Continuity of the Eulerian data-to-solution map

Establish continuity of the Eulerian data-to-solution map for the ideal incompressible magnetohydrodynamics Cauchy problem in the low-regularity range (n+1)/2 < s <= n/2+1, including control of changes in the differentiated fixed-point map and composition with different inverse Lagrangian flows at the top H^s regularity; consequently, determine whether a Hadamard well-posedness statement can hold uniformly on bounded H^s subsets in regime (R1).

Background

The paper proves local existence and uniqueness for ideal incompressible magnetohydrodynamics with a nonzero constant background magnetic field below the classical Lipschitz threshold. However, the construction is based on a Lagrangian fixed point and the Eulerian solution is recovered through composition with the inverse flow map.

The authors identify two unresolved difficulties in comparing solutions generated by different initial data: controlling the variation of the fixed-point map and controlling composition with two distinct inverse flows at the top Hs regularity. In regime (R1), the lifespan also depends on the datum-specific longitudinal modulus omega_{v_0}, which is not uniformly controlled on bounded Hs sets. Thus the paper establishes existence and uniqueness but does not establish continuous dependence on the data or a Hadamard well-posedness result.

References

Continuity of the data-to-solution map: the present argument does not prove continuity of the Eulerian data-to-solution map. Comparing solutions arising from two different data would require controlling both the change in the fixed-point map and composition with two different inverse flows at the top Hs regularity. We do not pursue this additional step here. Moreover, in regime~\textup{(R1)} the available lifespan is not uniform on bounded subsets of Hs, because the modulus \omega_{v_0} is not uniformly controlled there. Accordingly, no Hadamard well-posedness statement is asserted.

— Ideal MHD below the classical well-posedness threshold  (2609.01093 - Giardi, 1 Sep 2026) in Subsection “What is left open,” first issue

Determining the minimal regularity of B_0 for which the commutator estimates close and thus the present construction can be carried out in this generalised setup is left to future work.

— Ideal MHD below the classical well-posedness threshold  (2609.01093 - Giardi, 1 Sep 2026) in Subsection “What is left open,” second issue