On a model of axisymmetric flows in a free boundary domain

TL;DR

This paper models axisymmetric free boundary vortex flows, identifying vortex boundaries via variational methods and analyzing vortex dynamics through continuity equations, providing a mathematical framework for such fluid phenomena.

Contribution

It introduces a variational approach to determine vortex boundaries and reduces vortex dynamics to solvable continuity equations, advancing theoretical understanding of axisymmetric flows.

Findings

Vortex boundary characterized by a variational problem.

Constructed solutions for the continuity equations governing vortex dynamics.

Provides a mathematical foundation for analyzing free boundary axisymmetric flows.

Abstract

We consider a model of axisymmetric flows for a free boundary vortex embedded in a statically stable fluid at rest. We identify the boundary of the vortex by solving a variational problem. Then, we reduce the analysis of the dynamics of the vortex to the study of a class of continuity equations for which we construct a solution.

Equations

$$\{\begin{array}{ll}\frac{\mathrm{D}u}{\mathrm{D}t}+\frac{uv}{r}+2\Omega v+\frac{1}{r}\frac{\partial \phi }{\partial \lambda }& =\frac{1}{r}{F}_0,\\ \frac{\mathrm{D}\theta }{\mathrm{D}t}& =F_1,\\ -\frac{u^2}{r}-2\Omega u+\frac{\partial \phi }{\partial r}& =0,\\ -g\frac{\theta }{{\theta }_0}+\frac{\partial \phi }{\partial z}& =0,\\ \frac{\partial }{\partial \lambda }(u)+\frac{\partial }{\partial r}(rv)+\frac{\partial rw}{\partial z}& =0.\end{array}$$

$${\Gamma }_{{\varsigma }_t}:=\left\{(r,z):0\le z\le H,r_0\le r\le \varsigma (t,z)\right\}.$$

$$\frac{\mathrm{D}u}{\mathrm{D}t}+\frac{uv}{r}+2\Omega v$$

$$\frac{\mathrm{D}\theta }{\mathrm{D}t}$$

$$\frac{u^2}{r}+2\Omega -\frac{\partial \phi }{\partial r}$$

$$g\frac{\theta }{{\theta }_0}-\frac{\partial \phi }{\partial z}$$

$$\frac{1}{r}\frac{\partial }{\partial r}(rv)+\frac{\partial w}{\partial z}$$

$$\{\begin{array}{l}⟨(v_t,w_t),{\mathbf{n}}_{\mathbf{t}}⟩=0\text{ on }{\Lambda }_{rig},\\ \frac{\partial {\varsigma }_t}{\partial t}+w\frac{\partial {\varsigma }_t}{\partial z}=v\text{on}{\Lambda }_{\varsigma }.\end{array}$$

$$\phi (t,\varsigma (t,z),z)=0\text{for}\varsigma (t,z)>0.$$

$${\int }_{{\Gamma }_{\varsigma }}\left(\frac{u^2}{2}-\frac{g\theta }{{\theta }_0}z\right)rdrdz.$$

$$\mathcal{A}({\theta }_0(z)){\nabla }_{r,z}\left(\phi +{\Omega }^2\frac{r^2}{2}\right)\text{ is invertible.}$$

$$\mathcal{P}({\mathbb{R}}^2)\ni \sigma ⟼\mathcal{H}(\sigma ):=h(\sigma )+\underset{\varrho \in \mathcal{R}}{\inf }{W}_2^2(\sigma ,\mathbf{f}\#{\mu }_{\varrho })+{\int }_{{\mathbb{R}}^2}E(\mathbf{p}){\mu }_{\varrho }(d\mathbf{p})$$

$$E(\mathbf{p})=f_0(s)-\frac{1}{2}\left(s^2+z^2/{\theta }_0^2(z)\right)\text{with}\mathbf{p}=(s,z)\text{and}{f}_0(s)=\frac{r_0^2{\Omega }^2}{2(1-2r_0^{2}s)}.$$

$$\mathcal{H}(\sigma )=h(\sigma )+\sup \left({\int }_{{\mathbb{R}}^2}-\Psi (\mathbf{q})\sigma (d\mathbf{q})+\underset{\varrho \in \mathcal{R}}{\inf }{\int }_0^H{\int }_0^{\varrho (z)}\left(f_0(s)-P(\mathbf{p},m)\right){\delta }_{{\theta }_0(z)}(dm){\mu }_{\varrho }(d\mathbf{p})\right).$$

$$\mathcal{U}:=\Bigl{\{}(P,\Psi)\in C(\mathcal{W}\times{\mathcal{I}}_{0})\times C(\mathbb{R}^{2}_{+}):P({\bf p},m)+\Psi({\bf q})\geq c({\bf p},m,{\bf q})\text{ for all }({\bf p},{\bf q})\in\mathcal{W}\times\mathbb{R}_{+}^{2},\;m\in{\mathcal{I}}_{0}\Bigr{\}}$$

$$c(\mathbf{p},m,\mathbf{q})=s{\rm Y}+\frac{zZ}{m}with\mathbf{p}=(s,z)and\mathbf{q}=({\rm Y},Z).$$

$$\mathcal{T}[P^{\sigma }](\mathbf{p})=\left({\partial }_s{P}^{\sigma }(\mathbf{p},{\theta }_0(z)),{\theta }_0(z){\partial }_z{P}^{\sigma }(\mathbf{p},{\theta }_0(z))\right)\mathbf{p}=(s,z)$$

$$\mathcal{S}[{\Psi }^{\sigma }](\mathbf{q})=\left({\partial }_{{\rm Y}}{\Psi }^{\sigma }(\mathbf{q}),{\varphi }^{-1}({\partial }_Z{\Psi }^{\sigma })(\mathbf{q})\right)\varphi (z)=z/{\theta }_0(z)$$

$$\{\begin{array}{l}\frac{\partial \sigma }{\partial t}+\mathrm{div}(\sigma V_t[\Psi ])=0,{\mathcal{D}}^{\prime }\left((0,T)\times {\mathbb{R}}^2\right)\\ \sigma {|}_{t=0}={\sigma }_0,\end{array}$$

$$V_t[\Psi ]=\left(2\sqrt{{\rm Y}}{F}_0\left(t,\frac{1}{\Omega }\sqrt{2f_0(\frac{\partial \Psi }{\partial {\rm Y}})},{\varphi }^{-1}\left(\frac{\partial \Psi }{\partial Z}\right)\right),g{F}_1\left(t,\frac{1}{\Omega }\sqrt{2f_0(\frac{\partial \Psi }{\partial {\rm Y}})},{\varphi }^{-1}\left(\frac{\partial \Psi }{\partial Z}\right)\right)\right)and\varphi (z)=z/{\theta }_0(z)$$

$$\{\begin{array}{l}\mathcal{T}[P]\#{\mu }_{\varrho }=\sigma ,\\ \mathcal{S}[\Psi ]\circ \mathcal{T}[P]=\mathbf{id}{\mu }_{{\sigma }_{\varrho }},\\ P(\varrho (z),z,{\theta }_0(z))=\frac{{\Omega }^2{r}_0^2}{2(1-2r_0^2\varrho (z))}on\{\varrho >0\}.\end{array}$$

$$\{\begin{array}{l}{\theta }_t(r,z)={\theta }_0(z){\partial }_z{P}_t\circ {\mathbf{s}}_{{\theta }_0}(r,z)\\ {\phi }_t(r,z)=P_t\circ {\mathbf{s}}_{{\theta }_0}(r,z)-{\Omega }^2{r}^2/2\\ {\varsigma }_t(z)=\frac{1}{\Omega }\sqrt{2f_0({\varrho }_t(z))}\end{array}$$

$$\{\begin{array}{l}{u}_t=1/r\sqrt{{\partial }_s{P}_t\circ {\mathbf{s}}_{{\theta }_0}}-r\Omega \\ v_t=\left({\partial }_t{\mathbb{S}}_1\right)\circ \mathbb{T}+2\sqrt{{\partial }_s{P}_t\circ {\mathbf{s}}_{{\theta }_0}}{F}_{0t}\left({\partial }_{{\rm Y}}{\mathbb{S}}_1\right)\circ \mathbb{T}+g{F}_{1t}\left({\partial }_Z{\mathbb{S}}_1\right)\circ \mathbb{T}\\ w_t=\left({\partial }_t{\mathbb{S}}_2\right)\circ \mathbb{T}+2\sqrt{{\partial }_s{P}_t\circ {\mathbf{s}}_{{\theta }_0}}{F}_{0t}\left({\partial }_{{\rm Y}}{\mathbb{S}}_2\right)\circ \mathbb{T}+g{F}_{1t}\left({\partial }_Z{\mathbb{S}}_2\right)\circ \mathbb{T}\end{array}$$

$$\mathcal{W}:=[0,1/2r_0^2)\times [0,H]and{\mathcal{W}}_{\infty }:=[0,\infty )\times [0,H].$$

$$\mathrm{spt}(\mu ):=\left\{\mathbf{x}\in {\mathbb{R}}^D:\mu (O)>0\text{ for any open set }O\text{ of }{\mathbb{R}}^D\right\}.$$

$${\int }_{{\mathbb{R}}^D}|\mathbf{x}{|}^{p}d\mu (\mathbf{x})<\infty .$$

$$\mathcal{T}\#\mu (A):=\mu ({\mathcal{T}}^{-1}(A))\text{ for any Borel set }A\in {\mathbb{R}}^D.$$

$$W_p^p(\mu ,\nu )=\min \left\{{\int }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}|x-y{|}^{p}d\gamma :{\mathfrak{p}}^1\#\gamma =\mu \text{and}{\mathfrak{p}}^2\#\gamma =\nu \right\},$$

$$\mathbf{d}(\mathbf{s}(t),\mathbf{s}(s))\le {\int }_t^s\mathbf{p}(r)dr\text{for all }a<t\le s<b.$$

$$c(\mathbf{p},m,\mathbf{q})=s{\rm Y}+\frac{zZ}{m}.$$

$$\{\begin{array}{l}\frac{1}{r}{\partial }_r(rv)+{\partial }_{z}w=0\text{on}{\Gamma }_{{\varsigma }_t},\\ ⟨{\mathbf{v}}_t,{\mathbf{n}}_t⟩=0\text{on}{\Lambda }_{rig},\\ {\partial }_t{\varsigma }_t+w{\partial }_z{\varsigma }_t=v\text{on}{\Lambda }_{{\varsigma }_t}.\end{array}$$

$$\frac{\partial \sigma }{\partial t}+\mathrm{div}(\sigma V_t)=0,{\mathcal{D}}^{\prime }\left((0,T)\times {\mathbb{R}}^2\right).$$

Full text

African Institute for Mathematical Sciences Tanzania, (sedjro@aims.ac.tz)

1. Introduction

Axisymmetric flows are appropriate models to describe idealized tropical cyclones. Typically, they describe the evolution of a balanced vortex under the forcing effects of tangential momentum and heat sources. Though widely studied, these flows still present challenges as attested by recent studies and results; see [1], [2]. In [3], Craig derived a system of equations for flows that are almost circular in gradient balance, the so-called the almost axisymmetric flows. These flows model, in the absence of viscosity, the motion of a vortex in a rotating reference frame where the coriolis coeffficient is $\Omega>0$ and the gravity of earth is $g$. The vortex evolves at a velocity ${\bf u}=(u,v,w)$ in a domain where the potential temperature is $\theta$ and the pressure is $\varphi$, and is kept in an ambient fluid at a prescribed temperature $\theta_{0}$. Under the effects of forcing terms $F_{0}$ and $F_{1}$, the equations for the $3-$dimensional axisymmetric flows are given in cylindrical coordinates with standard variables $(\lambda,r,z)$ by:

[TABLE]

In the absence of forcing terms, the almost axisymmetric flows approximate the hydrostatic Boussinesq equations. Though a simpler model, almost axisymmetric flows still present some challenging regularity issues (see [4]). In [5], the authors introduced two-dimensional flows derived from (1.1). These flows provide axisymmetric solutions to (1.1) and share the same stability states as the almost axisymmetric flows as flow parcels follow displacements preserving angular momentum and potential temperatures (see [6]). Building on the work in [6], [7] and [8], they developed a procedure that uses the theory of optimal mass transport to construct a solution to the two-dimensional flows within a moving domain $\Gamma_{\varsigma_{t}}$ defined by

[TABLE]

One key assumption in the model considered in [5], is that the ambient temperature $\theta_{0}$ is constant. This assumption makes the problem tractable. However, when $\theta_{0}$ is constant, the ambient fluid loses its static stability. In this paper, we consider the more physically relevant model in which the ambient temperature varies in function of the height level of the vortex.

