Y. Brenier, R. Natalini and M. Puel have considered a ``relaxation" of the Euler equations in <b>R</b><sup>2</sup>.
After an approriate scaling, they have obtained the following hyperbolic version of the Navier-Stokes equations, which is similar to the hyperbolic version of the heat equation introduced by Cattaneo,
$$\varepsilon u_{tt}^\varepsilon + u_t^\varepsilon -\Delta u^\varepsilon
+P (u^\varepsilon \nabla u^\varepsilon) \, = \, Pf~, \quad
(u^\varepsilon(.,0), u_t^\varepsilon(.,0)) \, = \, (u_0(.),u_1(.))~,
\quad (1) $$
where $P$ is the classical Leray projector and $\varepsilon$ is a
small, positive number. Under adequate hypotheses on the forcing term
$f$, we prove global existence and uniqueness of a mild solution
$(u^\varepsilon,u_t^\varepsilon) \in C^0([0, +\infty), H^{1}({\bf R}^2) \times
L^2({\bf R}^2))$ of (1), for large initial data
$(u_0,u_1)$ in $H^{1}({\bf R}2) \times L^2({\bf R}2)$,
provided that $\varepsilon>0$ is small enough, thus improving the global existence results of Brenier, Natalini and Puel
(actually, we can work in less regular Hilbert spaces).
The proof uses appropriate Strichartz estimates, combined with energy estimates.
We also show that $(u^\varepsilon,u_t^\varepsilon)$ converges to $(v,v_t)$
on finite intervals of time $[t_0,t_1]$, $0 <t_0><+ \infty$, when $\varepsilon$ goes to $0$, where $v$ is the
solution of the corresponding Navier-Stokes equations
$$
v_t -\Delta v
+P (v\nabla v) \, = \, Pf~, \quad
v(.,0) \, = \, u_0~.
\quad (2)
$$
We also consider Equation (1) in the three-dimensional case.
Here we expect global existence results for small data.
Under appropriate assumptions on the forcing term, we prove global existence and uniqueness of a mild solution
$(u^\varepsilon,u_t^\varepsilon) \in C^0([0, +\infty), H^{1+\delta}({\bf R}^3) \times H^{\delta}({\bf R}^3))$ of (1),
for initial data $(u_0,u_1)$ in $H^{1 +\delta}({\bf R}^3) \times H^{\delta}({\bf R}^3)$
(where $\delta >0 $ is a small positive number), provided that $\varepsilon > 0$ is small enough and that $u_0$ and $f$ satisfy a smallness condition.
(Joint work with Marius Paicu)
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