$$\begin{aligned} \frac{\text {d}}{\text {d}s} \sum _{i=1}^d \Vert U_s^i \Vert _{L^p}^p \le \left( \sum _{i=1}^d \Vert U_s^i \Vert _{L^p}^p \right) \left( p C(d) \sup _i \sum _{j=1}^d \left\| \frac{\partial E^j_{T-s}}{\partial x_i} \right\| _{L^\infty } + \left\| \nabla \cdot \varvec{E}_{T-s} \right\| _{L^\infty } \right) \end{aligned}$$

Eventually, we have that

$$\begin{aligned} \left( \sum _{i=1}^d \Vert U^i_s \Vert _{L^p}^p \right) ^{\frac{1}{p}}\le & {} \left( \sum _{i=1}^d \Vert U^i _0\Vert _{L^p}^p \right) ^{\frac{1}{p}} \exp \left( C(d) \int _0^s \sup _i \sum _{j=1}^d \left\| \frac{\partial E^j_{T-\sigma }}{\partial x_i} \right\| _{L^\infty } \text {d}\sigma \right. \nonumber \\{} & {} \quad \left. + \frac{1}{p} \int _0^s \left\| \nabla \cdot \varvec{E}_{T-\sigma } \right\| _{L^\infty } \text {d}\sigma \right) . \end{aligned}$$

(3.10)

As $p \rightarrow \infty$, we obtain (3.2). Take $\eta (x) = (1 + |x|^2)^{\frac{1}{2}}$. Recalling (3.6) and applying Lemma 3.5, we deduce

Since $\eta$ is regular we eventually deduce (3.3) where C is such that $C^{-1} \le \frac{\eta (x)}{1+|x|} \le C$, and it does not depend even on the dimension. For the continuous dependence, we write (dropping s and $T-s$ from the subindex for convenience)

$$\begin{aligned} \partial _s (v - {\widehat{v}})&= \,v \frac{\varvec{E} - \varvec{\widehat{E}}}{\eta } \cdot \nabla \eta + \nabla v \cdot \frac{\varvec{E} - \varvec{\widehat{E}} }{\eta } + ({v - {\widehat{v}}}) \frac{ {\widehat{v}} \cdot \nabla \eta + D\Delta \eta }{\eta } \nonumber \\&\quad + \nabla (v - {\widehat{v}}) \cdot \frac{\varvec{\widehat{E}} +2 D\nabla \eta }{\eta } + D\Delta (v - {\widehat{v}}) . \end{aligned}$$

(3.11)

Then we have

$$\begin{aligned} |a_s| = \left| \frac{ {\widehat{v}} \cdot \nabla \eta }{\eta } \right| \le C(1 + |{\widehat{v}}|) \end{aligned}$$

and

$$\begin{aligned} |f_s| = \left| \frac{v}{\eta } \left( ( \varvec{E} - \varvec{\widehat{E}}) \cdot \nabla \eta \right) + \nabla v \cdot \frac{\varvec{E} - \varvec{\widehat{E}}}{\eta } \right| \le \left| \frac{ E- \varvec{\widehat{E}}}{\eta }\right| (|v| + |\nabla v| ). \end{aligned}$$

Notice that

$$\begin{aligned} \nabla v = \nabla \frac{\varphi }{\eta } = \frac{1}{\eta ^2} \left( \eta \nabla \psi - \psi \nabla \eta \right) = \frac{1}{\eta } \nabla \psi + \frac{v}{\eta } \nabla \eta . \end{aligned}$$

Hence, we can deduce using Lemma 3.5 that

$$\begin{aligned} \Vert v_s - {\widehat{v}}_s \Vert _{L^\infty ({\mathbb {R}^d})} \le C_1 \Vert v_0 - {\widehat{v}}_0 \Vert _{L^\infty } + C_2 \int _0^T \left\| \frac{\varvec{E}_{T-\sigma } - \varvec{\widehat{E}}_{T-\sigma }}{1+|x|} \right\| _{L^\infty } \text {d}\sigma \end{aligned}$$

(3.12)

where

$$\begin{aligned} C_1&= C\left( T,D,\left\| \psi / (1 + |x|)\right\| _{L^\infty }, \Vert \varvec{E} \Vert _{L^\infty } \right) , \nonumber \\ C_2&= \sup _{s \in [0,T] } \left( \Vert v_s \Vert _{L^\infty } \Vert \nabla \eta \Vert _{L^\infty } + \Vert \nabla v_s \Vert _{L^\infty } \right) , \end{aligned}$$

(3.13)

which can be estimated using (3.2) and (3.3).

Eventually, since $\psi _0^T - {\widehat{\psi }}_0^T = 0$, we recover (3.5). Notice that $C_1$ cannot be uniformly bounded over the set ${{\,\textrm{Lip}\,}}(\psi _0) \le 1$, where we can bound $C_2$. Here is where the assumptions that $\varvec{{\mathfrak {K}}}_i$ satisfies (2.5) and (2.6) will come into play in Sect. 5.

3.1.3 Time continuity

Taking $v_s = \psi _{s} / \eta ^{(1)}$ and ${\widehat{v}}_s = \psi _{s+h} / \eta ^{(1)}$ where $\eta ^{(1)} = (1 + |x|^2)^{\frac{1}{2}}$, we similarly deduce going back to (3.12) that

$$\begin{aligned} \left\| \frac{\psi _{s+h}- \psi _s}{1 + |x|} \right\| _{L^\infty } \le C_1 \left\| \frac{\psi _{h}- \psi _0}{1 + |x|} \right\| _{L^\infty } + C_2 \sup _{\sigma \in [0,T]} \left\| \frac{\varvec{E}_{T-(\sigma +h)} - \varvec{E}_{T-\sigma }}{1 + |x|} \right\| _{L^\infty } \end{aligned}$$

(3.14)

Therefore, $\psi$ inherits the time continuity of E. Then the only remaining difficulty is the time continuity at 0.

For $D > 0$ we use Duhamel’s formula for the heat equation $u_t - D \Delta u = f$, where we denote the heat kernel $K_D$. Notice that $K_D(t,z) = K_1 (Dt,z)$. For the first term we have that

$$\begin{aligned} \frac{\psi _s(x) - \psi _0(x)}{1+|x|}&=\, \frac{1}{1+|x|} \Bigg ( \int _{\mathbb {R}^d}{K_D(s,x-y) } {\psi _0(y)} \text {d}y - \psi _0(x) \\&\quad + \int _0^s \int _{\mathbb {R}^d}K_D(s - \sigma , x-y) {E_{T-\sigma }(y)} \cdot \nabla \psi _\sigma (y) \text {d}y \text {d}\sigma \Bigg ) \\&=\, \int _{\mathbb {R}^d}{K_D(t,x-y) } \frac{\psi _0(y) - \psi _0(x) }{1+|x|} \text {d}y\\&\quad + \int _0^s \int _{\mathbb {R}^d}K_D(s - \sigma , x-y) \frac{1+|y|}{1+|x|} \frac{E_{T-\sigma }(y)}{1+|y|} \cdot \nabla \psi _\sigma (y) \text {d}y \text {d}\sigma \,. \end{aligned}$$

We estimate as follows

$$\begin{aligned} \left| \int _{\mathbb {R}^d}{K_D(s,x-y) } \frac{\psi _0(y) - \psi _0(x) }{1+|x|} \text {d}y \right|&\le \Vert \nabla \psi _0 \Vert _{L^\infty ({\mathbb {R}^d})} \int _{\mathbb {R}^d}K_D(s,x-y)|x-y| \text {d}y \\&= \Vert \nabla \psi _0 \Vert _{L^\infty ({\mathbb {R}^d})} \int _{\mathbb {R}^d}|z| \frac{e^{-\frac{|z|^2}{4Ds}}}{(4\pi Ds)^{\frac{d}{2}}} \text {d}z \\&= C \Vert \nabla \psi _0 \Vert _{L^\infty ({\mathbb {R}^d})} (4Ds)^{\frac{1}{2}} \int _{\mathbb {R}^d}|z| \frac{e^{-|z|^2}}{\pi ^{\frac{d}{2}}} \text {d}z. \end{aligned}$$