1.1. Axisymmetric flows with forcing terms

The axisymmetric flows are derived from (1.1) by assuming that the quantities $u,v,w,\varphi,\theta$ and the operator ${D\over Dt}={\partial\over\partial t}+u{\partial\over\partial r}+v{\partial\over\partial r}+w{\partial\over\partial z}$ are all independent of the angular variable $\lambda$. These considerations lead to the following system:

[TABLE]

The equations (1.3e) are to be solved in the domain $\Gamma_{\varsigma_{t}}$. Thus, we supplement (1.3e) with a Neumann boundary condition on the rigid boundary $\Lambda_{rig}$ composed of sets $\{r=r_{0}\}$, $\{z=0\}$, and $\{z=H\}$ and a kinematic boundary condition on the free boundary $\Lambda_{\varsigma}$ representing $\{r=\varsigma(t,z)\}$ :

[TABLE]

Here ${\bf n_{t}}$ is the unit outward normal vector field at time $t$. On the free boundary, we impose the following condition on the pressure :

[TABLE]

The Hamiltonian relevant to the system (1.3e) is given by

[TABLE]

From a meteorological point of view, we are looking for solutions for which the vortex is stable with respect to perturbations. To that aim, a stability condition is imposed on the pressure. Notably,

[TABLE]

Here, $\mathscr{A}(m)$ denotes the $2\times 2$ diagonal matrix $\text{Diag}(1,m)$.

1.2. Hamiltonian and stable solutions

We discover that the Hamiltonian in (1.6) plays an important role in the construction of solution to stable axisymmtric flows. By making the change of coordinate system $\Upsilon=(ru+r^{2}\Omega r^{2})^{2}$ and $Z=g\theta$, this Hamiltonian can be written solely in terms of a measure $\sigma$ provided that the stability condition (1.7) is satisfied. Subsequently, the Hamiltonian takes the form of the following functional :

[TABLE]

Here, $W_{2}$ denotes the $2-$Wasserstein distance, ${\bf f}(s,z)=(s,z/\theta_{0}(z))$, $h(\sigma)=\int_{\mathbb{R}^{2}}\left({\Upsilon\over 2r_{0}^{2}}-\Omega\sqrt{\Upsilon}\right)\sigma(d{\bf q})$ with ${\bf q}=(\Upsilon,Z)$, and the function $E$ is defined by

[TABLE]

${\mathcal{R}}$ denotes the set of the Borel functions $\varrho:[0,H]\longmapsto[0,1/(2r_{0}^{2})$. For any map $\varrho\in{\mathcal{R}}$, we associate the Borel measure $\mu_{\varrho}$ which is absolutely continuous with respect to the Lebesgue measure and whose density is given by $\left(\frac{2f_{0}(s)}{\Omega^{2}}\right)^{2}\chi_{D_{\varrho}}(s,z)$ with $D_{\varrho}=\left\{(s,z):0\leq s\leq\varrho(z),z\in[0,H]\right\}$. While the $W_{2}(\sigma,\cdot)$ has some convexity properties, the function $E$ is not convex. To study the existence and the uniqueness of a minimizer in (1.8), we consider a dual formulation :

[TABLE]

The supremum in (1.9) is taken over the set

[TABLE]

where the cost function $c$ is given by

[TABLE]

Assume that $\varrho^{\sigma}_{0}$ is a minimizer in (1.8) and that $(P^{\sigma},\Psi^{\sigma})$ are c-transform of each other in the sense of definition 4.2 and are maximizers of (1.9). Then, $\varrho^{\sigma}_{0}$ is uniquely determined and the maps $\mathcal{T}[P^{\sigma}]$ and $\mathcal{S}[\Psi^{\sigma}]$ defined respectively by

[TABLE]

and

[TABLE]

are essentially injective functions and we have $\mathcal{S}[\Psi^{\sigma}]\circ\mathcal{T}[P^{\sigma}]={\bf id}\quad\mu_{\varrho_{0}^{\sigma}}$-a.e. Furthermore, we have that $\mathcal{T}[P^{\sigma}]=\mathscr{A}(\theta_{0}(z))\nabla P^{\sigma}$ pushes forward $\mu_{\varrho_{0}^{\sigma}}$ onto $\sigma$ and $2(1-2r^{2}_{0}\varrho_{0}(z))P^{\sigma}(\varrho_{0}(z),z,\theta_{0}(z))=r^{2}_{0}\Omega^{2}\text{ on }\{\varrho_{0}^{\sigma}>0\}$. If, in addition, $\sigma$ is absolutely continuous with respect to Lebesgue then $\mathcal{T}[P^{\sigma}]\circ\mathcal{S}[\Psi^{\sigma}]={\bf id}\;\sigma-$a.e.

1.3. Continuity equation corresponding to the 2D Axisymmetric Flows with Forcing Terms

A class of continuity equations plays a determining role in the construction of solutions to the axisymmetric flows (1.3e) satisfying the stability condition (1.7). Assume that $\sigma\in AC_{2}\left(0,T;\mathscr{P}\left(\mathbb{R}^{2}\right)\right)$ and that $t\longrightarrow\sigma_{t}$ satisfies

[TABLE]

with

[TABLE]

where $\Psi_{t}$ is such that $(P_{t},\;\Psi_{t})$ $c-$transforms of each other for each $t\in[0,T]$. Assume that $\varrho_{t}$ is monotone and that $(P_{t},\;\Psi_{t},\varrho_{t})$ solve uniquely

[TABLE]

Then, given enough regularity, we can construct a solution $u,v,w,\theta,\varphi,\varsigma$ to (1.3e)-(1.5) and (1.7). As shown in section 3, through the change of variable ${\bf s}_{\theta_{0}}(r,z)=(s[r],z,\theta_{0}(z))$ with $2s=2s[r]:=r_{0}^{-2}-r^{-2}$, the quantities $\theta$, $\varphi$ and $\varsigma$ are obtained by

[TABLE]

and the velocity field $(u,v,w)$ is given by

[TABLE]

Here, $\mathbb{T}=\mathcal{T}[P^{\sigma}]\circ{\bf s}$ and $\mathbb{S}={\bf r}\circ\mathcal{S}[\Psi^{\sigma}]$ with ${\bf s}(r,z)=(s[r],z)$ and its inverse ${\bf r}(s,z)=\left(\frac{1}{\Omega}\sqrt{2f_{0}(s)},z\right)$.

1.4. Plan of the paper

This paper is organized as follows: In section 2 we collect notation, definitions and key assumptions throughout the paper. In section 3, we explain how solutions to the axisymmetric flows can be constructed via the study of a class of continuity equations with enough regularity. In section 4, we study a variational problem that determines the free boundary, its regularity and the velocity fields associated to the class of continuity equations considered. In Section 5, we study the stability of the free boundaries and the velocity fields governing these continuity equations. In section 6, we follow a discretization scheme developped in [10] to construct solution for (1.13) and (1.14).

2. Notation, Definitions and Assumptions

Throughout this paper, we use the following notation, definitions and assumptions:

2.1. Notation and Definitions

• •

$g,r_{0}$ and $H$ are positive constants and we set

[TABLE]

$\overset{o}{{\mathcal{W}}}$ denotes the interior of $\mathcal{W}$.

• •

$f_{0}$ denotes the function on $[0,1/2(r_{0}^{2}))$ defined by $f_{0}(s)={r_{0}^{2}\Omega^{2}\over 2(1-2r_{0}^{2}s)}.$

• •

${\mathcal{R}}$ denotes the set of the Borel functions $\varrho:[0,H]\longmapsto[0,1/(2r_{0}^{2})$. For any map $\varrho\in{\mathcal{R}}$, we associate the set $D_{\varrho}:=\left\{(s,z):0\leq s\leq\varrho(z),z\in[0,H]\right\}$ and the Borel measure $\mu_{\varrho}$ which is absolutely continuous with respect to the Lebesgue measure in $\mathbb{R}^{2}$ and whose density is given by $\left(\frac{2f_{0}(s)}{\Omega^{2}}\right)^{2}\chi_{D_{\varrho}}(s,z)$.

• •

${\mathcal{R}}_{0}$ is the subset of ${\mathcal{R}}$ for which $\mu_{\varrho}$ is a Borel probability measure.

• •

${\mathcal{V}}:=(0,\infty)\times(0,\infty)$ and $B_{a}$ is the ball in $\mathbb{R}^{2}$ centered at $(0,0)$ and of radius $a$.

We denote by $B^{+}_{a}:=B_{a}\cap{\mathcal{V}}.$

• •

$\mathcal{I}_{0}$ is an open bounded interval of $\mathbb{R}_{+}$ such that $0\notin\bar{\mathcal{I}_{0}}$. Here, $\bar{\mathcal{I}_{0}}$ denotes the closure of $\mathcal{I}_{0}$.

• •

Let $D\in\mathbb{N}$. For any naturel numbers $1\leq i<j<k\leq D$, $\pi^{i},\;\pi^{i,j}\;\pi^{i,j,k}$ denote the projection operators on $\mathbb{R}^{D}$ defined respectively by $\pi^{i}(x_{1},\cdots x_{D})=x_{i},\quad\;\pi^{i,j}(x_{1},\cdots x_{D})=(x_{i},x_{j})$ and $\pi^{i,j,k}(x_{1},\cdots x_{D})=(x_{i},x_{j},x_{k})$. $D$ will be determined by the context.

• •

${\mathcal{L}}^{D}$ denotes the Lebesgue measure in $\mathbb{R}^{D}$.

• •

If $\sigma$ is a measure on $\mathbb{R}^{D}$ absolutely continuous with respect to ${\mathcal{L}}^{D}$ then we denote by $\dfrac{d\sigma}{d{\mathcal{L}}^{D}}$ the Radon-Nycodym derivative of $\sigma$ with respect to ${\mathcal{L}}^{D}$.

• •

$\mathscr{P}(\mathbb{R}^{D})$ is the set of all Borel probability measures on $\mathbb{R}^{D}$.

• •

For $\mu\in\mathscr{P}(\mathbb{R}^{D})$, we denote by ${\rm spt}(\mu)$ the support of $\mu$ defined by

[TABLE]

• •

$\mathscr{P}_{p}(\mathbb{R}^{D})$ ($1\leq p<\infty$) denotes the set of probability measures with finite $p-$ moments:

[TABLE]

• •

We denote by $\mathscr{P}^{ac}(\mathbb{R}^{D})$ the set of all elements of $\mathscr{P}(\mathbb{R}^{D})$ that are absolutely continuous with respect to Lebesgue.

• •

Let $\mu\in\mathscr{P}(\mathbb{R}^{D})$ and ${\mathcal{T}}:\mathbb{R}^{D}\longrightarrow\mathbb{R}^{D}$ a Borel map. The push-forward of $\mu$ through ${\mathcal{T}}$, denoted by ${\mathcal{T}}\#\mu\in\mathscr{P}(\mathbb{R}^{D})$, is defined by

[TABLE]

• •

Given $\mu,\;\nu\in\mathscr{P}_{p}\left(\mathbb{R}^{D}\right)$, the $p$-Wasserstein distance between $\mu$ and $\nu$ is defined as

[TABLE]

$\mathfrak{p}^{1}$ and $\mathfrak{p}^{2}$ denote respectively the first and second projections on $\mathbb{R}^{D}\times\mathbb{R}^{D}$.

• •

Let $\left({\mathcal{S}},\mathbf{d}\right)$ be a complete metric space, $a$ and $b$ be real numbers such that $a<b$. A curve $\mathbf{s}:(a,b)\longrightarrow{\mathcal{S}}$ is said to belong to $AC_{m}\left(a,b;\;{\mathcal{S}}\right)$ if there exists ${\bf p}\in L^{m}(a,b)$ such that

[TABLE]

Curves in $AC_{m}\left(a,b;{\mathcal{S}}\right)$ are said to be $m-$absolutely continuous, see [10].

• •

For any matrix $A$, we denote by $A^{\tau}$ the transpose of $A$.

• •

For time-dependent functions, we use the notation $S_{t}(\cdot,\cdot)=S(t,\cdot,\cdot)$ for convenience.

• •

${\mathcal{A}}:{\mathcal{I}}_{0}\longrightarrow\mathbb{R}^{2\times 2}$ defined by $m\longrightarrow\mathcal{A}(m)=\begin{bmatrix}1&0\\ 0&m\end{bmatrix}$.

• •

We use the notation ${\bf p}=(s,z)$, ${\bf q}=(\Upsilon,Z)$ and denote by $c$ the cost function on $\mathbb{R}^{2}\times{\mathcal{I}}_{0}\times\mathbb{R}^{2}$ defined by

[TABLE]

2.2. Assumptions

• •

The function $\theta_{0}:[0,H]\longrightarrow{\mathcal{I}}_{0}$ is assumed smooth on $(0,H)$ and satisfies the following conditions:

• (A1)

$\theta_{0}(z)-z\theta_{0}^{\prime}(z)>0$.

• (A1’)

$\inf_{0\leq z\leq H}\left\{\theta_{0}(z)-z\theta_{0}^{\prime}(z)\right\}>0$ for all $z\in(0,H)$.

• (A2)

$\theta_{0}$ is Lipschitz continuous.

