The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation
In the cylindrical domain \(\Omega_T=\Omega\times (0,T)\), \(T\leq\infty\), where \(\Omega\subset \mathbb R^d\), \(d\geq 1\), is a convex (possible unbounded) domain with exterior unit normal \(n\). The authors consider the initial-boundary value problem \[ \begin{aligned} \partial_t u+\operatorname{div} \left(u D\left(2 \frac{\Delta u^{1/2}}{u^{1/2}}-f\right)\right)&=0,\quad (x,t)\in \Omega_T,\\ \partial_n u&=0\quad\text{ on }\partial \Omega\times (0,T),\\ u\partial_n \left(2 \frac{\Delta u^{1/2}}{u^{1/2}}-f\right)&=0\quad\text{ on }\partial \Omega\times (0,T),\\ u(x,0)=u_0(x)&\geq 0\quad\text{ in }\Omega,\quad u_0\in L^1(\Omega), \quad \int_\Omega |x|^2 u_0 (x)\,dx<\infty. \end{aligned} \] Non-negative weak solutions of this problem are obtained as a limit of a variational approximation scheme, using the particular structure of the mentioned equation, which is the gradient flow of the Fisher information functional \[ F^f(u):=2^{-1}\int_\Omega|D\, \text{log}\,u|^2 u\,dx +\int_\Omega fu\,dx \] with respect to the Kantorovich-Rubinstein-Wasserstein distance between probability measures. The long-time behavior of the solution is studied, too. Particularly, the exponential decay to the equilibrium state \(g=\exp(-V)\) is proved, characterized by \[ -\Delta V+2^{-1}|DV|^2=f,\qquad \int_\Omega \exp (-V)\,dx =\int_\Omega u_0\,dx, \] when the potential \(V\) is uniformly convex. In this case the functional \(F^f\) coincides with the relative Fisher information \[ F^f(u)=2^{-1}\Phi (u/g)=\int_\Omega | D\, \log (u/g)|^2\,dx. \]
fourth-order quasilinear parabolic equation
scientific article; zbMATH DE number 5640827
scientific article