Now since $E(y)/(1+|y|)$ and $\nabla \psi$ are bounded, it leaves to integrate

$$\begin{aligned} \int _0^s \int _{\mathbb {R}^d}K_D(\sigma , x-y) \frac{1+|y|}{1+|x|} \text {d}y \text {d}\sigma&= \int _0^s \int _{\mathbb {R}^d}\frac{1}{(4\pi D \sigma )^{\frac{d}{2}}} e^{-\frac{|z|^2}{4 D \sigma }} \frac{1+|x|+|z|}{1+|x|} \text {d}z \text {d}\sigma \\&= \int _0^s \int _{\mathbb {R}^d}\frac{1}{(4\pi D \sigma )^{\frac{d}{2}}} e^{-\frac{|z|^2}{4 D \sigma }} (1 + |z|) \text {d}z \text {d}\sigma \\&= \int _0^s \int _{\mathbb {R}^d}\frac{1}{(4\pi D \sigma )^{\frac{d}{2}}} e^{-\frac{|z|^2}{4 D \sigma }} \text {d}z \text {d}\sigma \\ {}&\quad + \int _0^s \int _{\mathbb {R}^d}\frac{1}{(4\pi D \sigma )^{\frac{d}{2}}} e^{-\frac{|z|^2}{4 \sigma }} |z| \text {d}z \text {d}\sigma \\&= \int _0^s \int _{\mathbb {R}^d}\frac{1}{\pi ^{\frac{d}{2}}} e^{-{|w|^2}} \text {d}w \text {d}\sigma \\ {}&\quad + \int _0^s \int _{\mathbb {R}^d}(2D\sigma )^{\frac{1}{2}} \frac{1}{\pi ^{\frac{d}{2}}} e^{-|w|^2} |w| \text {d}w \text {d}\sigma \\&\le s + C D^{\frac{1}{2}} s^{\frac{3}{2}}. \end{aligned}$$

Eventually, we recover using (3.2) that

$$\begin{aligned} \left\| \frac{\psi _s(x) - \psi _0(x)}{1+|x|} \right\| _{L^\infty } \le C(d) \left( (Ds)^{\frac{1}{2}} + (s + D^{\frac{1}{2}} s^{\frac{3}{2}}) \sup _{[0,s]} \left\| \frac{\varvec{E}_{T-\sigma } }{1+|x|} \right\| _{L^\infty } \right) \Vert \nabla \psi _0 \Vert _{L^\infty } \end{aligned}$$

(3.15)

This result can also be deduced for $D = 0$ without involving any convolution. Also, it is recovered as a limit $D \searrow 0$. Joining (3.14) and (3.15) we recover (3.4).

Remark 3.6

(Time continuity at $s=0$ if $\Delta \psi _0$ is bounded or $D = 0$). To estimate the time derivative at time 0, we consider the candidate sub and super-solutions

$$\begin{aligned} {\underline{\psi }}_s = \psi _{0} - C_0 s, \qquad {\overline{\psi }}_s = \psi _{0} + C_0 s. \end{aligned}$$

We have that

$$\begin{aligned}&\partial _s {\underline{\psi }}_s - \varvec{E}_{T-s} \cdot \nabla {\underline{\psi }}_{s} - D\Delta {\underline{\psi }}_s = - C_0 - \varvec{E}_{T-s} \cdot \nabla \psi _{0} - D\Delta \psi _0, \\&\partial _s {\overline{\psi }}_s - \varvec{E}_{T-s} \cdot \nabla {\overline{\psi }}_{s} - D\Delta {\overline{\psi }}_s = C_0 - \varvec{E}_{T-s} \cdot \nabla \psi _{0} - D\Delta \psi _0\,. \end{aligned}$$

So we need

$$\begin{aligned} C_0 \ge \sup _{[0,T] \times {\mathbb {R}^d}} |\varvec{E}_{T-s} \cdot \nabla \psi _{0} + D\Delta \psi _0|. \end{aligned}$$

Then

$$\begin{aligned} \left\| \frac{\psi _s - \psi _0}{s} \right\| _{L^\infty ({\mathbb {R}^d})} \le \Vert \varvec{E}_{T-s} \cdot \nabla \psi _{0} + D\Delta \psi _0 \Vert _{L^\infty ([0,T] \times {\mathbb {R}^d})} . \end{aligned}$$

As $s \rightarrow 0$, we get an estimate of the time derivative at $s = 0$. A similar computation can be done for $\psi / (1 + |x|)$. This kind of result is useful for the case $D = 0$, since even as $D_k \searrow 0$, one can take approximate initial data $\psi _0^{(k)}$ so that the constant is uniformly bounded.

3.2 Proof of Proposition 3.1

First we prove uniqueness. Take $v^{(2)} = \psi / \eta ^{(2)}$ where $\eta ^{(k)}(x) = (1 + |x|^2)^{k/2}$. We observe that if $\frac{\psi _s}{1+|x|} \in L^\infty$ then $v_0^{(2)} \in L^\infty ({\mathbb {R}^d})$, with decay 1/|x| at infinity. By dividing viscosity test functions by $\eta ^{(2)}$, we observe that $v^{(2)}$ is a viscosity solution of (3.6). Notice since $v^{(2)}$ remains bounded, $\sup _{(0,T) \times \mathbb R^d {\setminus } B_R } \psi / (1 + |x|)$ is bounded by $R^{-\frac{1}{2}}$. The uniqueness of $\psi$ follows from the uniqueness of $v^{(2)}$ (see, e.g., [45, Theorem 3.1]).

Let us now construct the solution and prove its properties. If $D> 0$, $\psi _0 \in W_c^{2,\infty } ({\mathbb {R}^d})$, and $\varvec{E}$ is very regular, then existence is simple by standard arguments. For the problem in ${\mathbb {R}^d}$, regularity and decays as $|x| \rightarrow \infty$ for compactly supported initial data follow as in [34] and hence all the estimates above are justified. If $\psi _0 \ge 0$ we can proceed like in Lemma 3.5 to prove $\psi \ge 0$.

Now we consider the general setting, and we argue by approximation. Consider an approximating sequence $0 < D^{(k)} \rightarrow D$, satisfying

$$\begin{aligned} \begin{array}{ll} W_c^{2,\infty } ({\mathbb {R}^d}) \ni \frac{\psi _0^{(k)}}{(1+|x|^2)^{\frac{1}{2}}} \longrightarrow \frac{\psi _0}{(1+|x|^2)^{\frac{1}{2}}} &{}\quad \text { in } L^\infty ({\mathbb {R}^d}), \\ W^{2,\infty } ([0,T] \times {\mathbb {R}^d})^d \ni \varvec{E}^{(k)} \longrightarrow \varvec{E} &{} \quad \text { in } C ((0,T) \times {\mathbb {R}^d})^d. \end{array} \end{aligned}$$

We have the uniform continuity estimates (3.2) and (3.4). Due to the Ascoli–Arzelá theorem and a diagonal argument, $\psi ^{(k)} \rightarrow \psi$ uniformly over compacts of $[0,T] \times {\mathbb {R}^d}$. Applying Lemma 2.15, the limit is a viscosity solution with the limit coefficients. By the uniqueness above this is the solution we are studying. All estimates are stable by convergence uniformly over compact sets. $\square$

4 Dual-viscosity solutions of problem ($\hbox {P}_{\varvec{E}}$)

Now we focus on showing well-posedness and estimates for ($\hbox {P}_{\varvec{E}}$). We construct the solutions as the duals of those in Sect. 3.