Typically, $\theta_{0}(z)=\left(A+Bz^{\alpha}\right)^{\beta}$ with $A>0,\;B>0$, $\beta\geq 1$ and $\alpha\beta\leq 1$ satisfies conditions (A1’) and (A2). The condition (A1) implies that $\phi(z):=z/\theta_{0}(z)$ is strictly increasing.

• •

We assume that $F_{0}:=F_{0t}(r,z)$ and $F_{1}=F_{1t}(r,z)$ are such that $F_{0},F_{1}\in C^{1}((0,\infty)\times\mathbb{R}^{2})$ and satisfy the following conditions:

• (B1)

$0\leq F_{0},\;gF_{1}\leq M$ for some positive constant $M$.

• (B2)

$\partial_{z}F_{0}=\partial_{r}F_{1}=0.$

• (B3)

$\partial_{r}F_{0},\;\partial_{z}F_{1}>0.$

3. continuity equations and axisymmetric flows

In this section, we discuss how one can derive a solution for the axisymmetric flows from the study of a class of continuity equations. We point out that this derivation relies on the assumption that we have enough regularity for solutions to this class of continuity equations. Let $\textbf{v}=(v,w)$ be a smooth velocity field and $\varsigma_{t}$ a smooth function such

[TABLE]

Here, ${\bf n}_{t}$ is the outward unit normal vector to the rigid boundary $\Lambda_{rig}$ for each $t$ fixed. The following lemma is proved in dimension 3 in [4]. We reproduce the proof in dimension 2 for the reader’s convenience.

Lemma 3.1**.**

Let $T>0$ and $\sigma\in AC_{2}\left(0,T;\mathscr{P}\left(\mathbb{R}^{2}\right)\right)$ and $V\in C^{1}\left((0,T)\times\mathbb{R}^{2}\right)$ such that

[TABLE]

Let $\mathcal{G}=\left(\mathcal{G}^{1},\mathcal{G}^{2}\right)$ be a smooth function on $(0,T)\times\mathbb{R}^{2}$ such that $\mathcal{G}_{t}$ is invertible with inverse $\mathcal{F}_{t}$ for each $t\in[0,T]$. Assume that there exists $\varsigma$ such that for $t\in(0,T)$ we have $\mathcal{G}_{t}\#\sigma_{t}=(r\chi_{\Gamma_{\varsigma_{t}}}{\mathcal{L}}^{2})$. Define $v,w$ respectively by

[TABLE]

Then $v,w,\varsigma$ solve (3.1)

Proof: For each $t\in[0,T]$ fixed, let $\mathcal{G}_{t}$ be the inverse of $\mathcal{F}_{t}$. Let $\psi\in C_{c}\left((0,T)\times\mathbb{R}^{2}\right)$ and set $\eta_{t}=\psi_{t}\circ\mathcal{G}_{t}$. We note that

[TABLE]

The last equality in (3.3) is obtained by using $\mathcal{G}_{t}\#\sigma_{t}=(r\chi_{\Gamma_{\varsigma_{t}}}{\mathcal{L}}^{2})$. The equations in (3.2) can be rewritten in the vectorial form as $V\circ\mathcal{F}_{t}=\partial_{t}\mathcal{F}_{t}+\nabla\mathcal{F}_{t}\begin{bmatrix}v\\ w\end{bmatrix}$ so that (3.3) becomes

[TABLE]

In the second line of (3.4), we have used the fact that $\mathcal{G}_{t}\circ\mathcal{F}_{t}={\bf id}$ implies that $\partial_{t}\mathcal{G}_{t}\circ\mathcal{F}_{t}+[\nabla\mathcal{G}_{t}]\circ\mathcal{F}_{t}\partial_{t}\mathcal{F}_{t}=0$ and $[\nabla\mathcal{G}_{t}]\circ\mathcal{F}_{t}[\nabla\mathcal{F}_{t}]=\mathbf{I}$. Applying the divergence theorem in space-time, we obtain that

[TABLE]

Here $\mathbf{H}^{2}$ denotes the 2-dimensional Hausdorff measure. As $A=0$ and $\psi$ is arbitrary, (3.5) implies that $v,w,\varsigma$ solve (3.1).  $\square$

We define ${\bf s}:{\mathcal{W}}_{\infty}\longrightarrow\mathcal{W}$ by ${\bf s}(r,z)=\left(s[r],z\right)$ where $2s=2s[r]:=r_{0}^{-2}-r^{-2}$. ${\bf s}$ is invertible with inverse ${\bf r}$ defined by ${\bf r}(s,z)=\left(\frac{1}{\Omega}\sqrt{2f_{0}(s)},z\right)$. If $\varrho\in{\mathcal{R}}_{0}$ and $\varsigma$ is defined by the third equation of (1.16) then

[TABLE]

$\Gamma_{\varsigma}$ is defined in (1.2). For $\theta_{0}$ fixed, we define $\mathbf{s}_{\theta_{0}}$ on ${\mathcal{W}}_{\infty}$ by $\mathbf{s}_{\theta_{0}}(s,z)=(s[r],z,\theta_{0}(z)).$ To any $P\in C\left({\mathcal{W}}\times{\mathcal{I}}_{0}\right)$ such that $P(\cdot,z,m)$ and $P(s,\cdot,m)$ are differentiable we associate the function $\mathcal{T}[P]$ defined by

[TABLE]

Similarly, to any function $\Psi\in C(B_{l}^{+})$ such that $\Psi(\Upsilon,\cdot)$ and $\Psi(\cdot,Z)$ are differentiable, we associate the function $\mathcal{S}[\Psi]$ defined by

[TABLE]

Proposition 3.2**.**

Let $l>0$ and assume that (A1) holds. Let $T>0$ and $\sigma\in AC_{2}\left(0,T;\mathscr{P}\left(\mathbb{R}^{2}\right)\right)$ and $V\in C^{1}\left((0,T)\times\mathbb{R}^{2}\right)$ such that

[TABLE]

Let $P\in C^{1}\left((0,T)\times\overset{o}{{\mathcal{W}}}\times{\mathcal{I}}_{0}\right)$, $\Psi\in C^{1}\left((0,T)\times B_{l}^{+}\right)$ and $\varrho\in C^{1}\left((0,T)\times(0,H)\right)$ such that $\varrho_{t}\in{\mathcal{R}}_{0}$ for each $t$ fixed. Assume that $\mathcal{T}[P_{t}]$ and $\mathcal{S}[\Psi_{t}]$ as defined in respectively in (3.7) and (3.8) are inverse of each other in the interior of their domains, that $\partial_{s}P>0$ and that $\mathcal{S}[\Psi_{t}^{\sigma}]$ pushes forward $\sigma_{t}$ onto $\mu_{\varrho_{t}}$ with

[TABLE]

for each $t\in[0,T]$. Define $\varphi$, $\varsigma$ and $\theta$ respectively through (1.16) $u$ $v$ and $w$ through (1.17) and set $\frac{D}{Dt}:=\partial_{t}+v\partial_{r}+w\partial_{z}.$ Assume $V$ is the velocity field as in (1.14). Then $(u,v,w)$, $\theta$, $\varphi$ and $\varsigma$ solve (1.3e)-(1.5) and (1.7).

Proof: The first equations of (1.16) and (1.17) imply that

[TABLE]

These, in light of the second equation of (1.16), yield (1.3c) and (1.3d). We define the function $\mathbb{T}=\left(\mathbb{T}_{1},\mathbb{T}_{2}\right)$ by $\mathbb{T}_{t}:={\mathcal{T}}[P_{t}]\circ{\bf s}$ and $\mathbb{S}=\left(\mathbb{S}_{1},\mathbb{S}_{2}\right)$ by $\mathbb{S}_{t}:={\bf r}\circ{\mathcal{S}}[\Psi_{t}]$ for each $t\in(0,T)$. As ${\mathcal{T}}[P_{t}]$ and ${\mathcal{S}}[\Psi_{t}]$ are inverse of each other for each $t\in(0,T)$, so are $\mathbb{T}_{t}$ and $\mathbb{S}_{t}$ for $t\in(0,T)$. We notice that we can rewrite (3.10) as

[TABLE]

Therefore,

[TABLE]

and

[TABLE]

The two last equations of (1.17) actually read in vectorial form

[TABLE]

In view of (3.14),

[TABLE]

As

[TABLE]

we use (3.15) to get

[TABLE]

Since $\mathbb{T}_{t}$ and $\mathbb{S}_{t}$ are inverse of each other for each $t\in(0,T)$, we have $\partial_{t}\mathbb{T}_{t}+[\nabla_{r,z}\mathbb{T}_{t}]\left(\partial_{t}\mathbb{S}_{t}\right)\circ\mathbb{T}_{t}=0$ and $[\nabla_{r,z}\mathbb{T}][\nabla_{\Upsilon,Z}\mathbb{S}_{t}]\circ\mathbb{T}_{t}={\bf I}$ so that (3.16) reduces to

[TABLE]

As $\partial_{s}P>0$ we have that $\mathbb{T}_{1}>0$. Subsequently, we combine (3.12), (3.13) and (3.17) to obtain (1.3a) and (1.3b). In light of (3.6), we observe that

[TABLE]

for each $t\in[0,T]$. We use lemma 3.1 to obtain (1.3e) and (1.4). We combine the second and third equations in (1.16) with (3.9) to get (1.5). The invertibility of $\mathbb{T}$ and the second equation in (1.16) yield (1.7).   $\square$

4. Minimization problem and Duality Method

In this section, we prove the existence and uniqueness of a variational solution for problem (4.1). This result is obtained by investigating c-subdifferential of maximizers in (4.8) with respect to the cost function $c$ as defined in (2.1) and by establishing subsequently a duality between problem (4.4) and problem (4.8).

Let $l>0$ and $\sigma\in\mathscr{P}\left(\mathbb{R}^{2}\right)$ such that ${\rm spt}(\sigma)\subset B_{l}^{+}$. We consider the following system of equations where the unknowns are $P\in C({\mathcal{W}}\times{\mathcal{I}}_{0})$, $\Psi\in C(B_{l})$ and $\varrho\in{\mathcal{R}}_{0}$. We require that $P$ and $\Psi$ satisfy (4.23) and (4.24) and solve

[TABLE]

Remark 1**.**

The maps $\mathcal{S}[\Psi]$ and $\mathcal{T}[P]$ in (4.1) are defined respectively in (3.7) and (3.8). We note that if $\sigma\in\mathscr{P}^{ac}\left(\mathbb{R}^{2}\right)$ then the system of equations (4.1) is equivalent to

[TABLE]

4.1. Primal and Dual formulation of the problem

Let $\sigma\in\mathscr{P}\left(\mathbb{R}^{2}\right)$. We define the functional ${\mathcal{K}}[\sigma]$ on ${\mathcal{R}}$ as follows :

[TABLE]

Here, $\mathbf{f}$ is defined on $\mathcal{W}$ by $\mathbf{f}(s,z)=\left(s,\frac{z}{\theta_{0}(z)}\right)$. We consider the variational problem

[TABLE]

To study the minimization problem in (4.4), we investigate a dual formulation through the functional

[TABLE]

where ${\mathcal{J}}[\sigma]$ is defined on

[TABLE]

and the functional $\mathscr{S}$ is defined by

[TABLE]

for $(\varrho,z)\in{\mathcal{W}}.$ The dual problem we will be looking at is the following:

[TABLE]

Set

[TABLE]

and

[TABLE]

for all ${\bf p}=(s,z)\in{\mathcal{W}}$, $m\in{\mathcal{I}}_{0}$ and ${\bf q}=(\Upsilon,Z)\in\mathbb{R}^{2}_{+}$. We recall that

[TABLE]

Note that the cost function $c$ can be expressed as

[TABLE]

and the second moment of $\sigma$ is given by

[TABLE]

Proposition 4.1**.**

Let $l>0$, $\sigma\in\mathscr{P}\left(\mathbb{R}^{2}\right)$ and assume that (A1) holds. Then,

1. (1)

We have ${\mathcal{K}}[\sigma](\varrho)\geq{\mathcal{J}}[\sigma](P,\Psi)+\mathbf{m}_{2}[\sigma]$ for all $\varrho\in{\mathcal{R}}$ and all $(P,\Psi)\in{\mathcal{U}}$ 2. (2)

Let $(P_{0},\Psi_{0})\in{\mathcal{U}}_{0}$ and $\varrho_{0}\in{\mathcal{R}}_{0}$. Then, the following hold:

${\mathcal{K}}[\sigma](\varrho_{0})={\mathcal{J}}[\sigma](P_{0},\Psi_{0})+\mathbf{m}_{2}[\sigma]$* if and only if there exists $\alpha_{0}\in\mathscr{P}\left(\mathbb{R}^{2}\times\mathbb{R}\times\mathbb{R}^{2}\right)$ such that $\pi^{1,2,3}\#\alpha_{0}=\left({\bf id},\theta_{0}\circ\pi^{2}\right)\#\mu_{\varrho_{0}}$ and $\pi^{3,4}\#\alpha_{0}=\sigma$ for which $P_{0}({\bf p},m)+\Psi_{0}({\bf q})=c({\bf p},m,{\bf q})$ $\alpha_{0}-$ a.e and*

[TABLE]

In that case,

[TABLE]

Proof: (1) Let $\varrho\in{\mathcal{R}}_{0}$ and $(P,\Psi)\in C(\bar{\mathcal{W}}\times\bar{\mathcal{I}}_{0})\times C(\bar{B}_{l}^{+})$ be such that

[TABLE]

for all $({\bf p},m)\in\mathcal{W}\times{\mathcal{I}}_{0}$ and ${\bf q}\in B_{l}^{+}$. Then,

[TABLE]

This implies that

[TABLE]

for any $\alpha\in\mathscr{P}\left(\mathbb{R}^{2}\times\mathbb{R}\times\mathbb{R}^{2}\right)$ such that $\pi^{1,2,3}\#\alpha=\left({\bf id},\theta_{0}\circ\pi^{2}\right)\#\mu_{\varrho}$, $\pi^{4,5}\#\alpha=\sigma$. We have used (4.13) in the last inequality of (4.15).

