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Article Dans Une Revue Journal de Mathématiques Pures et Appliquées Année : 2023

Elsässer formulation of the ideal MHD and improved lifespan in two space dimensions

Dimitri Cobb
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Résumé

In the present paper, we show an improved lower bound for the lifespan of the solutions to the ideal MHD equations in the case of space dimension d = 2. In particular, for small initial magnetic fields b0 of size (say) ε > 0, the lifespan Tε > 0 of the corresponding solution goes to +∞ in the limit ε → 0 +. Such a result does not follow from standard quasi-linear hyperbolic theory. For proving it, three are the crucial ingredients: first of all, to work in endpoint Besov spaces B s ∞,r , under the condition s > 1 and r ∈ [1, +∞] or s = r = 1; moreover, to use the Elsässer formulation of the ideal MHD, recasted in its vorticity formulation; finally, to take advantage of the special structure of the non-linear terms. We also rigorously establish the equivalence between the original formulation of the ideal MHD and its Elsässer formulation for a large class of weak solutions. The construction of explicit counterexamples shows the sharpness of our assumptions. Related non-uniqueness issues are discussed as well. 2010 Mathematics Subject Classification: 35Q35 (primary); 76W05, 76B03, 35L60 (secondary).
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Dates et versions

hal-02948690 , version 1 (24-09-2020)

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Dimitri Cobb, Francesco Fanelli. Elsässer formulation of the ideal MHD and improved lifespan in two space dimensions. Journal de Mathématiques Pures et Appliquées, 2023, 169, pp.189-236. ⟨10.1016/j.matpur.2022.11.012⟩. ⟨hal-02948690⟩
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