Periodic solutions for a 1D-model with nonlocal velocity via mass transport (Q264451)

The authors look for \(2\pi \)-periodic solutions to the 1D evolution problem \( u_{t}+(\mathcal{H}(u)u)_{x}=0\) where \(\mathcal{H}\) is the Hilbert periodic transform defined as NEWLINE\[NEWLINE\mathcal{H}(u)(x)=\frac{1}{2\pi }P.V.\int_{-\pi }^{\pi }\cot (\frac{x-y}{2})u(y)dy.NEWLINE\]NEWLINE The initial condition \(u(x,0)=u_{0}(x)\) is imposed. The authors introduce a mass transport and gradient-flow framework. They first recall the mass transport theory in the unit circle \( \mathbb{S}^{1}\) introducing the space \(\mathcal{P}(\mathbb{S}^{1})\) of periodic probability measures equipped with the 2-Wasserstein distance. They define the notion of geodesics with constant speed and they build an example of such geodesics. They then define a free energy functional \(\mathcal{F} _{\nu }\) on \(\mathcal{P}(\mathbb{S}^{1})\) and they establish its lower semicontinuity, of coercivity and of convexity properties. The main result of the paper proves the convergence of an approximate discrete solution of the gradient flow equation to a locally Lipschitz curve which is the unique gradient flow of \(\mathcal{F}_{\nu }\). They also establish properties of this limit solution.