[TABLE]

and

[TABLE]

In view of (4.11), we combine (4.15)-(4.17) to obtain

[TABLE]

We observe that $\pi^{1,2}\#[\Phi\#\alpha]=\sigma$ and $\pi^{3,4}\#[\Phi\#\alpha]={\bf f}\#\mu_{\varrho}$. By taking the infimum in (4.18) over $\alpha$ we obtain that

[TABLE]

That is,

[TABLE]

(2) We have $(P,\Psi)\in{\mathcal{U}}_{0}$ and $\varrho\in{\mathcal{R}}$ satisfy the equality in (4.20) if and only if they satisfy the equality in (4.14) and in the second line of (4.15). The equality is satisfied in (4.14) if and only if (4.12) holds. The equality in the second line of (4.15) if and only if there exists $\alpha_{0}\in\mathscr{P}\left(\mathbb{R}^{2}\times\mathbb{R}\times\mathbb{R}^{2}\right)$ such that $\pi^{1,2,3}\#\alpha_{0}=\delta_{\theta_{0}(z)}\mu_{\varrho}$ and $\pi^{4,5}\#\alpha_{0}=\sigma$ , and $P_{0}({\bf p},m)+\Psi_{0}({\bf q})=c({\bf p},m,{\bf q})$ hold $\alpha_{0}$ almost everywhere. In that case, the equality holds in (4.18) and then in (4.19). As a result,

[TABLE]

$\square$

4.2. c-transforms and c-subdifferentials

Definition 4.2**.**

Let $l>0$, $\Psi\in C(B_{l}^{+})$ and $P\in C({\mathcal{W}}\times{\mathcal{I}}_{0})$. We define the $c-$transform of $\Psi$, denoted $\Psi^{c}$, by

[TABLE]

Similarly, we define the $c-$transform of $P$, denoted $P_{c}$, by

[TABLE]

We note that $c-$transform functions enjoy some regularity properties. Indeed, the functions $\Psi^{c}(p,\cdot),\Psi^{c}(\cdot,m)$ and $P_{c}$ are convex as supremum of convex functions. As a consequence, they are locally Lipschitz and thus differentiable almost everywhere with respect to the Lebesgue measure. We consider the set ${\mathcal{U}}_{0}$ of functions $(P,\Psi)$ defined by

[TABLE]

and

[TABLE]

Definition 4.3**.**

Let $l>0$ and $(P,\Psi)\in{\mathcal{U}}_{0}$. For any $({\bf p},m)\in\mathcal{W}\times\mathcal{I}_{0}$, we define

[TABLE]

In a similar way, for any ${\bf q}\in B_{l}$ we define

[TABLE]

Lemma 4.4**.**

Let $l>0$ and assume the condition (A1) holds.

1. (i)

*There exists $k_{0}>0$ such that whenever $(P,\Psi)\in{\mathcal{U}}_{0}$ we have $\Psi$ is *$k_{0}-$*Lipschitz continuous on $B_{l}$ and $P$ is *$k_{0}-$Lipschitz continuous on ${\mathcal{W}}\times{\mathcal{I}}_{0}$. 2. (ii)

Let $P$ be a $c$-transform of some $\Psi\in C(\bar{B}_{l})$. If $m_{0}\in\mathcal{I}_{0}$ and ${\bf p}_{0}=(s_{0},z_{0})$ a point of differentiability of $P(\cdot,m_{0})$ then

[TABLE]

As a consequence, the function $\mathcal{T}[P]({\bf p})={\mathcal{A}}[\theta_{0}(z)]\nabla_{{\bf p}}P({\bf p},\theta_{0}(z))$ is well-defined Lebesgue almost everywhere. If, in addition, $P({\bf p}_{0},\cdot)$ is differentiable at $m_{0}$ then

[TABLE] 3. (iii)

Let $\Psi$ be a $c$-transform of some $P\in C(\bar{{\mathcal{W}}}\times\bar{\mathcal{I}}_{0})$. If ${\bf q}_{0}$ be a point of differentiability of $\Psi$ then,

[TABLE]

If we assume furthermore that ${\bf q}_{0}\in\partial_{c}P({\bf p}_{0},m_{0})$, that ${\bf p}_{0}=(s_{0},z_{0})$ is a point of differentiability of $P(\cdot,m_{0})$ and that $m_{0}=\theta_{0}(z_{0})$ then the function $\mathcal{S}[\Psi]$ as defined in (3.8) is defined almost everywhere with respect to Lebesgue.

Proof:

1. Let ${\bf q}_{1},{\bf q}_{2}\in B_{l}$. Choose $({\bf p}_{1},m_{1})\in{\mathcal{W}}\times\mathcal{I}_{0}$ such that

[TABLE]

As $({\bf p}_{1},m_{1})\in{\mathcal{W}}\times\mathcal{I}_{0}$, we have

[TABLE]

Set

[TABLE]

We combine (4.30) and (4.30) to get

[TABLE]

By permuting the roles of ${\bf q}_{1}$ and ${\bf q}_{2}$ is the above reasoning, we obtain that

[TABLE]

It follows that $\Psi$ is $k_{0}$-lipschitz continuous on $B_{l}$. A similar argument shows that $P$ is $k_{0}$-lipschitz continuous on $\mathcal{W}\times\mathcal{I}_{0}$.

2. Let $m_{0}\in\mathcal{I}_{0}$ and ${\bf p}_{0}=(s_{0},z_{0})\in{\mathcal{W}}$. Let ${\bf q}_{0}\in\partial_{c}P({\bf p}_{0},m_{0})$, that is,

[TABLE]

Consider the map $(u,t)\longrightarrow B(u,t):=c(u,t,{\bf q}_{0})-P(u,t)-\Psi({\bf q}_{0})$. Assume that ${\bf p}_{0}=(s_{0},z_{0})\in{\mathcal{W}}$ is a point of differentiability of $P(\cdot,m_{0})$. Then, $B(\cdot,m_{0})$ is differentiable at ${\bf p}_{0}$ and attained its maximum at ${\bf p}_{0}$. Thus, $\nabla_{{\bf p}}B({\bf p}_{0},m_{0})=0$, that is, ${\mathcal{A}}(1/m_{0}){\bf q}_{0}=\nabla P(p_{0},m_{0})$. Hence ${\bf q}_{0}={\mathcal{A}}(m_{0})\nabla P({\bf p}_{0},m_{0})$. It follows that $\partial_{c}P({\bf p}_{0},m_{0})$ is given by (4.27). Since $P(\cdot,m_{0})$ is convex, it is locally lipschitz and thus differentiable Lebesgue almost everywhere. This implies that the map $\mathcal{T}[P]({\bf p})={\mathcal{A}}[\theta_{0}(z)]\nabla_{{\bf p}}P({\bf p},\theta_{0}(z))$ is well-defined Lebesgue almost everywhere. Assume in addition that $P(p_{0},\cdot)$ is differentiable at $m_{0}$ then $\partial_{t}V({\bf p}_{0},m_{0})=0$, that is, (4.28) holds.

3. Let ${\bf q}_{0}\in B_{l}$ and $({\bf p}_{0},m_{0})\in\partial^{c}\Psi({\bf q}_{0})$ with ${\bf p}_{0}=(s_{0},z_{0})$. Then, $P(p_{0},m_{0})=c({\bf p}_{0},m_{0},{\bf q}_{0})-\Psi({\bf q}_{0})$ and so the map ${\bf q}\longrightarrow E({\bf q})=c({\bf p}_{0},m_{0},{\bf q})-P({\bf p}_{0},m_{0})-\Psi({\bf q})$ attains its maximum at ${\bf q}_{0}$. Assume $q_{0}$ is a point of differentiability of $\Psi$. We have $\nabla E({\bf q}_{0})=0$, that is, $s_{0}=\partial_{\Upsilon}\Psi({\bf q}_{0})$ and $z_{0}=m_{0}\partial{Z}\Psi({\bf q}_{0})$. Thus, (4.29) holds. Assume $m_{0}=\theta_{0}(z_{0})$ and that (A1) holds. Since $\Psi$ is differentiable almost everywhere with respect to Lebesgue, the map $\mathcal{S}[\Psi]$ is well-defined almost everywhere with respect to Lebesgue. $\mathcal{S}[\Psi]$ is defined almost everywhere with respect to Lebesgue.   $\square$

Remark 2**.**

Let $(P,\Psi)\in{\mathcal{U}}_{0}$. By the characterization of $\partial_{c}P$ provided in lemma 4.4, we conclude that $0\leq\partial_{z}P,\;\partial_{s}P\leq l\quad{\mathcal{L}}^{3}-a.e$.

4.3. Existence of a maximizer in the dual problem

Let $l>0$. We recall that ${\mathcal{U}}_{0}$ denotes the subset of ${\mathcal{U}}$ consisting of $(P,\Psi)$ satisfying (4.23)-(4.24).

For $A>0$ we denote by

[TABLE]

Lemma 4.5**.**

Let $A>0$. Then

[TABLE]

Proof: As $P\leq A$ we have

[TABLE]

Since

[TABLE]

the result follows from (4.34).   $\square$

Lemma 4.6**.**

Let $A>0$ and assume that conditions (A1) and (A2) hold.

1. (i)

For each $z\in[0,H]$ fixed and $P\in\mathcal{E}_{A}$, $\mathscr{S}_{\theta_{0}}[P](\cdot,z)$ has a minimizer over $[0,1/(2r_{0}^{2}))$. 2. (ii)

There exists $M_{*}$ such that the following holds : for any $P\in\mathcal{E}_{A}$ and $z\in[0,H]$ if $\bar{\varrho}$ is a minimizer of $\mathscr{S}_{P}(\cdot,z)$ over $[0,1/(2r_{0}^{2}))$ then

[TABLE] 3. (iii)

*Assume, additionally, that $P$ is Lipschitz continuous and that $\partial_{z}P\geq 0\quad\mathcal{L}^{3}-a.e$. Let $z_{1},z_{2}\in[0,H]$ and $\varrho_{1},\varrho_{2}\in[0,1/(2r_{0}^{2}))$ be such that $\varrho_{i}$ is the minimizer of $\mathscr{S}_{\theta_{0}}[P](\cdot,z_{i})$ over $[0,1/(2r_{0}^{2}))$ $i=1,2$. If $z_{1}\leq z_{2}$, then $\varrho_{1}\leq\varrho_{2}.$ *

Proof: (i) follows from the continuity of $\mathscr{S}_{\theta_{0}}[P](\cdot,z)$ over $[0,1/(2r_{0}^{2}))$ and Lemma 4.5.

(ii) Let $\left\{M_{n}\right\}_{n=1}^{\infty}$ be such that $0<M_{n}\leq 1/(2r_{0}^{2})$ and $\left\{M_{n}\right\}_{n=1}^{\infty}$ converges to $1/(2r_{0}^{2})$. Assume there exist $\left\{P_{n}\right\}_{n=1}^{\infty}\subset\mathcal{E}_{A}$ and $\left\{z_{n}\right\}_{n=1}^{\infty}\subset[0,H]$ such that $\bar{\varrho}_{n}$ is a minimizer of $\mathscr{S}_{\theta_{0}}[P_{n}](\cdot,z_{n})$ over $[0,1/(2r_{0}^{2}))$ but $M_{n}\leq\bar{\varrho}_{n}\leq 1/(2r_{0}^{2})$. Then

[TABLE]

This contradicts Lemma 4.5.

