First and foremost, it remains to pursue the strategy outline in Section 2.7 to extend our results to general dimensions.
In Section 2.5, we showed, assuming initial convergence of the microscopic entropy towards the macroscopic entropy, this convergence is propagated along the evolution of the system.
It would be interesting to prove uniformity in time of this convergence. The difficulty here lies in the fact that in general, the microscopic relative entropy HN(µNt |νfN∞) does not decay in t, since the weight of PµNt (P
xη(x) = K) on the hyperplanes of constant particles is invariant under the evolution of the particle system. Thus as long as µN0 is not chosen exactly such that
PµNt
X
x
η(x) = K
=Pνf∞N
X
x
η(x) =K ,
we cannot take advantage of decay of the microscopic entropy. On the other hand, by the equivalence of ensembles, we expect that
PµNt
X
x
η(x) = K
≈Pνf∞N
X
x
η(x) = K
and that we can still deduce a uniform-in-time convergence of the entropy. This remains to be clarified in future work. Another question, answered in Kosygina [50] for simple exclusion processes, is whether the entropies converge for all positive times even if we only assume a hydrodynamic limit (macroscopic profile) initially and no convergence of the initial entropies.
It should be possible to prove a strong conservation of local equilibrium using our tech- niques, see Remark 2.3.3 (4). In the case of attractive processes, this is a known result - however, our method has the advantage of yielding explicit uniform-in-time bounds on the rate of convergence.
Finally, it remains to be seen if this method can be extended to problems where the hydrodynamic limit is not yet known, especially limit equations that can exhibit shocks.
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