Proposition 4.1

For every $D, T \ge 0$ and $\varvec{E} \in C([0,T] ; {{\,\textrm{Lip}\,}}_0 ({\mathbb {R}^d}, {\mathbb {R}^d})$, and $\mu _0 \in \mathcal {P}_1 ({\mathbb {R}^d})$ there exists exactly one dual viscosity solution $\mu \in C([0,T]; \mathcal {P}_1 ({\mathbb {R}^d}))$ of ($\hbox {P}_{\varvec{E}}$). It is also a distributional solution. Furthermore, the map

$$\begin{aligned} S_T : (\mu _0, \varvec{E}) \in \mathcal {P}_1 ({\mathbb {R}^d}) \times C([0,T] ; {Lip}_0({\mathbb {R}^d}, {\mathbb {R}^d}))&\longmapsto \mu _T \in \mathcal {P}_1 ({\mathbb {R}^d}) \end{aligned}$$

is continuous with the following estimate

$$\begin{aligned} \begin{aligned} d_1 (S_T[\mu _0, \varvec{E}], S_T[{\widehat{\mu }}_0, \varvec{\widehat{E}}])&\le C\left( d_1 (\mu _0, {\widehat{\mu }}_0) + \int _0^T \left\| \frac{\varvec{E}_{\sigma } - \varvec{\widehat{E}}_{\sigma }}{1+|x|} \right\| _{L^\infty } \text {d}\sigma \right) \\ \text {where }C&= C\left( d, D, \int _0^T \left\| \nabla \varvec{E}_{\sigma } \right\| _{L^\infty } \text {d}\sigma , \int _{\mathbb {R}^d}|x| \text {d}\mu _0 \right) \end{aligned} \end{aligned}$$

(4.1)

depends monotonically on each entry. The semigroup property holds in the sense that

$$\begin{aligned} S_{{\widehat{t}} + t} [\mu _0, \varvec{E}] = S_{{\widehat{t}}} \Bigg [S_{ t} \Big [\mu _0, \varvec{E}|_{[0, t]}\Big ], \varvec{E}|_{[t, t+{\widehat{t}}]} (\cdot -t) \Bigg ]. \end{aligned}$$

Therefore, we have constructed a continuous flow

$$\begin{aligned} S : (\mu _0, \varvec{E}) \in \mathcal {P}_1 ({\mathbb {R}^d}) \times C([0,\infty ) ; {Lip}_0({\mathbb {R}^d}, {\mathbb {R}^d})) \longmapsto \mu \in C([0,\infty );\mathcal {P}_1 ({\mathbb {R}^d})) . \end{aligned}$$

Proof of Proposition 4.1

We begin by proving uniqueness and continuous dependence. For $D \ge 0$ and any two weak dual viscosity solutions $\mu$ and ${\widehat{\mu }}$ corresponding to $(\mu _0, \varvec{E})$ and $({\widehat{\mu }}_0, \varvec{\widehat{E}})$ we have, applying Lemma 2.11, (3.3), and (3.5), that

$$\begin{aligned} d_1 (\mu _T, {\widehat{\mu }}_T)&\le d_1(\mu _0 , {\widehat{\mu }}_0) \sup _{{{\,\textrm{Lip}\,}}(\psi _0) \le 1} {{\,\textrm{Lip}\,}}(\psi _T^T) \\ {}&\quad + \left( 1 + \int _{\mathbb {R}^d}|x| \text {d}{\widehat{\mu }}_0(x) \right) \sup _{ \begin{array}{c} {{\,\textrm{Lip}\,}}(\psi _0) \le 1 \\ \psi _0(0) = 0 \end{array}} \left\| \frac{\psi _T^T - {\widehat{\psi }}_T^T}{{1+|x|}} \right\| _{L^\infty } \\&\le C(d) d_1(\mu _0 , {\widehat{\mu }}_0) \exp \left( \int _0^T \left( D+ \left\| \nabla {\varvec{E} _\sigma } \right\| _{L^\infty } \right) \text {d}\sigma \right) \\&\quad + C\left( d, D, \int _0^T \left\| \nabla \varvec{E}_{T-\sigma } \right\| _{L^\infty } \right) \left( 1 + \int _{\mathbb {R}^d}|x| \text {d}{\widehat{\mu }}_0(x) \right) \\ {}&\qquad \int _0^T \left\| \frac{\varvec{E}_{T-\sigma } - \varvec{\widehat{E}}_{T-\sigma }}{1+|x|} \right\| _{L^\infty } \! \! \! \!\text {d}\sigma . \end{aligned}$$

The second constant obtained from (3.13) by taking the supremum when ${{\,\textrm{Lip}\,}}(\psi _0) \le 1$. Eventually, we can simplify this expression to (4.1). Therefore, we have uniqueness of dual viscosity solutions for any $D \ge 0$.

For $D> 0, \mu _0 \in H^1({\mathbb {R}^d})$ and $\varvec{E}_t \in C([0,T]; W^{2,\infty } ({\mathbb {R}^d}; {\mathbb {R}^d}))$, the theory of existence and regularity of weak solutions is nowadays well known (see, e.g., [9, 10] and the references therein). The dual problem is decoupled from ($\hbox {P}_{\varvec{E}}$), and we have already constructed the unique viscosity solutions of the dual problem (see Proposition 3.1), that are also weak solutions when $\varvec{E}$ is regular. For regular enough datum $\psi _0$, we can use $\psi$ as test function in the weak formulation of ($\hbox {P}_{\varvec{E}}$) to deduce ($\hbox {D}_i$). Hence, any weak solution of ($\hbox {P}_{\varvec{E}}$) is a dual viscosity solution.

Let us now show the time continuity with respect to the $\mathcal {P}_1 ({\mathbb {R}^d})$ distance in space. We take $\psi _0$ such that ${{\,\textrm{Lip}\,}}(\psi _0) \le 1$, and we estimate

$$\begin{aligned} \left| \int _{\mathbb {R}^d}{\psi _0} \text {d}(\mu _{t+h} - \mu _t) \right| \le \left| \int _{\mathbb {R}^d}( \psi ^{t+h}_{t+h} - \psi ^t_t ) \text {d}\mu _0 \right| \le \int _{\mathbb {R}^d}(1 + |x|) \text {d}\mu _0 \left\| \frac{\psi _{t+h}^{t+h} - \psi _t^t}{1+|x|} \right\| _{L^\infty ({\mathbb {R}^d})}. \end{aligned}$$

This last quantity is controlled by continuous dependence on $\varvec{E}$ and time continuity of ($\hbox {P}_{\varvec{E}}^*$). First, letting ${\widehat{\psi }}_s = \psi ^{t+h}_s$ with $\psi ^{t+h}_0 = \psi _0$, which corresponds to $\varvec{\widehat{E}}_s = \varvec{E}_{t+h-s}$, we have that

$$\begin{aligned} \left\| \frac{\psi ^{t+h}_{t} - \psi _t^t}{1+|x|} \right\| _{L^\infty ({\mathbb {R}^d})} \le C \sup _{\sigma \in [0,t]} \Vert \varvec{E}_{t+h - \sigma } - \varvec{E}_{t-\sigma }\Vert _{Lip_0}. \end{aligned}$$

The right-hand side of this equation is a modulus of continuity, which we denote $\omega _E$. Now we use the time continuity of ($\hbox {P}_{\varvec{E}}^*$) given by (3.4) to deduce that

$$\begin{aligned} \left\| \frac{\psi ^{t+h}_{t+h} - \psi _t^{t+h}}{1+|x|} \right\| _{L^\infty ({\mathbb {R}^d})} \le \omega _D (h), \end{aligned}$$

where C and $\omega _D$ are given by the right-hand side of (3.4). The constants are uniform for $\psi _0$ with ${{\,\textrm{Lip}\,}}(\psi _0) \le 1$. Eventually, taking the supremum on $\psi _0$ and applying (2.2) we deduce

$$\begin{aligned} d_1 (\mu _{t+h},\mu _t) \le \omega _D(h) + \omega _E(h). \end{aligned}$$

Let us now consider the general case for $D\ge 0$, $\mu _0$ and $\varvec{E}_0$. The uniform estimates (4.1) shows that, if $0 < D^n \rightarrow D, H^1({\mathbb {R}^d}) \ni \mu _0^n \rightarrow \mu _0$ in $\mathcal {P}_1 ({\mathbb {R}^d})$ and $C([0,T]; W^{2,\infty } ({\mathbb {R}^d}; {\mathbb {R}^d})) \ni \varvec{E}^n \rightarrow \varvec{E}$ in $C([0,T]; {Lip}_0 ({\mathbb {R}^d},{\mathbb {R}^d}))$, then the sequence $\mu ^n$ is Cauchy in the metric space $C([0,T]; \mathcal {P}_1 ({\mathbb {R}^d}))$. Since this is a Banach space, the sequence $\mu ^n$ converges to a unique limit $\mu$. Due to the stability of the dual problem, we can pass to the limit in ($\hbox {D}_i$) to check that $\mu$ is a dual viscosity solution. We have already shown the uniqueness at the beginning of the proof. Due to the approximation, the solution constructed is also a distributional solution.

