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Let $M$ be a finite dimensional manifold of constant curvature $\kappa$. Consider a solution of the Hamilton--Jacobi equation $$ \partial_t u + |\nabla u|^2 = 0. $$ Can we give a precise estimate of a ...

It is known that if a function (which we shall call the lagrangian flow, or lagrangian trajectory) $$X:(\mathbb{R}/\mathbb{Z})^3 \times [0,T] \to \mathbb{R}^3$$ with $X \in H^1_t$ (i.e. has weak time ...