(iii) For $z\in[0,H]$ fixed, $\mathscr{S}_{\theta_{0}}[P](\cdot,z)$ is differentiable on $(0,1/(2r_{0}^{2}))$ and we have

[TABLE]

Note that, by (A2), $(s,z)\longrightarrow(s,\theta_{0}(z))$ is Lipschitz continuous. As $P$ is Lipschitz continuous, $(s,z)\longrightarrow P(s,z,\theta(z))$ is Lipschitz and therefore differentiable Lebesgue almost everywhere on ${\mathcal{W}}$. The mixed partial derivatives of $\mathscr{S}_{\theta_{0}}[P]$ give

[TABLE]

for a.e $(\varrho,z)\in{\mathcal{W}}$. In light of (4.28), we have

[TABLE]

for a.e $(\varrho,z)\in{\mathcal{W}}$. We recall that $\phi(z)=\frac{z}{\theta_{0}(z)}$ and note that

[TABLE]

for all $z\in(0,H)$. It follows that

[TABLE]

for almost every $(\varrho,z)\in{\mathcal{W}}$. In light of the assumption (A1) and the fact that $\partial_{z}P\geq 0\quad\mathcal{L}^{3}-a.e$, we have

[TABLE]

Let $z_{i}\in[0,H]$ and $\varrho_{i}$ be a minimizer $\mathscr{S}_{\theta_{0}}[P](\cdot,z_{i})$ over $[0,1/(2r_{0}^{2}))$, $i=1,2.$ We exploit this minimality condition on $\varrho_{1}$, $\varrho_{2}$ to obtain

[TABLE]

In light of (4.41), the equation (4.42) implies the following: if $z_{1}<z_{2},$ then $\varrho_{1}\leq\varrho_{2}.$   $\square$

Lemma 4.7**.**

Let $A>0$ and $P\in\mathcal{E}_{A}$. Let $z_{0}\in(0,H)$. Assume that condition (A2) holds and that $P$ is Lipschitz continuous such that $\partial_{z}P\geq 0\quad{\mathcal{L}}^{3}-a.e$. Let $\varrho_{1},\varrho_{2}:[0,H]\longrightarrow[0,1/(2r_{0}^{2}))$ be two maps defined in such a way that ${\varrho}_{i}(z)$ are minimizers of $\mathscr{S}_{\theta_{0}}[P](\cdot,z)$ over $[0,1/(2r_{0}^{2}))$. Then, the following hold:

1. (i)

$\varrho_{1}(z_{0})$* and $\varrho_{2}$ are monotone.* 2. (ii)

Assume ${\varrho}_{i}$ are continuous at $z_{0}$. Then, $\varrho_{1}(z_{0})=\varrho_{2}(z_{0})$. 3. (iii)

$\mathscr{S}_{\theta_{0}}[P](\cdot,z)$* has a unique minimizer over $[0,1/(2r_{0}^{2}))$ for almost every $z$ with respect to Lebesgue.*

Proof: (i) follows from Lemma (4.6) (iii). Since ${\varrho}_{1}$ is continuous at $z_{0}$, $\lim_{\delta\rightarrow 0}\varrho_{1}(z_{0}-\delta)=\varrho_{1}(z_{0})$. In light of Lemma (4.6) (iii), $\varrho_{1}(z_{0}-\delta)\leq\varrho_{2}(z_{0})$ for $\delta$ small and positive. It follows that $\varrho_{1}(z_{0})\leq\varrho_{2}(z_{0})$. An analogous reasonning leads to $\varrho_{1}(z_{0})\geq\varrho_{2}(z_{0})$ which proves (ii). As $\varrho_{1},\varrho_{2}$ are monotone, they have a countable number of discontinuous points. Thus, by (ii), $\varrho_{1}(z_{0})=\varrho_{2}(z_{0})$ for almost every $z$ with respect to Lebesgue. This proves (iii).   $\square$

Lemma 4.8**.**

Let $l>0$ and $c_{0}\in\mathbb{R}$. Then, the following hold:

• (i)

The set of all $(P,\Psi)\in{\mathcal{U}}$ such that

[TABLE]

is precompact in $C(\bar{\mathcal{W}}\times\bar{\mathcal{I}}_{0})\times C(\bar{B}_{l})$.

• (ii)

The set ${\mathcal{M}}$ of $P\in C(\bar{\mathcal{W}}\times\bar{\mathcal{I}}_{0})$ such that $(P,\Psi)$ satisfies (4.43) for some $\Psi\in C(\bar{B}_{l})$ is contained in $\mathcal{E}_{A}$ for some $A>0$.

Proof: Fix $(\mathfrak{p},\mathfrak{m})\in{\mathcal{W}}\times{\mathcal{I}}$ and let $(P,\Psi)\in{\mathcal{U}}_{0}$. By lemma 4.4, $P$ is $k_{0}-$Lipschitz continuous. It follows that

[TABLE]

for all $({\bf p},m)\in{\mathcal{W}}\times{\mathcal{I}}_{0}$. As $\Psi+P(\mathfrak{p},\mathfrak{m})=\left(P-P(\mathfrak{p},\mathfrak{m})\right)^{c}$ and $\bar{k}_{1}:=\sup\left\{c({\bf p},m,{\bf q}):({\bf p},m)\in{\mathcal{W}}\times{\mathcal{I}}_{0},\;{\bf q}\in B_{l}\right\}$ is finite, we have that

[TABLE]

for all ${\bf q}\in B_{l}$. We observe that

[TABLE]

for $(P,\Psi)\in{\mathcal{U}}_{0}$ and for any constant $\bar{\varrho}\in[0,1/(2r_{0}^{2})))$. In light of (4.44) and (4.45), the estimate (4.46) implies

[TABLE]

for $(P,\Psi)\in{\mathcal{U}}_{0}$ and for any constant $\bar{\varrho}\in[0,1/(2r_{0}^{2})))$. For $(P,\Psi)\in{\mathcal{U}}_{0}$ such that $\mathcal{J}[\sigma](P,\Psi)\geq c_{0}$, we have

[TABLE]

for any constant $\bar{\varrho}\in[0,1/(2r_{0}^{2})))$. By choosing $\bar{\varrho}=0$ and then $\bar{\varrho}=\bar{\varrho}_{0}$ where $\bar{\varrho}$ is such that

$\left(1-H\int_{0}^{\bar{\varrho}_{0}}(2f_{0}(s)/\Omega^{2})^{2}dsdz\right)<0$ we obtain

[TABLE]

with

[TABLE]

As $(\mathfrak{p},\mathfrak{m})$ is an arbitrary point in ${\mathcal{W}}\times{\mathcal{I}}_{0}$, it follows from (4.47) that the set ${\mathcal{M}}$ is uniformly bounded with respect to the uniform norm and, in particular (4.47) implies that $P\in\mathcal{E}_{A}$ whenever $(P,\Psi)\in{\mathcal{M}}$. This proves (ii). As ${\mathcal{M}}$ is uniformly bounded the estimate (4.45) implies that the set of $\Psi$ such that $(P,\Psi)\in{\mathcal{U}}_{0}$ and $\mathcal{J}[\sigma](P,\Psi)\geq c_{0}$ is uniformly bounded with respect to the uniform norm. Using the uniform Lipschitz constant established in lemma 4.4 we have that the set of $(P,\Psi)\in{\mathcal{U}}_{0}$ such that $\mathcal{J}[\sigma](P,\Psi)\geq c_{0}$ is precompact which proves (i). Whenever $(P,\Psi)\in{\mathcal{U}}_{0}$, $P$ is Lipschitz -thus, differentiable Lebesgue almost everywhere- and $\partial_{c}P\subset B_{l}^{+}$.   $\square$

Lemma 4.9**.**

Let $l>0$ and $\left\{\sigma_{n}\right\}_{n=0}^{\infty}$ such that ${\rm spt}(\sigma_{n})\subset B_{l}^{+}$ and $\left\{\sigma_{n}\right\}_{n=1}^{\infty}$ converges narrowly to $\sigma_{0}$. Let $(P_{0},\Psi_{0})\subset{\mathcal{U}}$ and $\left\{(P_{n},\Psi_{n})\right\}_{n=1}^{\infty}\subset{\mathcal{U}}_{0}$ such $\left\{P_{n}\right\}_{n=1}^{\infty}$ converges uniformly to $P_{0}$ on compact subsets of ${\mathcal{W}}\times{\mathcal{I}}_{0}$ and $\left\{\Psi_{n}\right\}_{n=1}^{\infty}$ converges uniformly to $\Psi_{0}$ on $B_{l}$. Then, $\left\{{\mathcal{J}}[\sigma_{n}]\left(P_{n},\Psi_{n}\right)\right\}_{n=1}^{\infty}$ converges to ${\mathcal{J}}[\sigma_{0}]\left(P_{0},\Psi_{0}\right)$.

Proof: Let $\varrho_{n}\in{\mathcal{R}}$ such that $\varrho_{n}$ is monotone and $\varrho_{n}(z)$ is the minimizer of $\mathscr{S}_{\theta_{0}}[P_{n}](\cdot,z)$ over $[0,1/(2r_{0}^{2}))$ for $n\geq 0$ and for each $z\in[0,H]$ fixed, as provided by lemma 4.6(i) and lemma 4.7 (i). By lemma 4.6(ii) there exists $M_{*}>0$ such that

[TABLE]

for $n\geq 0$ and $z\in[0,H]$. Helly’s theorem ensures that $\left\{\varrho_{n}\right\}_{n=1}^{\infty}$ converges pointwise -up to a subsequence denoted again $\left\{\varrho_{n}\right\}_{n=1}^{\infty}$- to $\bar{\varrho}$. We set ${\mathcal{W}}_{M_{*}}=[0,M_{*}]\times[0,H]$. Note that $\mathscr{S}_{\theta_{0}}[P_{n}](\cdot,z)$ is uniformly bounded on ${\mathcal{W}}_{M_{*}}$. As $\left\{P_{n}\right\}_{n=1}^{\infty}$ converges uniformly to $P_{0}$ on ${\mathcal{W}}_{M_{*}}\times{\mathcal{I}}_{0}$, we easily check that $\left\{\mathscr{S}_{\theta_{0}}[P_{n}]\right\}_{n=1}^{\infty}$ converges uniformly to $\mathscr{S}_{\theta_{0}}[P_{0}]$ on ${\mathcal{W}}_{M_{*}}$. As a result, $\bar{\varrho}$ minimizes $\mathscr{S}_{\theta_{0}}[P_{0}](\cdot,z)$ over $[0,1/(2r_{0}^{2}))$. In light of lemma 4.7 (iii), it follows that $\bar{\varrho}=\varrho_{0}$ almost everywhere with respect to Lebesgue. By the definition of $\varrho_{n}$, it is straightforward that ${\mathcal{H}}(P_{n})=\int_{0}^{H}\mathscr{S}_{\theta_{0}}[P_{n}](\varrho_{n}(z),z)dz$ and so, the Lebesgue dominated convergence ensures that $\left\{{\mathcal{H}}(P_{n})\right\}_{n=1}^{\infty}$ converges to ${\mathcal{H}}(P_{0})$. Thus,

[TABLE]

As $\left\{\sigma_{n}\right\}_{n=0}^{\infty}$ converges narrowly to $\sigma_{0}$ and $\left\{\Psi_{n}\right\}_{n=1}^{\infty}$ converges uniformly to $\Psi_{0}$ on $B_{l}^{+}$, we get that

[TABLE]

It follows from (4.49) and (4.50) that $\left\{{\mathcal{J}}[\sigma_{n}]\left(P_{n},\Psi_{n}\right)\right\}_{n=1}^{\infty}$ converges to ${\mathcal{J}}[\sigma_{0}]\left(P_{0},\Psi_{0}\right)$.   $\square$

Proposition 4.10**.**

Let $l>0$ and $\sigma\in\mathscr{P}\left(\mathbb{R}^{2}\right)$ such that ${\rm spt}(\sigma)\subset B_{l}$. ${\mathcal{J}}[\sigma]$ admit a maximizer over ${\mathcal{U}}_{0}$.

Proof: Note that ${\mathcal{J}}[\sigma]\not\equiv\infty$. Indeed, set

[TABLE]

Then, $(P_{00},\Psi_{00})\in{\mathcal{U}}$ and $\mathfrak{c}_{00}:={\mathcal{J}}[\sigma](P_{00},\Psi_{00})$ is finite. Let $\{(P_{n},\Psi_{n})\}_{n=1}^{\infty}\subset{\mathcal{U}}$ be a maximizing sequence of ${\mathcal{J}}[\sigma]$. One can easily check that $P_{n}\leq(P_{n\;c})^{c}$, $\Psi_{n}\leq(P_{n\;c})$ and that ${\mathcal{J}}[\sigma](P_{n},\Psi_{n})\leq{\mathcal{J}}[\sigma](P_{n\;c})^{c},P_{n\;c})$. As $\{(P_{n\;c})^{c},(P_{n\;c})\}\in{\mathcal{U}}_{0}$, we assume without loss of generality that the maximizing sequence $\{(P_{n},\Psi_{n})\}_{n=1}^{\infty}\subset{\mathcal{U}}_{0}$. Therefore, $\mathcal{J}[\sigma](P_{n},\Psi_{n})>\mathfrak{c}_{00}$ for $n\geq n_{0}$ for some positive integer $n_{0}$. In light of lemma 4.8, there exists a subsequence of $\{(P_{n},\Psi_{n})\}_{n=1}^{\infty}$ that we denote again by $\{(P_{n},\Psi_{n})\}_{n=1}^{\infty}$ that converges uniformly to $(P_{0},\Psi_{0})$. By lemma 4.9, we have that $\{{\mathcal{J}}[\sigma](P_{n},\Psi_{n})\}_{n=1}^{\infty}$ converges to ${\mathcal{J}}[\sigma](P_{0},\Psi_{0})$. As a result, $(P_{0},\Psi_{0})$ is a maximizer of ${\mathcal{J}}[\sigma]$ over ${\mathcal{U}}$ and we have ${\mathcal{J}}[\sigma](P_{0},\Psi_{0})\leq{\mathcal{J}}[\sigma](P_{0\;c})^{c},P_{0\;c})$. This concludes the proof   $\square$

4.4. Existence of a minimizer in the primal problem

In this section, we show the existence and uniqueness of the minimizer in variational problem. This result is achieved through the study of dual problem. Subsequently, we obtain a solution for problem (4.1).