The semigroup property holds in the regular setting, so we can pass to the limit. This completes the proof. $\square$

5 Existence and uniqueness for (1.1). Proof of Theorem 2.10

We prove the existence by using Banach’s fixed-point contraction theorem. We construct the map

$$\begin{aligned} \mathcal {T}_t : C([0,t] ; \mathcal {P}_1 ({\mathbb {R}^d})^n ) \longrightarrow \mathcal {P}_1 ({\mathbb {R}^d})^n \end{aligned}$$

(5.1)

by

$$\begin{aligned} \mathcal {T}_t[\varvec{\mu }] = \begin{pmatrix} S_t[ \mu _0^1 , \varvec{{\mathfrak {K}}}^1 [\varvec{\mu }] ] \\ \vdots \\ S_t[ \mu _0^n , \varvec{{\mathfrak {K}}}^n [\varvec{\mu }]] \end{pmatrix}. \end{aligned}$$

We must work on the bounded cubes $Q(R) = B_{\mathcal {P}_1} (R)^n$ where we recall the definition of ball in Wasserstein space given by (2.9). Hence, we take

$$\begin{aligned} R > \max _{i=1, \cdots , n} d_1(\mu _0^i, \delta _0). \end{aligned}$$

To get a very rough estimate of $d_1 (\mathcal {T}_T[\varvec{\mu }], \delta _0)$ we first indicate that when $\varvec{E} = 0$ we recover the heat kernel at time $Dt$

$$\begin{aligned} S_t[\delta _0,0] (x) = H_{Dt} (x) = \frac{1}{(4 \pi Dt )^{-\frac{d}{2}}} \exp \left( -\frac{|x|^2}{4Dt} \right) . \end{aligned}$$

Hence, using the triangle inequality

$$\begin{aligned} d_1 (\delta _0, S_T[\mu _0, \varvec{E}] )&\le d_1(\delta _0, S_T[\delta _0,\varvec{0}]) + d_1 (S_T[\delta _0, \varvec{0}], S_T[\mu _0, \varvec{E}]). \end{aligned}$$

For the first term, we simply write

$$\begin{aligned} d_1(\delta _0, S_T[\delta _0,\varvec{0}]) = d_1 (\delta _0, H_{Dt}) \le \omega (Dt). \end{aligned}$$

To get a uniform constant in (4.1) define

$$\begin{aligned} C_R (T)= \max _{\begin{array}{c} t\in [0,T] \\ i=1, \cdots , n \\ \varvec{\mu } \in C([0,T]; Q (R)) \end{array}} \Vert \nabla \varvec{{\mathfrak {K}}}^i [\varvec{\mu }]_t \Vert _{L^\infty ({\mathbb {R}^d})} < \infty \end{aligned}$$

(5.2)

by the assumptions. As a consequence, for any $\mathbf \mu$ such that $d_1 (\mu _t^i, \delta _0) \le R$ for all $i = 1, \cdots , n$ and $t \le T_1$, then $d_1 (\delta _0, \mathcal {T}_t[\varvec{\mu }]^i) \le R$ for all $i = 1, \cdots , n$ and $t \le T_1$, i.e.,

$$\begin{aligned} \mathcal {T} : C\Big ([0,T_1] ; Q(R) \Big ) \longmapsto C\Big ([0,T_1] ; Q(R) \Big ). \end{aligned}$$

Notice that, since we are constructing solutions using $S_T$, the constructed solution is also a distributional solution.

Applying again (4.1) we have that for every $i = 1, \cdots , n$

$$\begin{aligned} \sup _{t \in [0,T]} d_1 (\mathcal {T}^i_t[\varvec{\mu }], \mathcal {T}^i_t[\widehat{ \varvec{\mu }}] )&= \sup _{t \in [0,T]} d_1 \Bigg ( S_t\Big [\mu _0^i, \varvec{{\mathfrak {K}}}^i [\varvec{\mu }]\Big ], S_t\Big [\mu _0^i, \varvec{{\mathfrak {K}}}^i[\widehat{\varvec{\mu }}]\Big ] \Bigg )\\&\le C(T_1,R) \int _0^T \left\| \frac{ \varvec{{\mathfrak {K}}}^i[\varvec{\mu }]_\sigma - \varvec{{\mathfrak {K}}}^i[\widehat{\varvec{\mu }}]_\sigma }{1 +|x| } \right\| _{L^\infty } \text {d}\sigma \\&\le C(T_1,R) T L \sup _{\begin{array}{c} \sigma \in [0,T] \\ j = 1, \cdots , n \end{array} } d_1 (\mu ^j_\sigma , {\widehat{\mu }}^j_\sigma ), \end{aligned}$$

where $C(T_1,R)$ is recovered again through (4.1). Given that R is fixed, we can find $T_2 < T_1$ so that the map $\mathcal {T}_{T_2}$ is contracting with the norm

$$\begin{aligned} d^{n,T_2}_1 (\varvec{\mu }, \widehat{\varvec{\mu }} ) = \sup _{\begin{array}{c} t \in [0,T_2]\\ i=1,\cdots ,n \end{array}} d_1 (\mu ^i_t, {\widehat{\mu }}^i_t). \end{aligned}$$

Therefore, we can apply Banach’s fixed-point theorem to proof existence and uniqueness for short times. Since $\mathfrak {K}$ is uniformly Lipschitz, we can extend the existence time to infinity applying the classical argument. To prove the continuous dependence on $\mu _0$, we apply (2.8), (3.5) and (2.6). This completes the proof of existence and uniqueness. $\square$

Remark 5.1

Following the non-explicit constant in (3.5), it would be possible to recover quantitative dependence estimates.

Remark 5.2

(Numerical analysis when $D= 0$: the particle method). Our notion of solution justifies the convergence of the particle method when $D= 0$. The aim of the particle method is to consider an approximation of the initial datum given by finitely many isolated particles

$$\begin{aligned} \mu _0^{i,N} = \sum _{j=1}^N a^{ijN} \delta _{X_0^{ijN}} . \end{aligned}$$

Then, it is not difficult to see that, for $D= 0$ the solution is given by particles

$$\begin{aligned} \mu _t^{iN} = \sum _{j=1}^N a^{ijN} \delta _{X_t^{ijN}} . \end{aligned}$$

The evolution of these particles is given by a system of ODEs for the particles

$$\begin{aligned} \partial _t X^{ijN}_t = - \sum _{j} a^{ijN} { \varvec{K}}^{i}_t [\varvec{\mu }^N_t ] (X^{ijN}_t) \end{aligned}$$

This system is well posed. Due to the continuous dependence

$$\begin{aligned} \sup _{t \in [0,T]} d_1 (\varvec{\mu }_t, \varvec{\mu }_t^N) \le C(T , \varvec{\mu }_0) d_1 (\varvec{\mu }_0, \varvec{\mu }_0^N) \end{aligned}$$

It is easy to see that the finite combinations of Dirac deltas is dense in 1-Wasserstein distance, and estimates for convergence are well known (see, e.g., [37]).

6 Gradient flows of convex interaction potentials and $D= 0$

The aim of this section is to give an application of our results to the classical aggregation equation

$$\begin{aligned} \partial _t \mu _t = \nabla \cdot (\mu _t \nabla W * \mu _t). \end{aligned}$$

(6.1)

This problem falls into the well-posedness theory developed in Theorem 2.10 provided that $D^2 W \in L^\infty ({\mathbb {R}^d})^{d\times d}$ for the $\mathcal {P}_1 ({\mathbb {R}^d})$ theory. In the context of gradient-flow solutions (see, e.g., [2]) we are able to weaken the hypothesis on $\mathfrak {K}$.

Theorem 6.1

Let $\mu _0 \in \mathcal {P}_2({\mathbb {R}^d})$ be compactly supported. Assume W is convex and that $W \in C^{1,s}_{loc} ({\mathbb {R}^d})$. Then, the gradient flow solution $\mu \in C([0,T]; \mathcal {P}_2 ({\mathbb {R}^d}))$ is a dual-viscosity solution.

We can approximate W by a sequence of convex functions $W^k$, so that $\nabla W^k \rightarrow \nabla W$ uniformly over compacts but satisfies $D^2 W^k \in L^\infty ({\mathbb {R}^d})$. We construct $\mu ^k_t$ through Theorem 2.10. Since $\mu _0$ is compactly supported and $W^k$ is convex, if we pick $A_0$ the convex envelope of ${{\,\textrm{supp}\,}}\mu _0$, then

$$\begin{aligned} {{\,\textrm{supp}\,}}\mu _t^k \subset A_0, \qquad \forall t > 0. \end{aligned}$$

This is easy to see, for example, looking at the solution by characteristics. We set ourselves in the hypothesis of Theorem 2.14 by proving below that

• 1. $\psi ^{k,T}$ for every $\psi _0 \in C_c^\infty ({\mathbb {R}^d})$ are uniformly equicontinuous.

• 2. $\mu ^k \rightarrow \mu$ in $C([0,T]; \mathcal {P}_2 ( {\mathbb {R}^d}))$.

