\int_0^m\psi_a'\big(X_{\mu(t)}(z)-Y(\zeta)\big){\, \mathrm{d}}\zeta & \text{if $q_a>1$}, \\
              2G\big(X_{\mu(t)}(z)\big)-m & \text{if $q_a=1$},
            \end{cases}
            \\
            &\quad +
            \begin{cases}
              \int_0^1\psi_r'\big(X_{\mu(t)}(z)-X_{\mu(t)}(\zeta)\big){\, \mathrm{d}}\zeta & \text{if $q_r>1$}, \\
              2z-1 & \text{if $q_r=1$}
            \end{cases}
          \end{split} 
        \end{aligned}$$ holds at every $t\ge0$ and every $z\in[0,1)$.

Note that continuous differentiability as a map into ${\mathrm{D}}([0,1))$ implies that $t\mapsto X_{\mu(t)}(z)$ is a differentiable real function for every $z\in[0,1)$.

In the cases $q_r=1$ or $q_r=2$, the energy functional $\mathcal E$ from is $\lambda$-convex for some $\lambda\in{\mathbb{R}}$, and therefore it possesses a unique $\lambda$-contractive gradient flow in the Wasserstein $W_2$ metric — we refer to [@AGS08] for the case $q_r=2$, and to [@BS; @BCDP] for the case $q_r=1$. It is easily seen that these gradient flow solutions are transient solutions in the sense of Definition \[dfn:transient\] above, and vice versa.

\[thm:existence\] Let an initial condition $\mu_0\in{\mathcal{P}}({\mathbb{R}})$ with density in $L^\infty_c(\mathbb{R})$ be given. Then, there is a unique transient solution $\mu:[0,\infty)\to{\mathcal{P}}_2({\mathbb{R}})$ in the sense of Definition \[dfn:transient\] with $\mu(0)=\mu_0$. In particular, $\mu$ is a distributional solution of the original equation , i.e., for all $\varphi \in C^{\infty}_{c}([0,\infty) \times \mathbb{R})$, it satisfies $$\begin{aligned}
    \label{eq:weak} 
    \begin{split}
      &-\int_{0}^{\infty} \int_{\mathbb{R}} \partial_{t}\varphi(t,x) \, \mathrm{d} \mu(t,x) \, \mathrm{d} t - \int_{\mathbb{R}} \varphi(0,x) \, \mathrm{d} \mu_0(x)\\
      ={}&\int_{0}^{\infty} \int_{\mathbb{R}} \partial_{x}\varphi(t,x) \cdot \left(\psi_r' \ast 