Proposition 4.11**.**

Let $c_{0},\;l>0$ and $\sigma\in\mathscr{P}(\mathbb{R}^{2})$ such that ${\rm spt}(\sigma)\subset B^{+}_{l}$. Assume the condition (A1) and (A2) hold.

1. (i)

${\mathcal{K}}[\sigma]$* admits a unique minimizer $\varrho_{0}$ over ${\mathcal{R}}_{0}$. Furthermore, if $(P_{0},\Psi_{0})\in{\mathcal{U}}_{0}$ is a maximizer of ${\mathcal{J}}[\sigma]$ on ${\mathcal{U}}$, then ${\mathcal{T}}[P_{0}]:={\mathcal{A}}(\theta_{0})\nabla P_{0}$ pushes $\mu_{\varrho_{0}}$ forward onto $\sigma$ so that ${\mathcal{J}}[\sigma](P_{0},\Psi_{0})={\mathcal{K}}[\sigma](\varrho_{0})+\mathbf{m}_{2}[\sigma]$ and $\varrho_{0}$ is monotone non decreasing on $[0,H]$ satisfying*

[TABLE]

If, additionally, we assume that $\sigma$ is absolutely continuous with respect to the Lebesgue measure then ${\mathcal{S}}[\Psi_{0}]$, defined in (3.8), pushes $\sigma$ forward onto $\mu_{\varrho_{0}}$ and we have

[TABLE] 2. (ii)

Assume $\sigma$ is absolutely continuous with respect to the Lebesgue measure such that $\frac{\partial\sigma}{\partial{\mathcal{L}}^{2}}>c_{0}\quad{\mathcal{L}}^{2}-a.e$ and that ${\rm spt}(\sigma)=B_{l}^{+}$. If $(P_{0},\Psi_{0}),\;(P_{1},\Psi_{1})\in{\mathcal{U}}_{0}$ are such that $(P_{0},\Psi_{0})$ is a maximizer of ${\mathcal{J}}[\sigma]$ and ${\mathcal{J}}[\sigma](P_{0},\Psi_{0})={\mathcal{J}}[\sigma](P_{1},\Psi_{1})$ then we have that $P_{1}=P_{0}\quad\text{on}\quad{\mathcal{W}}\times{\mathcal{I}}_{0}$ and $\Psi_{1}=\Psi_{0}\quad\text{on}\quad B_{l}^{+}$. 3. (iii)

Assume that (A1’) holds and that $(P_{0},\Psi_{0})$ is a maximizer of ${\mathcal{J}}[\sigma]$ such that $\partial_{z}P_{0}\geq b_{0}\quad{\mathcal{L}}^{3}-a.e$ for some $b_{0}>0$. For any $z_{1},z_{2}\in[0,H]$ such that $\varrho_{0}(z_{1}),\varrho_{0}(z_{2})>0$, there exists $C>0$ such that

[TABLE]

Remark 3**.**

• •

If ${\rm spt}(\sigma)\subset B_{l}^{b_{0}+}:=B_{l}^{+}\cap{\mathcal{V}}_{b_{0}}$ where ${\mathcal{V}}_{b_{0}}=(0,\infty)\times(b_{0},\infty)$ with $0<b_{0}<l$ then ${\mathcal{J}}[\sigma]$ admits a maximizer $(P_{0},\Psi_{0})$ satisfying (4.23) and (4.24) with $B_{l}^{+}$ replaced by $B_{l}^{b_{0}+}$. As a result, we obtain $\partial_{z}P_{0}\geq b_{0}\quad{\mathcal{L}}^{3}-a.e$.

• •

The estimate (4.53) implies that the boundary of the domain $D_{\varrho_{0}}$ is piecewise Lipschitz continuous. This result can be found in **[5]**.

Proof: 1. Assume that $(P_{0},\Psi_{0})\in{\mathcal{U}}_{0}$ is a maximizer of ${\mathcal{J}}[\sigma]$ over ${\mathcal{U}}$. Let $\varrho_{0}\in{\mathcal{R}}$ such that for each $z\in[0,H]$ $\varrho_{0}(z)$ is a minimizer of $\mathscr{S}_{\theta_{0}}[P_{0}](\cdot,z)$ over $[0,1/(2r_{0}^{2}))$. Then, if $\varrho_{0}(z)>0$ by differentiating $\mathscr{S}_{\theta_{0}}[P_{0}](\cdot,z)$ at $\varrho_{0}(z)$ we get (4.51). Using the minimizing property of $\varrho_{0}$, we have

[TABLE]

for all $\varrho\in{\mathcal{R}}$. As a result,

[TABLE]

For $h\in C_{c}(\mathbb{R}^{2})$ and $\kappa\in(-1,1)$, we set

[TABLE]

We note that $\left\{P_{\kappa}\right\}_{-1<\kappa<1}\subset C(\bar{\mathcal{W}}\times{\mathcal{I}}_{0}).$ One can show that (cfr[15]) the following holds:

[TABLE]

for Lebesgue almost every ${\bf p}\in\mathbb{R}^{2}$. Let $\left\{\kappa_{n}\right\}^{\infty}_{n=1}$ a sequence of $(-1,1)$ that converges to [math]. Let $\varrho_{\kappa_{n}}\in{\mathcal{R}}$ such that $\varrho_{\kappa_{n}}(z)$ a minimizer of $\mathscr{S}_{\theta_{0}}[P_{\kappa_{n}}](\cdot,z)$ over $[0,1/(2r_{0}^{2}))$. It follows from lemma 4.6 (i) and lemma 4.7 (i) that $\left\{\varrho_{\kappa_{n}}\right\}_{n=1}^{\infty}$ is a sequence of monotone functions of ${\mathcal{R}}$ uniformly bounded away from $1/(2r_{0}^{2})$. By Helly’s theorem there exists a subsequence of $\left\{\varrho_{\kappa_{n}}\right\}_{n=1}^{\infty}$ still denoted $\left\{\varrho_{\kappa_{n}}\right\}_{n=1}^{\infty}$ such that $\left\{\varrho_{\kappa_{n}}\right\}_{n=1}^{\infty}$ converges to some $\bar{\varrho}\in{\mathcal{R}}$. In view of the first equation of (4.55), $\left\{P_{\kappa_{n}}\right\}_{n=1}^{\infty}$ is a sequence of continuous functions that converges uniformly to $P_{0}$ on compact subsets of ${\mathcal{W}}\times{\mathcal{I}}_{0}$ and so, $\left\{\mathscr{S}_{\theta_{0}}[P_{\kappa_{n}}]\right\}_{n=1}^{\infty}$ is a sequence of continuous functions that converges uniformly to $\mathscr{S}_{\theta_{0}}[P_{0}]$ on compact subsets of ${\mathcal{W}}$. As a result, $\bar{\varrho}$ is a minimizer of $\mathscr{S}_{\theta_{0}}[P_{\kappa_{0}}](\cdot,z)$ over $[0,1/(2r_{0}^{2}))$. Using lemma 4.7 (iii), we conclude that $\varrho_{0}=\bar{\varrho}$ Lebesque almost everywhere on $[0,H]$. And so,

[TABLE]

for Lebesgue almost all $z\in[0,H]$. We exploit the minimizing property of $\varrho_{0}(z)$ to get

[TABLE]

Analogously, we use the minimizing property of $\varrho_{\kappa}(z)$ to obtain

[TABLE]

We combine (4.57) and (4.58) to get that

[TABLE]

with

[TABLE]

By lemma 4.6, we choose $M_{*}$ such that

[TABLE]

for $z\in[0,H]$ and $n\geq 1$. As $f_{0}$ is bounded on $[0,M_{*}]$,

[TABLE]

It follows that

[TABLE]

By the Lebesgue dominated convergence theorem, (4.55) implies that

[TABLE]

We combine (4.62) and (4.63) to get that

[TABLE]

We note that

[TABLE]

We use (4.64) and (4.65) to get

[TABLE]

Since $(P_{0},\Psi_{0})$ maximizes $\mathcal{J}[\sigma]$ over ${\mathcal{U}}$ and $(P_{\kappa},\Psi_{\kappa})\in{\mathcal{U}}$, (4.66) implies that

[TABLE]

As $h\in C_{c}(\mathbb{R}^{2})$ is arbitrary, we have that (4.67) implies that ${\mathcal{T}}[P_{0}]\#\mu_{\varrho_{0}}=\sigma$. By lemma 4.4, ${\mathcal{T}}[P_{0}]({\bf p})\in\partial^{c}P_{0}({\bf p},\theta_{0}(z))$ for almost every ${\bf p}\in{\mathcal{W}}$. As $\mu_{\varrho_{0}}$ is absolutely continuous with respect to Lebesgue, we have

[TABLE]

that is,

[TABLE]

This, combined with (4.54) yields ${\mathcal{K}}[\sigma](\varrho_{0})+\mathbf{m}_{2}[\sigma]={\mathcal{J}}[\sigma](P_{0},\Psi_{0})$ in light of Proposition 4.1. As a result, $\varrho_{0}$ is a minimizer of ${\mathcal{K}}[\sigma]$ over ${\mathcal{R}}_{0}$.

5. Assume $\sigma$ is absolutely continuous with respect to Lebesgue. A similar reasoning as above yields that ${\mathcal{S}}[\Psi_{0}]\#\sigma=\mu_{\varrho_{0}}$. As $\left({\mathcal{S}}[\Psi_{0}]({\bf q}),\theta_{0}\circ\phi^{-1}(\partial_{Z}\Psi_{0}({\bf q}))\right)$ belongs to $\partial^{c}\Psi({\bf q})$ for Lebesgue almost every ${\bf q}\in B_{l}$ we have

[TABLE]

Using lemma 4.4, the results in (4.68) and (4.69) imply that ${\mathcal{T}}[P_{0}]\circ{\mathcal{S}}[\Psi_{0}]({\bf q})={\bf q}\quad\sigma-a.e$ and ${\mathcal{S}}[\Psi_{0}]\circ{\mathcal{T}}[P_{0}]({\bf p})={\bf p}\quad\mu_{\varrho_{0}}-a.e$. It follows that $\alpha_{0}=\left({\mathcal{S}}[\Psi_{0}],\;\theta_{0}\circ{\mathcal{S}}_{1}[\Psi_{0}],\;{\bf id}\right)\#\sigma$. We note that

[TABLE]

By lemma 4.1, $\Phi\#\alpha_{0}$ is the unique optimal plan between $\sigma$ and $\mathbf{f}\#\mu_{\varrho_{0}}$ with respect to the quadratic distance. 6. Assume $(P_{1},\Psi_{1})$ is another maximizer of ${\mathcal{J}}[\sigma]$ in ${\mathcal{U}}_{0}$. In light of (4.70), we have

[TABLE]

$\Phi$ here is defined in (4.10). As $\mathbf{f}$ is bijective, (4.71) implies that ${\mathcal{S}}[\Psi_{0}]={\mathcal{S}}[\Psi_{1}]\quad\sigma-$a.e, that is, $\partial_{\Upsilon}\Psi_{0}=\partial_{\Upsilon}\Psi_{1}$ and $\partial_{Z}\Psi_{0}=\partial_{Z}\Psi_{1}\;\sigma-$a.e. As $\sigma$ is absolutely continuous with respect to the Lebesgue measure with $\frac{\partial\sigma}{\partial{\mathcal{L}}^{2}}>c_{0}\quad{\mathcal{L}}^{2}-a.e$ and $\Psi_{0},\;\Psi_{1}$ are Lipschitz continuous, we have $\Psi_{1}=\Psi_{0}+k$ on $B_{l}$ for some $k\in\mathbb{R}$. Since $(P_{0},\Psi_{0}),(P_{1},\Psi_{1})\in{\mathcal{U}}_{0}$ we get

[TABLE]

In light of (4.51), the equation (4.72) yields that $P_{1}=P_{0}$.