• 3. $W^k * \mu ^k \rightarrow W*\mu$ uniformly over compacts.

Remark 6.2

We point that the result in Theorem 6.1 can be extended from (6.1) to a system of equations coming from the gradient flow of interaction potentials between the different components. To fix ideas, we can treat systems of two species like

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t \rho _1 = \nabla \cdot ( \rho _1 \nabla ( H_1 * \rho _1 + K * \rho _2 ) \\ \partial _t \rho _2 = \nabla \cdot ( \rho _2 \nabla ( H_2 * \rho _2+ K * \rho _1 ) \end{array}\right. \end{aligned}$$

See, e.g., [31].

6.1 Regularity in space

Constructing solutions by characteristics it is well known that

$$\begin{aligned} (\varvec{E}_{T-s} (x) - \varvec{E}_{T-s} (y)) \cdot (x-y) \ge 0, \end{aligned}$$

then characteristics grow apart. Then, the solution of (2.3) satisfies

$$\begin{aligned} \Vert \nabla \psi _s \Vert _{L^\infty ({\mathbb {R}^d})} \le \Vert \nabla \psi _0 \Vert _{L^\infty ({\mathbb {R}^d})}. \end{aligned}$$

(6.2)

We check that this holds for $\varvec{E} = \nabla W * \mu$. Since W is convex, then

$$\begin{aligned} (\nabla W (x) - \nabla W (y)) \cdot (x-y) = (x-y) \cdot D^2 W(\xi (x,y)) (x-y) \ge 0. \end{aligned}$$

Then, the convolution with a non-negative measure is also convex

$$\begin{aligned} \Big (\nabla W * \mu (x) - \nabla W * \mu (y)\Big ) \cdot \Big (x-y \Big )&= \int _{\mathbb {R}^d}\Big ( \nabla W (x-z) - \nabla W (y-z) \Big ) \\&\quad \cdot \Big ( (x-z) - (y-z) \Big ) \text {d}\mu _0 (z) \\&\ge 0, \end{aligned}$$

and so (6.2) holds.

6.2 Regularity in time

First, we look at the evolution of the support. When $\varvec{E}_{T-s}$ is locally bounded, we can construct a super-solutions by characteristics. We define

$$\begin{aligned} A(s_0,s_1) = \bigcup _{s \in [s_0,s_1]} {{\,\textrm{supp}\,}}\psi _s . \end{aligned}$$

If we assume that $A(s,0) \subset B(x_0,R_s)$ then

$$\begin{aligned} \partial _s R_s \le \sup _{B(x_0,R_s)} |\varvec{E}_{T-s}|. \end{aligned}$$

So we end up with an estimate

$$\begin{aligned} A(s,0) \subset {{\,\textrm{supp}\,}}\psi _0 + B_{R_s}. \end{aligned}$$

(6.3)

Assume that $\psi _0 \in C_c^\infty ({\mathbb {R}^d})$ with $\Vert \nabla \psi _0\Vert _{L^\infty ({\mathbb {R}^d})} \le 1$. Take $s_0 < s$. Consider

$$\begin{aligned} {\underline{\psi }}_s = \psi _{s_0} - C_0 (s-s_0), \qquad {\overline{\psi }}_s = \psi _{s_0} + C_0 (s-s_0). \end{aligned}$$

We have that

$$\begin{aligned} \partial _s {\underline{\psi }}_s - \varvec{E}_{T-s} \cdot \nabla {\underline{\psi }}_{s} = - C_0 - \varvec{E}_{T-s} \cdot \nabla \psi _{s_0}, \qquad \partial _s {\overline{\psi }}_s - \varvec{E}_{T-s} \nabla {\overline{\psi }}_{s} = C_0 - \varvec{E}_{T-s} \nabla \psi _{s_0}. \end{aligned}$$

They are a sub and super-solution in $[s_0,s_1]$ if

$$\begin{aligned} C_0 = \Vert \nabla \psi _{s_0}\Vert _{L^\infty } \sup _{[s_0,s_1] \times A(s_0,s_1) } |\varvec{E}|. \end{aligned}$$

Hence, we deduce that

$$\begin{aligned} \left\| \frac{\psi _{s_1} - \psi _{s_0}}{s_1-s_0} \right\| _{L^\infty ({\mathbb {R}^d})} \le \Vert \nabla \psi _{s_0}\Vert _{L^\infty } \sup _{[s_0,s_1] \times A(s_0,s_1) } |\varvec{E}| . \end{aligned}$$

(6.4)

This implies that a uniform bound on the time continuity based on T, local bounds of $\varvec{E}$ and the support of $\psi _0$.

6.3 Convergence of the convolution

Following [2, Theorem 11.2.1] the $\Gamma$-convergence of uniformly $\lambda$-convex interaction free energy functionals is sufficient so that

$$\begin{aligned} \sup _{[0,T]} d_2 (\mu _t^k , \mu _t)\rightarrow 0, \qquad \text { as } k \rightarrow \infty . \end{aligned}$$

(6.5)

In particular, if $W \in C^s$ is convex and $W^k (x) (1 +|x|)^{-2} \rightarrow W(x) (1 + |x|)^{-2}$ uniformly, (6.5) holds. It follows that the sequence $\mu ^k$ is uniformly continuous, i.e., the function

$$\begin{aligned} \omega (h) = \sup _k \sup _{t \in [0,T-h]} d_2 (\mu _{t+h}^k , \mu ^k_t) \end{aligned}$$

is a modulus of continuity.

On the other hand, for the convergence of $\nabla W^k * \mu ^k$ we prove the following result.

Lemma 6.3

Assume that

• 1. $\nabla W \in C^s_{loc}$

• 2. $\nabla W^k \rightarrow \nabla W$ uniformly over compacts of ${\mathbb {R}^d}$.

• 3. $\mu _t^k, \mu _t \in \mathcal {P} ({\mathbb {R}^d})$, and, for every $t > 0$, $\mu ^k_t \rightharpoonup \mu _t$ weak-$\star$ in $\mathcal {M} ({\mathbb {R}^d})$

• 4. There exists $A_0 \subset {\mathbb {R}^d}$ convex bounded such that for all $t > 0$, ${{\,\textrm{supp}\,}}\mu _t^k \subset A_0$

• 5. $\mu ^k \in C([0,T]; \mathcal {P}_2 ({\mathbb {R}^d}))$ are uniformly continuous.

Then

$$\begin{aligned} \nabla W^k * \mu ^k \rightarrow \nabla W * \mu \text { uniformly over compacts of } [0,T] \times {\mathbb {R}^d}. \end{aligned}$$

Proof

We use the intermediate element $\nabla W * \mu ^k$. First, for $A \subset {\mathbb {R}^d}$ compact

$$\begin{aligned} \sup _{x \in A} \left| \int _{\mathbb {R}^d}(\nabla W^k (x-z) - \nabla W (x-z) ) \text {d}\mu _s^k \right| \le \sup _{A + A_0} |\nabla W^k -\nabla W |. \end{aligned}$$

Hence, we have that

$$\begin{aligned} \sup _{\begin{array}{c} t \in [0,T] \\ x \in A \end{array} } \left| \nabla W^k * \mu _t^k (x) - \nabla W * \mu _t^k (x) \right| \le \sup _{A + A_0} | \nabla W^k - \nabla W|. \end{aligned}$$

(6.6)

Due to weak-$\star$ convergence, if $\nabla W \in C^s_{loc}$ then

$$\begin{aligned} \nabla W * \mu ^k_t (x) \rightarrow \nabla W * \mu _t (x) \text { for each } (t,x) \in [0,T] \times {\mathbb {R}^d}\end{aligned}$$

Now we prove uniform continuity. First in space. Let $A \subset {\mathbb {R}^d}$ be compact. Take

$$\begin{aligned} C_K = \sup _{ \begin{array}{c} x, y \in A - A_0 \\ x \ne y \end{array} } \frac{|\nabla W(x) - \nabla W(y)|}{|x-y|^s} \end{aligned}$$

Now, for $x,y \in A$ we can compute

$$\begin{aligned} \left| \nabla W* \mu ^k (x) - \nabla W*\mu ^k (y) \right| \le \int _{A_0} |\nabla W(x-z) - \nabla W(y-z)| \text {d}\mu ^k (z)\le C_A |x-y|^s. \end{aligned}$$

So $\nabla W * \mu ^k$ is uniformly continuous in x over compacts of $[0,T] \times {\mathbb {R}^d}$.

Lastly, let $\pi$ be optimal plan between $\mu ^k_t$ and $\mu ^k_{\tau }$. Due to assumption 4, the optimal plan can be selected so that ${{\,\textrm{supp}\,}}\pi \subset A_0 \times A_0$. Let $z \in A$. We have that

$$\begin{aligned}&\quad \left| \int _{\mathbb {R}^d}\nabla W(z-y) \text {d}\mu ^k_t(x) - \int _{\mathbb {R}^d}\nabla W(z-y) \text {d}\mu ^k_{\tau }(y) \right| \\&= \left| \iint ( \nabla W(z-x) - \nabla W(z-y) ) \text {d}\pi (x,y) \right| \\&\le C_A \iint |x-y|^s \text {d}\pi (x,y) \\&\le C_A \left( \iint |x-y|^2 \text {d}\pi (x,y) \right) ^{\frac{s}{2}}\\&= C_A d_2 (\mu ^k_t, \mu ^k_{\tau })^s \le C_A \omega (h)^s. \end{aligned}$$

Hence, we have the uniform estimate of continuity in time

$$\begin{aligned} \sup _{x \in K} |\nabla W * \mu ^k_t - \nabla W * \mu ^k_{\tau }| \le C_A \omega (|t-\tau |)^s. \end{aligned}$$

And we finally deduce that

$$\begin{aligned} \sup _{\begin{array}{c} t,\tau \in [0,T] \\ x, y \in K \end{array}} |\nabla W * \mu ^k_t (x) - \nabla W * \mu ^k_\tau (y)| \le C_A (|x-y| + \omega (|t - \tau |)^s). \end{aligned}$$

In particular, by the Ascoli–Arzelá theorem, there is a subsequence converging uniformly in [0, T]. Since we have characterised the point-wise limit, every convergent subsequence does so to $\nabla W * \mu$. Hence, the whole sequence converges uniformly over compacts, i.e.,

$$\begin{aligned} \sup _{\begin{array}{c} t \in [0,T] \\ x \in A \end{array}} |\nabla W * \mu ^k_t (x) - \nabla W * \mu _t (x)| \rightarrow 0 , \qquad \text {as } k \rightarrow \infty . \end{aligned}$$

(6.7)

Using the triangular inequality, (6.6), and (6.7) the result is proven. $\square$

Proof of Theorem 6.1

When $W^k$ is $C^2$, we can construct a unique classical solution as the push-forward of regular characteristics. This solution is well-known to be the gradient flow solution. By construction, it coincides with the unique dual viscosity solution that exists by Theorem 2.10.

First, we showed in (6.5) that $\mu ^k$ converges to the gradient flow solution in $C([0,T]; \mathcal {P}_2 ({\mathbb {R}^d}))$, and let us denote it by ${\overline{\mu }}$. Now we apply Theorem 2.14 where the hypothesis have been check in (6.4) (using (6.3)), and Lemma 6.3 to show that ${\overline{\mu }}$ is a dual viscosity solution. This completes the proof. $\square$

7 A $\dot{H}^{-1}$ and 2-Wasserstein theory when $D> 0$

7.1 Notion of solution and well-posedness theorem

Many of cases of (1.1) studied in the literature are 2-Wasserstein gradient flow. Our situation is more general. Unfortunately, the 2-Wasserstein distance, $d_2$, does not have a duality characterisation similar to $d_1$. It is known (see, e.g., [46]) that it can be one-side compared with the $\dot{H}^{-1}({\mathbb {R}^d})$

$$\begin{aligned} d_2 (\mu , {\widehat{\mu }}) \le 2 [ \mu - {\hat{\mu }} ]_{\dot{H}^{-1} ({\mathbb {R}^d})}, \end{aligned}$$

(7.1)

where, for $\mu \in D' ({\mathbb {R}^d})$ we define the norm

$$\begin{aligned} \Vert \mu \Vert _{\dot{H}^{-1}({\mathbb {R}^d})} = \sup _{ \begin{array}{c} f \in C_c^\infty ({\mathbb {R}^d}) \\ \Vert \nabla f \Vert _{L^2} \le 1 \end{array} } |\mu (f)|. \end{aligned}$$

The converse inequality to (7.1) only holds for absolutely continuous measures, and the constant depends strongly on the uniform continuity.

We consider the Sobolev semi-norm $[f]_{H^1} = \Vert \nabla f\Vert _{L^2}.$ The space $(C_c^\infty ({\mathbb {R}^d}) , [ \cdot ]_{H^1})$ is a normed space. Notice that, if $[f]_{H^1} = 0$ then f is constant. But since it has compact supported, the value of the constant is 0. This allows to define the dual space

$$\begin{aligned} \dot{H}^{-1} ({\mathbb {R}^d}) = (C_c^\infty ({\mathbb {R}^d}), [ \cdot ]_{H^1({\mathbb {R}^d})})'. \end{aligned}$$

Since it is the dual of a normed space, $\dot{H}^{-1} ({\mathbb {R}^d})$ is a Banach space.

Remark 7.1

The space $\dot{H}^1({\mathbb {R}^d})$ is defined as the completion of $(C_c({\mathbb {R}^d}), [ \cdot ]_{H^1({\mathbb {R}^d})})$, which is easy to see is not complete itself. This completion can be complicated (see, e.g., [12]). Hence, with our construction $\dot{H}^{-1} ({\mathbb {R}^d})$ is not the dual of $\dot{H}^{1} ({\mathbb {R}^d})$.

Similarly to above, we define

Definition 7.2

We say that $(\varvec{\mu }, \{\varvec{\Psi }^T\}_{T\ge 0})$ is an $\dot{H}^{-1}$ entropy pair if:

• 1. For every $T \ge 0$,

  $$\begin{aligned} \varvec{\Psi }^T: X = \{ \psi _0 \in C_c({\mathbb {R}^d}): \nabla \psi _0 \in L^2 ({\mathbb {R}^d}) \} \longrightarrow C([0,T]; X )^n \end{aligned}$$

  is a linear map with the following property: for every $\psi _0$ and $i = 1, \cdots , n$ we have $\Psi ^{T,i}[\psi _0]$ is a viscosity solution of ($\hbox {P}^*_i$).

• 2. For each i and $T \ge 0$, $\mu _T^i \in \dot{H}^{-1} ({\mathbb {R}^d})$ and satisfies the duality condition ($\hbox {D}_i$)

Remark 7.3

Notice that in this section we require that $\psi _0$ is compactly supported, and thus the unique viscosity solutions will satisfy $\psi (x) \rightarrow 0$ as $|x| \rightarrow \infty$.

The main result of this section is

Theorem 7.4

Let $D> 0$, $\mu _0 \in \dot{H}^{-1} ({\mathbb {R}^d})$ and assume that

$$\begin{aligned} \varvec{\mathfrak {K}}^i : C([0,T], \dot{H}^{-1} ({\mathbb {R}^d})) \rightarrow C([0,T], W^{1,\infty } ({\mathbb {R}^d})) \end{aligned}$$

(7.2)

is Lipschitz with data in $\dot{H}^{-1} ({\mathbb {R}^d})$ in the sense that, for any $\varvec{\mu }$ and $\widehat{\varvec{\mu }}$ in $C([0,T]; \dot{H}^{-1} ({\mathbb {R}^d}))$

$$\begin{aligned} \sup _{\begin{array}{c} t \in [0,T] \\ i = 1, \cdots , n \end{array}} \left\| \varvec{{\mathfrak {K}}^i[\varvec{\mu }]_t - \varvec{\mathfrak {K}}^i[ \widehat{\varvec{\mu }}]}_t \right\| _{W^{1,\infty }({\mathbb {R}^d})} \le L \sup _{\begin{array}{c} t \in [0,T] \\ i=1,\cdots ,n \end{array}} \Vert \mu ^i_t - {\widehat{\mu }} ^i_t \Vert _{\dot{H}^{-1}({\mathbb {R}^d})}. \end{aligned}$$

(7.3)

Then there exists exactly one dual viscosity solution $\varvec{\mu } \in C([0,T], \dot{H}^{-1} ({\mathbb {R}^d})^n)$.

Furthermore, if $\varvec{\mathfrak {K}}^i$ is autonomous (i.e., $\varvec{\mathfrak {K}}^i[\varvec{\mu }]_t = \varvec{K}^i[\varvec{\mu }_t]$), then the map $S_T : \varvec{\mu }_0 \in \dot{H}^{-1} ({\mathbb {R}^d})^n \mapsto \varvec{\mu }_T \in \dot{H}^{-1} ({\mathbb {R}^d}) ^n$ is a continuous semigroup.