7. Set $Q(s,z)=f_{0}(s)-P_{0}(s,z,\theta_{0}(z))$. Then, by lemma 4.8 (iii)

[TABLE]

and

[TABLE]

We use (4.73) to obtain

[TABLE]

We recall that $\theta_{0}$ has values in the bounded interval ${\mathcal{I}}_{0}$. By condition (A1’), there exists $b_{1}>0$ such that $\theta_{0}\geq b_{1}$ and $\phi^{\prime}\geq b_{1}$. Thus, (4.74) implies that $\partial_{z}Q(\bar{s},z_{1})\geq b_{0}b_{1}^{2}=:c_{2}$. It follows that

[TABLE]

In view of (4.51), $Q(\varrho_{0}(z_{1}),z_{1})=Q(\varrho_{0}(z_{2}),z_{2})=0$ so that by combining (4.75) and (4.76), we obtain

[TABLE]

We obtain (4.53) by interchanging $z_{1}$ and $z_{2}$ in (4.77).   $\square$

5. Stability of the optimal transports

Let $l>0$, $\sigma\in\mathscr{P}\left(\mathbb{R}^{2}\right)$ such that ${\rm spt}(\sigma)\subset B_{l}$ and $\theta_{0}:[0,H]\longrightarrow{\mathcal{I}}_{0}$. We recall that

[TABLE]

for any $\varrho\in{\mathcal{R}}_{0}$, the set of all $\varrho$ for which $\mu_{\varrho}$ is a probability measure. Here, $\mathbf{f}(s,z)=\left(s,z/\theta_{0}(z)\right)$ for any $(s,z)\in{\mathcal{W}}$. As, $\theta_{0}$ is of values in ${\mathcal{I}}_{0}$, $\phi$ is bounded and

[TABLE]

for all $\varrho\in{\mathcal{R}}_{0}$ such that $0\leq 2r_{0}^{2}\varrho(z)\leq 2r_{0}^{2}K<1$ for all $z\in[0,H]$.

Lemma 5.1**.**

Let $l>0$ and $\left\{\sigma_{n}\right\}_{n=0}^{\infty}\subset\mathscr{P}\left(\mathbb{R}^{2}\right)$ such that ${\rm spt}(\sigma_{n})\subset B_{l}$ for all $n\geq 0$ and $\left\{\varrho_{n}\right\}_{n=0}^{\infty}\subset\mathcal{R}_{0}$. Assume that $\left\{\sigma_{n}\right\}_{n=1}^{\infty}$ converges narrowly to $\sigma_{0}$ and that $\left\{\varrho_{n}\right\}_{n=1}^{\infty}$ converges pointwise to $\varrho_{0}$. Then, $\left\{\mathcal{K}[\sigma_{n}](\varrho_{n})\right\}_{n=1}^{\infty}$ converges to $\mathcal{K}[\sigma_{0}](\varrho_{0})$.

Proof: As $\mathbf{f}$ is bounded continuous and $\left\{\mu_{\varrho_{n}}\right\}_{n=1}^{\infty}$ converges narrowly to $\mu_{\varrho_{0}}$, we have that $\mathbf{f}\#\mu_{\varrho_{n}}$ is supported in a fixed bounded domain for $n\geq 1$ and $\left\{\mathbf{f}\#\mu_{\varrho_{n}}\right\}_{n=0}^{\infty}$ converges narrowly to $\mathbf{f}\#\mu_{\varrho_{0}}$. We then use the continuity of the Wasserstein distance $W_{2}(\cdot,\cdot)$ to get the result.   $\square$

Proposition 5.2**.**

Let $c_{0}\in\mathbb{R}$, $l>0$ and $\left\{\sigma_{n}\right\}_{n=0}^{\infty}\subset\mathscr{P}\left(\mathbb{R}^{2}\right)$ such that ${\rm spt}(\sigma_{n})\subset B_{l}$ for all $n\geq 0$. Let $\left\{(P_{n},\Psi_{n})\right\}_{n=0}^{\infty}\in{\mathcal{U}}_{0}$ such that ${\mathcal{J}}[\sigma_{n}](P_{n},\Psi_{n})\geq c_{0}$ and let $\left\{\varrho_{n}\right\}_{n=0}^{\infty}\subset\mathcal{R}_{0}$ be a sequence of monotone functions such that

[TABLE]

for all $n\geq 0$. If $\left\{\sigma_{n}\right\}_{n=1}^{\infty}$ converges narrowly to $\sigma_{0}$ then the following holds :

• (i)

$\left\{\mu_{\varrho_{n}}\right\}_{n=1}^{\infty}$* converges narrowly to $\mu_{\varrho_{0}}$.*

• (ii)

$\left\{{\mathcal{T}}[P_{n}]\right\}_{n=1}^{\infty}$* converges pointwise to ${\mathcal{T}}[P_{0}]$ Lebesgue almost everywhere.*

• (iii)

$\left\{{\mathcal{S}}[\Psi_{n}]\right\}_{n=1}^{\infty}$* converges pointwise to ${\mathcal{S}}[\Psi_{0}]$ Lebesgue almost everywhere.*

Proof: 1. Lemma 4.8 ensures that $P_{n}\in\mathcal{E}_{A}$ for all $n\geq 0$ for some $A>0$. Using lemma 4.6(ii), there exists $M_{*}>0$ such that

[TABLE]

for all $n\geq 0$ and $z\in[0,H]$. In light of Helly’s theorem, we assume that $\left\{\varrho_{n}\right\}_{n=1}^{\infty}$ converges to some monotone function $\bar{\varrho}_{0}$. As a result, it is straightforward that $\left\{\mu_{\varrho_{n}}\right\}_{n=1}^{\infty}$ converges weakly∗ to $\mu_{\bar{\varrho}_{0}}$. Note that

[TABLE]

And so,

[TABLE]

Thus, $\left\{\mu_{\varrho_{n}}\right\}_{n=1}^{\infty}$ is tight and without loss of generality, we assume that $\left\{\mu_{\varrho_{n}}\right\}_{n=1}^{\infty}$ converges narrowly to $\mu_{\bar{\varrho}_{0}}$. We next show that $\varrho_{0}=\bar{\varrho}_{0}\quad{\mathcal{L}}^{1}-a.e$. In light of lemma 4.9 and lemma 5.1, (5.3) becomes in the limit:

[TABLE]

In view of lemma 4.1, the equality in (5.5) implies that $\bar{\varrho}_{0}$ is a minimizer of ${\mathcal{K}}[\sigma_{0}]$. The uniqueness result established in proposision 4.11 thus guarantees that $\bar{\varrho}_{0}=\varrho_{0}\quad{\mathcal{L}}^{1}-a.e$. The reasoning above applies to any subsequence of $\left\{\mu_{\varrho_{n}}\right\}^{\infty}_{n=1}$. As the limit is unique, we conclude that (i) holds.

2. Let ${\bf p}_{0}=(s_{0},z_{0})$ be a point of ${\mathcal{W}}$ such that $P_{n}$ is differentiable at $({\bf p}_{0},\theta(z_{0}))$ for $n\geq 0$. Let ${\bf q}_{n}\in\partial P_{n}({\bf p}_{0},\theta_{0}(z_{0}))$. As $({\bf p}_{0},\theta(z_{0}))$ is a point of differentiability of $P_{n}$, we have ${\bf q}_{n}={\mathcal{T}}[P_{n}]({\bf p}_{0})$ by lemma 4.4(ii). Since $\partial P_{n}({\bf p}_{0},\theta_{0}(z_{0}))\subset\bar{B}_{l}$, up to a subsequence, $\left\{{\bf q}_{n}\right\}^{\infty}_{n=1}$ converges to some ${\bf q}_{0}\in\bar{B}_{l}$. By definition of $\partial P_{n}({\bf p}_{0},\theta_{0}(z_{0}))$, we have $P_{n}({\bf p}_{0},\theta_{0}(z_{0}))+\Psi_{n}({\bf q}_{n})=c({\bf p}_{0},m,{\bf q}_{n})$. The continuity of $\Psi_{n}$ and $c$ and uniform convergence of $\left\{P_{n}\right\}^{\infty}_{n=1}$ and $\left\{\Psi_{n}\right\}^{\infty}_{n=1}$ yield $P_{0}({\bf p}_{0},\theta_{0}(z_{0}))+\Psi_{0}({\bf q}_{0})=c({\bf p}_{0},m,{\bf q}_{0})$. Thus, ${\bf q}_{0}={\mathcal{T}}[P_{0}]({\bf p}_{0})$, which is independent of subsequences of $\left\{{\bf q}_{n}\right\}^{\infty}_{n=1}$. As $P_{n},\;n\geq 0$ is differentiable almost everywhere, (ii) holds. (iii) holds by similar arguments.   $\square$

6. Existence of solutions for Continuity equations associated with the Axisymmetric Model

In section 3, we identified a class of continuity equations which yield solutions to the axisymmetric flows provided the velocity field associated with this of continuity equations is smooth enough. In this section, we construct solutions to such continuity equations. We point out, however, that the solution constructed are not smooth enough to generate a solution to the axisymmetric flow.

Assume (A1) holds and let $T>0$. For any $\Psi:{\mathcal{V}}\longrightarrow\mathbb{R}$ convex such that $\nabla\Psi({\bf q})\in{\mathcal{W}}$ a.e, for all ${\bf q}\in{\mathcal{V}}$ and $t\in[0,T]$ we associate the velocity field

[TABLE]

Let $l>0$. Under condition (B1),

[TABLE]

for all $t\in[0,T]$ and ${\bf q}\in B_{l}$.

Lemma 6.1**.**

Assume conditions (A1), (B1), (B2) and (B3) hold. Let $l>0$ and $\Psi:{\mathcal{V}}\longrightarrow\mathbb{R}$ convex such that $\nabla\Psi({\bf q})\in{\mathcal{W}}$ a.e for all ${\bf q}\in{\mathcal{V}}$. There exists a sequence of convex smooth functions $\left\{\Psi_{n}\right\}_{n\geq 1}$ $\Psi_{n}:B_{l}\longrightarrow\mathbb{R}$ such that ${\rm div}\left(V_{t}[\Psi_{n}]\right)\geq 0$ and $\nabla\Psi_{n}({\bf q})\in{\mathcal{W}}$ a.e for all ${\bf q}\in B_{l}$ for all $n\geq 1$ and $\left\{V_{t}[\Psi_{n}]\right\}_{n\geq 1}$ converges to $V_{t}[\Psi]$ almost everywhere with respect to the Lebesgue measure.

Proof: Since $\Psi$ be a convex function, $\Psi$ is locally Lipschitz and thus differentiable almost everywhere with respect to Lebesgue. Let $j$ be a smooth probability density contained with support contained in the unit ball. We consider the functions $\Psi_{n}:{\mathcal{V}}_{n}\longrightarrow\mathbb{R}$ defined by $\Psi_{n}=j_{n}*\Psi$, with $j_{n}=\frac{1}{n^{2}}j(\frac{\cdot}{n})$ and ${\mathcal{V}}_{n}=\left\{{\bf q}\in{\mathcal{V}}:dist({\bf q},\partial{\mathcal{V}})\right\}$. It follows that $\left\{\nabla\Psi_{n}\right\}_{n=1}^{\infty}$ converges to $\nabla\Psi$ in $L_{loc}^{1}({\mathcal{V}})$. Thus, there exists a subsequence of $\left\{\nabla\Psi_{n}\right\}_{n=1}^{\infty}$ denoted again by $\left\{\nabla\Psi_{n}\right\}_{n=1}^{\infty}$ that converges to $\nabla\Psi$ almost everywhere with respect to the Lebesgue measure. As $\phi^{-1}$ is continuous, $\left\{\phi^{-1}\left(\frac{\partial\Psi_{n}}{\partial Z}\right)\right\}_{n=1}^{\infty}$ converges to $\left\{\phi^{-1}\left(\frac{\partial\Psi}{\partial Z}\right)\right\}$ almost everywhere with respect to the Lebesgue measure. As a consequence, $\left\{V_{t}[\Psi_{n}]\right\}_{n\geq 1}$ converges to $V_{t}[\Psi]$ almost everywhere with respect to the Lebesgue measure. As $\Psi_{n}$ is smooth, we have

[TABLE]

Here, for simplicity, we make the following identifications:

[TABLE]

[TABLE]

$i=0,1$. In light of the convexiy of $\Psi_{n}$ and conditions (A1), (B1), (B2) and (B3), the equation (6.3) implies that ${\rm div}\ [V_{t}[\Psi_{n}]]\geq 0$.   $\square$

Lemma 6.2**.**

*Let $l_{0}>0$, $\tau>0$. $C_{0}$ is as defined in (6.2). Assume conditions (A1), (A2), (B1), (B2) and (B3). Let $t_{0}>0$ and $\sigma_{t_{0}}\in\mathscr{P}^{ac}\left(\mathbb{R}^{2}\right)$ such that ${\rm spt}(\sigma_{t_{0}})\subset B^{+}_{l_{0}}$. Let $\Psi$ a convex function on ${\mathcal{V}}$ such that $\nabla\Psi({\bf q})\in{\mathcal{W}}$ ${\mathcal{L}}^{2}$-a.e, for all ${\bf q}\in{\mathcal{V}}$. Then, there exists $\sigma_{t}\in\mathscr{P}^{ac}\left(\mathbb{R}^{2}\right)$ such that ${\rm spt}(\sigma_{t})\subset B^{+}_{l_{t}}$ with $l_{t}\leq l_{0}+C_{0}(l_{0})(t-t_{0})$ for $t\in[t_{0},t_{0}+\tau)$ satisfying :

• (a)

$\int_{\mathbb{R}^{2}}\left(\frac{\partial\sigma_{t}}{\partial{\mathcal{L}}^{2}}\right)^{r}d{\bf q}\leq\int_{\mathbb{R}^{2}}\left(\frac{\partial\sigma_{t_{0}}}{\partial{\mathcal{L}}^{2}}\right)^{r}d{\bf q}$* for any $r\geq 1$ and $t\in[t_{0},t_{0}+\tau)$.*

• (b)

$t\longmapsto\sigma_{t}\in AC_{1}\left(t_{0},t_{0}+\tau;\mathscr{P}(\mathbb{R}^{2})\right)$* and*

[TABLE]

• (c)

$t\longmapsto\sigma_{t}$* is Lipschitz continuous with respect to the *$1-$Wasserstein distance and satisfies

[TABLE]

for all $t_{0}\leq\bar{t},t\leq t_{0}+\tau_{0}$.