We point that we can only get the $\dot{H}^{-1}({\mathbb {R}^d})$ theory when diffusion is present (i.e., $D> 0$). When $D = 0$ we cannot get an estimate of $\Vert \nabla \psi _s\Vert _{L^2}$.

Proof of Theorem 7.4

The proof follows the blueprint of the proof of Theorem 2.10 using a fixed-point argument. We need an adapted version of (2.8) given by

$$\begin{aligned} \Vert \mu _T - {\widehat{\mu }}_T\Vert _{\dot{H}^{-1}({\mathbb {R}^d})}&\le \, \Vert \mu _0 - {\widehat{\mu }}_0\Vert _{\dot{H}^{-1}({\mathbb {R}^d})} \sup _ {\begin{array}{c} \psi _0 \in C_c^\infty ({\mathbb {R}^d}) \\ \Vert \nabla \psi _0 \Vert _{L^2 ({\mathbb {R}^d})} \le 1 \end{array}} \Vert \nabla \psi _T^T \Vert _{L^2({\mathbb {R}^d})} \nonumber \\&\quad + \Vert {\widehat{\mu }}_0\Vert _{\dot{H}^{-1}({\mathbb {R}^d})} \sup _{\begin{array}{c} \psi _0 \in C_c^\infty ({\mathbb {R}^d}) \\ \Vert \nabla \psi _0 \Vert _{L^2 ({\mathbb {R}^d}) } \le 1 \end{array}} \Big \Vert \nabla ( \psi _T^T - {\widehat{\psi }}_T^T ) \Big \Vert _{L^2({\mathbb {R}^d})} . \end{aligned}$$

(7.4)

This is shown by a suitable modification of the argument in (2.7). Lastly, we use the bounds and continuous dependence proved below in Propositions 7.6 and 7.9. $\square$

Lastly, due to (7.1) we have the following partial result in 2-Wasserstein space.

Corollary 7.5

Let $D> 0$, $\varvec{\mu }_0 \in \dot{H}^{-1} ({\mathbb {R}^d})^n \cap \mathcal {P}({\mathbb {R}^d})^n$, then the unique solution constructed in Theorem 7.4 is $C([0,T]; \mathcal {P}_2 ({\mathbb {R}^d})^n)$.

Proof

Due to the previous theorem, we only need to show that probability distributions stay probability distributions. The non-negativity is trivial, since by Proposition 3.1 the test functions preserve the non-negativity. Hence, we get

$$\begin{aligned} \int _{\mathbb {R}^d}\psi _0 \text {d}\mu _T = \int _{\mathbb {R}^d}\psi _T \text {d}\mu _0 \ge 0, \qquad \text { for all } 0 \le \psi _0 \in C_c^\infty ({\mathbb {R}^d}). \end{aligned}$$

Thus $\mu _T \ge 0$ for all $T \ge 0$.

We now prove the conservation of total mass. Consider a sequence of $0 \le \psi _{0}^{(k)} \in X$ point-wise increasing k and converging uniformly over compacts to 1. Then, by the comparison principle, $0 \le \psi ^{(k)} \le 1$ are point-wise increasing in k. By Dini’s theorem, functions $\psi ^{(k)}$ converges uniformly over compacts. Let $\psi ^{(\infty )}$ be its limit. By stability of viscosity solutions $\psi ^{(\infty )}$ solves the same equation. By the uniform convergence $\psi ^{(\infty )}_0 = \lim _n \psi _0^{(k)} = 1$. Then $\psi ^{(\infty )} = 1$. By definition

$$\begin{aligned} \int _{\mathbb R^d} \psi ^{(k)}_0 d \mu _T = \int _{\mathbb R^d} \psi ^{(k)}_T d \mu _0. \end{aligned}$$

We can apply the monotone convergence theorem on both sides to deduce

$$\begin{aligned} \int _{\mathbb R^d} d \mu _T = \int _{\mathbb R^d} d \mu _0 = 1. \end{aligned}$$

$\square$

7.2 Study of ($\hbox {P}_{\varvec{E}}^*$)

To deal with the $\dot{H}^{-1}({\mathbb {R}^d})$ estimates, we must get $L^2({\mathbb {R}^d})$ of $\nabla \psi$.

Proposition 7.6

Under the hypothesis of Proposition 3.1 together with $\psi _0 \in C_c ({\mathbb {R}^d})$, then the unique viscosity solution of (2.3) constructed in Proposition 3.1 also satisfies

$$\begin{aligned} \Vert \nabla \psi _s\Vert _{L^2({\mathbb {R}^d})} \le \Vert \nabla \psi _0\Vert _{L^2({\mathbb {R}^d})} \exp \left( C(d) \int _0^s \Vert \nabla \varvec{E}_{T-\sigma } \Vert _{L^\infty ({\mathbb {R}^d})} \text {d}\sigma \right) . \end{aligned}$$

(7.5)

Lastly, given ${\widehat{\psi }}_0 = \psi _0$ and $\varvec{\widehat{E}}$ in the same hypotheses above, there exists a corresponding solution of (2.3), denoted by ${\widehat{\psi }}$, and we have the continuous dependence estimate

$$\begin{aligned} \Vert \nabla (\psi _s - {\widehat{\psi }}_s) \Vert _{L^2({\mathbb {R}^d})}^2 \le C_1 e^{C_2 s} \left( 1 + \frac{1}{D}\right) \int _0^s \Vert \varvec{E}_{T-\sigma } - \varvec{\widehat{E}}_{T-\sigma } \Vert _{W^{1,\infty }({\mathbb {R}^d}) } \text {d}\sigma . \end{aligned}$$

(7.6)

Remark 7.7

Notice that for $\nabla \psi _s \in L^2({\mathbb {R}^d})$ we do not use that $\varvec{E} \in L^\infty ({\mathbb {R}^d})$, or the ellipticity constant.

Proof

We begin with the $L^2({\mathbb {R}^d})$ estimates for smooth $\psi _0 \in C_c^\infty ({\mathbb {R}^d})$ and $\varvec{E} \in C_c^\infty ([0,T] \times {\mathbb {R}^d})$, where the solutions are classical and differentiable. For our duality characterisation in $\dot{H}^{-1}({\mathbb {R}^d})$ we do not need $L^2({\mathbb {R}^d})$ estimates on $\psi$, only on $\nabla \psi$. Notice that (7.5) is simply (3.10) when $p = 2$.

For the continuous dependence, we note that

$$\begin{aligned} \partial _s (U^i_s - {\widehat{U}}^i_s)&= \, \nabla (U^i_s - {\widehat{U}}^i_s ) \varvec{E}_{T-s} + \nabla (\psi _s - {\widehat{\psi }}_s) \frac{\partial \varvec{E}}{\partial x_i} \\&\quad +\nabla {\widehat{U}}^i_s (\varvec{E}_{T-s} - \varvec{\widehat{E}}_{T-s}) + \nabla {\widehat{\psi }}_s \frac{\partial }{\partial x_i} (\varvec{E}_{T-s} - \varvec{\widehat{E}}_{T-s}) \\&\quad + D\Delta (U^i_s - {\widehat{U}}^i_s). \end{aligned}$$

Multiplying by $U^i_s - {\widehat{U}}^i_s$ and integrating, we get

$$\begin{aligned} \partial _s \Vert U^i_s - {\widehat{U}}^i_s\Vert _{L^2 ({\mathbb {R}^d})}^2 = I_1 + \cdots + I_5\,. \end{aligned}$$