Proof: By lemma 6.1, there exist a sequence $\left\{\Psi_{n}\right\}_{n=1}^{\infty}$ such that ${\rm div}(V_{t}[\Psi_{n}])\geq 0$ and $V_{t}[\Psi_{n}]$ converges to $V_{t}[\Psi]$ ${\mathcal{L}}^{2}$-a.e. For each $n$ fixed, let $w^{n}_{t}$ be the flow associated to the vector field $V_{t}[\Psi_{n}]$ defined by $\dot{w}^{n}_{t}=V_{t}[\Psi_{n}](w^{n})$ and $w_{t_{0}}^{n}(t_{0})={\bf id}$. Then, $\sigma^{n}_{t}=w^{n}_{t}\#\bar{\sigma}_{t_{0}}$ solves $\eqref{eq:cont.Eq0}$ when $\Psi$ is replaced by $\Psi_{n}$. Since ${\rm div}(V_{t}[\Psi_{n}])\geq 0$, we have $\det\left(\nabla w^{n}\right)\geq\det\left(\nabla w^{n}_{t_{0}}\right)=1$ for any $n\geq 1$, and $t\in[t_{0},t_{0}+\tau)$. It follows that

[TABLE]

for any $n\geq 1$, $r\geq 1$ and $t\in[t_{0},\;t_{0}+\tau)$. This ensures that (a) holds for $\sigma^{n}$. In view of (6.2), we have

[TABLE]

Therefore,

[TABLE]

for all $t\in[t_{0},t_{0}+\tau)$ and ${\bf q}\in B_{l_{0}}^{+}$. It follows that $w_{t}^{n}(B_{l_{0}}^{+})\subset B_{l_{t}}^{+}$ where $l_{t}\leq l_{0}+C_{0}(l_{0})(t-t_{0})$. As ${\rm spt}(\sigma_{t_{0}})\subset B_{l_{0}}^{+}$ and $w^{n}_{t}$ is continuous, we have that ${\rm spt}(\sigma_{t})\subset B_{l_{t}}^{+}$ for all $t\in[t_{0},t_{0}+\tau)$. By [Theorem 8.3.1, [9]],

[TABLE]

Consequently, $t\longrightarrow\sigma^{n}_{t}$ is $C_{0}(l_{0})$-Lipschitz continuous on $[t_{0},\;t_{0}+\tau)$ for all $n\geq 1$. Thus,

[TABLE]

for all $t\in[t_{0},t_{0}+\tau]$ $n\geq 1$ . We conclude that $\left\{\sigma^{n}_{t}\right\}_{n=1}^{\infty}$ is uniformly bounded in the $1-$Wasserstein space. It follows from the $C_{0}(l_{0})-$Lipschitz continuity and uniform boundness of $\left\{\sigma^{n}_{t}\right\}_{n=1}^{\infty}$ that there exists a subsequence of $\left\{\sigma^{n}\right\}_{n=1}^{\infty}$ still denoted $\left\{\sigma^{n}_{t}\right\}_{n}$ ($n$ is independent of $t$) such that $\left\{\sigma^{n}_{t}\right\}_{n=1}^{\infty}$ converges narrowly to some $\sigma_{t}$. In light of (6.6), the Dunford Pettis theorem ensures that $\sigma_{t}$ is absolutely continuous with respect to Lebesgue and the weak lower semicontinuity of the $L^{r}-$ norm establishes (a). We note that, in view of (6.7), (c) is guaranteed by the lower semicontinuity of the Wasserstein distance with respect to the narrow convergence.

As $\left\{\sigma^{n}\right\}_{n=1}^{\infty}$ converges narrowly to $\sigma_{t}$ , $\left\{V_{t}[\Psi_{n}]\right\}_{n=1}^{\infty}$ converges to $V_{t}[\Psi]$ a.e and $\left\{V_{t}[\Psi_{n}]\right\}_{n=1}^{\infty}$ is bounded. These, combined with the fact that $\left\{\sigma^{n}_{t}\right\}_{n=1}^{\infty}$ satisfies (a), yield that $V_{t}[\Psi_{n}]\sigma^{n}_{t}$ converges to $V_{t}[\Psi]\sigma_{t}$ in the sense of distributions fot $t$ fixed by standard convergence results.  $\square$

Theorem 6.3**.**

Assume the conditions (A1), (A2), (B1), (B2) and (B3) hold. Let $l>0$, $l_{0}>0$ and $T>0$ such that $e^{4MT}(4l_{0}+1)<l+1$. Let $c$ be as defined in (2.1). Let $\sigma_{0}\in\mathscr{P}^{ac}\left(\mathbb{R}^{2}\right)$ with ${\rm spt}(\sigma_{0})\subset B^{+}_{l_{0}}$. Let $\Psi_{0}\in C(B_{l_{0}})$ with and $\varrho_{0}$ a monotone function such that $(\Psi_{0}^{c})_{c}=\Psi_{0}$ and $((\Psi_{0}^{c}),\Psi_{0},\varrho_{0})$ solves (4.2). Then, there exist $\left\{\sigma_{t}\right\}_{t\in[0,\;T]}\subset\mathscr{P}^{ac}\left(\mathbb{R}^{2}\right)$ with ${\rm spt}(\sigma_{t})\subset B_{l}$, $\left\{\Psi_{t}\right\}_{t\in(0,\;T]}\subset C\left(\mathbb{R}^{2}\right)$ with $(\Psi_{t}^{c})_{c}=\Psi_{t}$ and a sequence of monotone functions $\left\{\varrho_{t}\right\}_{t\in(0,\;T]}$ such that $((\Psi_{t}^{c}),\Psi_{t},\varrho_{t})$ solves (4.2) for each $t\in[0,T]$. Moreover, $t\longmapsto\sigma_{t}$ is Lipschitz continuous on $[0,\;T]$, belongs to $AC_{1}\left(0,T;\mathscr{P}(\mathbb{R}^{2})\right)$ and satisfies

[TABLE]

Proof: Let $N$ be a positive integer. We divide the interval $[0,T]$ into $N$ sub-intervals, each of length $\tau=\frac{T}{N}$. We consider $\sigma_{t}^{N}$ on $[0,T]$ defined as follows: $\sigma^{N}_{|t=0}=\bar{\sigma}_{0}$ and $\sigma^{N}$ solves (6.4) on $[0,\tau)$ for $t_{0}=0$ and $\Psi=\Psi_{0}$ thanks to lemma 6.2. To construct $\sigma_{t}^{N}$ on $[\tau,2\tau)$ we first choose $\Psi_{\tau}$ and $\varrho_{\tau}$ such that $(\Psi_{\tau}^{c},\Psi_{\tau},\varrho_{\tau})$ solves (4.2) when $\sigma$ is replaced by $\bar{\sigma}_{0}$. Then, $\sigma_{t}^{N}$ is obtained on $[\tau,2\tau)$ as a solution of (6.4) for $t_{0}=\tau$ and $\Psi=\Psi_{\tau}$ thanks to lemma 6.2. We repeat this process $(N-2)$ more times on the intervals $[k\tau,(k+1)\tau)$, $2\leq k\leq N-1$ by choosing $(\Psi_{k\tau}^{c},\Psi_{k\tau},\varrho_{k\tau})$ as a solution to (4.2) when $\sigma$ is replaced by $\sigma_{k\tau}$. We point out that proposition 4.11 guarantees the existence of $(\Psi_{k\tau},\varrho_{k\tau})$ provided that the support of $\sigma_{k\tau}$ is bounded. We also point out that lemma 6.2 ensures that $\left\{\sigma_{k\tau}^{N}\right\}_{k=1}^{N}\subset\mathscr{P}^{ac}\left(\mathbb{R}^{2})\right)$ with ${\rm spt}(\sigma_{k\tau})\subset B_{l_{k\tau}}$, where $l_{(k+1)\tau}\leq l_{k\tau}+C(l_{k\tau})\tau$ for $0\leq k\leq N-1$. In light of this construction, $\sigma_{t}^{N}$ satisfies

[TABLE]

Here, ${\bf v}^{N}_{t}=V_{t}[\Psi_{k\tau}]$ for $k\tau\leq t<(k+1)\tau$. We next show that the support of $\sigma^{N}$ is uniformly bounded independently of $N$ provided that $e^{4MT}(4l_{0}+1)<l+1$. To that aim, we write $l_{k}=l_{k\tau}$ for simplicity. We thus have

[TABLE]

By an inductive argument, we easily show that

[TABLE]

For $k=N$, we get

[TABLE]

In light of lemma 6.2 (a), the construction above yields

[TABLE]

for any $r\geq 1$ and $t\in[0,T]$. We use lemma 6.2(c) to obtain that $\sigma_{N}$ is $C_{0}(l)$-Lipschitz continuous on $[0,T]$. Since $\sigma^{N}$ satisfies (iii) and $\sigma^{N}_{|t=0}=\bar{\sigma}_{0}$ for all $N>0$, in light of standard compactness results, we assume without loss of generality that $\left\{\sigma_{t}^{N}\right\}_{N=1}^{\infty}$ converges narrowly to some $\sigma_{t}\in\mathscr{P}\left(\mathbb{R}^{2}\right)$. As $\sigma_{t}^{N}$ satisfies (6.12), the Dunford-Pettis theorem yields that $\sigma_{t}\in\mathscr{P}^{ac}\left(\mathbb{R}^{2}\right)$. The narrow convergence of $\left\{\sigma_{t}^{N}\right\}_{N=1}^{\infty}$ combined with weak semi-continuity of the $W_{1}$ leads to $C_{0}(l)$-Lipschitz continuous on $[0,T]$. Define $\bar{\sigma}_{t}^{N}$ by $\bar{\sigma}_{t}^{N}:=\sigma_{k\tau}$ for $t\in[k\tau,(k+1)\tau)$, $0\leq k\leq N-1$. We note that

[TABLE]

It follows that, as $\left\{\sigma_{t}^{N}\right\}_{N=0}^{\infty}$ converges narrowly to $\sigma_{t}$, $\left\{\bar{\sigma}_{t}^{N}\right\}_{N=1}^{\infty}$ converges narrowly to $\sigma_{t}$ for $t\in[0,T]$. Consequently, $\left\{{\bf v}_{t}^{N}\right\}_{N=0}^{\infty}$ converges to $V_{t}[\Psi]\quad{\mathcal{L}}^{1}-a.e\;t\in[0,T]$, by using proposition 5.2. As $\left\{{\bf v}^{N}_{t}\right\}_{N=1}^{\infty}$ is uniformly bounded in the $L^{\infty}(\mathbb{R}^{2})$ and $\left\{\sigma_{t}^{N}\right\}_{N=0}^{\infty}$ satisfies (6.12), we have that $\left\{{\bf v}^{N}_{t}\sigma_{t}^{N}\right\}_{N=1}^{\infty}$ converges in the sense of distributions to $V_{t}[\Psi_{t}]\sigma_{t}$ for a.e $t\in[0,T]$. Thus, we have that $t\rightarrow\sigma_{t}$ solves (6.9).   $\square$

Acknowledgments

Marc Sedjro is supported by the Alexander Von Humblodt Foundation. The project on which this publication is based has been carried out with funding from the German Federal Ministry of Education and Research under project reference number 01DG15010.