We estimate term by term the integrals $I_i$, $i=1,\dots ,5$, as

$$\begin{aligned} \left| I_1 \right|&= \frac{1}{2} \left| \int (U^i_s - {\widehat{U}}^i_s )^2 \nabla \cdot \varvec{E}_{T-s} \right| \le \frac{1}{2} \Big \Vert U^i_s - {\widehat{U}}^i_s \Big \Vert _{L^2({\mathbb {R}^d})}^2 \left\| \nabla \cdot \varvec{E}_{T-s} \right\| _{L^\infty ({\mathbb {R}^d})} ,\\ \left| I_2\right|&\le \sum _j \left| \int (U^j_s - {\widehat{U}}^j_s ) \frac{\partial \varvec{E}^j}{\partial x_i} (U^i_s - {\widehat{U}}^i_s )\right| \le \Big \Vert U^i_s - {\widehat{U}}^i_s \Big \Vert _{L^2} \sum _{j} \Big \Vert U^j_s - {\widehat{U}}^j_s \Big \Vert _{L^2({\mathbb {R}^d})} \left\| \frac{\partial \varvec{E}^j}{\partial x_i} \right\| _{L^\infty } ,\\ \left| I_4 \right|&\le \sum _j \left| \int {\widehat{U}}^j_s \frac{\partial }{\partial x_i} (\varvec{E}_{T-s}^j - \varvec{\widehat{E}}_{T-s}^j) (U^i_s - {\widehat{U}}^i_s) \right| \\&\le \Vert U^i_s - {\widehat{U}}^i_s \Vert _{L^2({\mathbb {R}^d})} \sum _j \left\| \frac{\partial }{\partial x_i} \Big (\varvec{E}_{T-s}^j - \varvec{\widehat{E}}_{T-s}^j\Big ) \right\| _{L^\infty } \Vert {\widehat{U}}^j_s \Vert _{L^2({\mathbb {R}^d})} \\&\le \frac{1}{2} \Vert U^i_s - {\widehat{U}}^i_s \Vert _{L^2({\mathbb {R}^d})}^2 + \frac{1}{2} \left( \sum _j \left\| \frac{\partial }{\partial x_i} \Big (\varvec{E}_{T-s}^j - \varvec{\widehat{E}}_{T-s}^j\Big ) \right\| _{L^\infty ({\mathbb {R}^d})} \Vert {\widehat{U}}^j_s \Vert _{L^2({\mathbb {R}^d})} \right) ^2 , \end{aligned}$$

and

$$\begin{aligned} I_5&= - D\int | \nabla (U^i_s - {\widehat{U}}^i_s)|^2 . \end{aligned}$$

There is only one problematic term that requires the use of the ellipticity condition

$$\begin{aligned} \left| I_3 \right|&\le \, \left| \int U^i_s (U^i_s - {\widehat{U}}^i_s) \nabla \cdot (\varvec{E}_{T-s} - \varvec{\widehat{E}}_{T-s}) \right| + \left| \int U^i_s (\varvec{E}_{T-s} - \varvec{\widehat{E}}_{T-s}) \nabla (U^i_s - {\widehat{U}}^i_s) \right| \nonumber \\&\le \,\frac{1}{2} \Vert U^i_s - {\widehat{U}}^i_s \Vert _{L^2({\mathbb {R}^d})}^2 + \frac{1}{2}\Vert U^i_s \Vert _{L^2({\mathbb {R}^d})}^2 \Vert \nabla \cdot (\varvec{E}_{T-s} - \varvec{\widehat{E}}_{T-s}) \Vert _{L^\infty ({\mathbb {R}^d})}^2 \nonumber \\&\quad + \frac{1}{4D} \Vert U^i_s \Vert _{L^2({\mathbb {R}^d})}^2 \Vert \varvec{E}_{T-s} - \varvec{\widehat{E}}_{T-s} \Vert _{L^\infty ({\mathbb {R}^d})}^2 + D\Vert \nabla (U^i_s - {\widehat{U}}^i_s) \Vert _{L^2({\mathbb {R}^d})}^2. \end{aligned}$$

(7.7)

Notice that $I_5$ cancels out the last term in $|I_3|$. Arguing as above we recover

$$\begin{aligned} \partial _s \Vert \nabla (\psi _s - {\widehat{\psi }}_s) \Vert _{L^2({\mathbb {R}^d})}^2&\le \, C(d) \Bigg ( (1 + \Vert \nabla \varvec{E}_{T-s} \Vert _{L^\infty ({\mathbb {R}^d})} ) \Vert \nabla (\psi _s^T - {\widehat{\psi }}_s^T) )\Vert _{L^2({\mathbb {R}^d})}^2 \\&\quad + \left( 1 + \frac{1}{D}\right) \Big ( \Vert \nabla \psi _s\Vert _{L^2({\mathbb {R}^d})}^2 + \Vert \nabla {\widehat{\psi }}_s\Vert _{L^2({\mathbb {R}^d})}^2 \Big ) \Vert \varvec{E}_{T-s} \\ {}&\quad - \varvec{\widehat{E}}_{T-s}\Vert _{W^{1,\infty }({\mathbb {R}^d})}^2\Bigg ). \end{aligned}$$

Eventually, since $\psi _0^T - {\widehat{\psi }}_0^T = 0$ we recover (7.6) where the constants depend on d, $\Vert \nabla \psi _0 \Vert _{L^2}$, $\Vert \nabla {\widehat{\psi }}_0 \Vert _{L^2}$ and $\int _0^T\Vert \varvec{E}_{T-\sigma }\Vert _{W^{1,\infty }} \text {d}\sigma$. This completes the $L^2$ estimates.

When $\psi _0$ is a general initial datum in $C_c ({\mathbb {R}^d})$ we proceed by an approximation argument as in Proposition 3.1. As for the $L^\infty ({\mathbb {R}^d})$ estimates, we can first assume that $\varvec{E} \in C^\infty _c ([0,T] \times {\mathbb {R}^d})$ and $\psi _0 \in C^\infty _c ({\mathbb {R}^d})$ are smooth, recover the $L^2$ estimates for the gradient, and then pass to the limit. $\square$

Remark 7.8

Notice that in (7.7) we use strongly the fact that $D > 0$, and the estimates are not uniform as $D \searrow 0$.

7.3 Study of ($\hbox {P}_{\varvec{E}}$)

Similarly, applying (7.4), (7.5) and (7.6), we deduce that

Proposition 7.9

For every $D, T > 0$, $\varvec{E} \in C([0,T]; W^{1,\infty } ({\mathbb {R}^d}; {\mathbb {R}^d}))$ and $\mu _0 \in \dot{H}^{-1} ({\mathbb {R}^d})$ there exists exactly one $\dot{H}^{-1} ({\mathbb {R}^d})$ dual viscosity solution $\mu \in C([0,T]; \dot{H}^{-1} ({\mathbb {R}^d}))$ of ($\hbox {P}_{\varvec{E}}$). Furthermore, the map

$$\begin{aligned} S_T : \dot{H}^{-1} ({\mathbb {R}^d}) \times C([0,T] ; W^{1,\infty }({\mathbb {R}^d}, {\mathbb {R}^d})) \longmapsto \mu _T \in \dot{H}^{-1} ({\mathbb {R}^d}) \end{aligned}$$

is continuous with the following estimate

$$\begin{aligned} \Big \Vert S_T [\varvec{E}, \mu _0] - S_T[ \varvec{\widehat{E}}, {\widehat{\mu }}_0]\Big \Vert _{\dot{H}^{-1}}&\le C\left( d, \tfrac{1}{D}, T, \int _0^T \Vert \nabla \varvec{E}_{T-\sigma } \Vert _{L^\infty } \text {d}\sigma \right) \nonumber \\ {}&\qquad \int _0^T \Vert \varvec{E} _\sigma - \varvec{\widehat{E}} _\sigma \Vert _{W^{1,\infty }} \text {d}\sigma , \end{aligned}$$

(7.8)

where C depends monotonically on each entry. Lastly, the semigroup property holds, i.e.,

$$\begin{aligned} S_{{\widehat{t}} + t} [\mu _0, \varvec{E}] = S_{{\widehat{t}}} \Bigg [S_{ t} \Big [\mu _0, \varvec{E}|_{[0, t]}\Big ], \varvec{E}|_{[t, t+{\widehat{t}}]} \Bigg ]. \end{aligned}$$

Remark 7.10

(Solutions by characteristics when $D= 0$). For $D= 0$ our notion of solution is $\mu _t = X_t \# \mu _0$ where $X_t$ is the unique solution of the flow equation

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial X_t}{\partial t} = -\varvec{E}_t (X_t), \\ X_0 (x) = x. \end{array}\right. \end{aligned}$$

(7.9)

A unique solution of this pointwise-decoupled problem exists via the Picard-Lindelöf theorem. Since we have forwards and backwards uniqueness, for each $t > 0$ we know that $X_t$ is a bijection. It is easy to show that the unique viscosity solution is given by

$$\begin{aligned} \psi _s^T = \psi _0 \circ X_T^{-1} \circ X_{T-s} \end{aligned}$$

And the duality condition ($\hbox {D}_i$) is trivially satisfied. The problem is that we do not have suitable $H^{1}$ in this theory.