Local existence and blow‐up criterion for the Euler equations in the Besov spaces

By Dongho Chae. Asymptotic Analysis, 2004

146 citations recorded by the source corpus.

We prove the local in time existence and a blow‐up criterion of solutions in the Besov spaces for the Euler equations of inviscid incompressible fluid flows in Rn, n≥2. As a corollary we obtain the persistence of Besov space regularity for the solutions of the 2‐D Euler equations with initial velocity belonging to the Besov spaces. For the proof of the results we establish a logarithmic inequality of the Beale–Kato–Majda type and a Moser type of inequality in the Besov spaces.

Subfields: Applied Mathematics, Earth-Surface Processes

View Paper: https://www.semanticscholar.org/paper/11e36a6cc4bd612a4d5c1d663d6d36756d4e2495

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In Applied Mathematics

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Citations in the corpus

Local existence for the non-resistive MHD equations in Besov spaces (2015, methodology): “The Lagrangian approach, most notably applied to the Euler equations by Yudovich (1963), has yielded significant results in Besov spaces for both the Euler equations (due to Chae (2004)) and for MHD.”
The three-dimensional Euler equations : Where do we stand? (2008, methodology): “For literature on variations of the BKM theorem see Ponce [31] and Chae [32] [33] [34] [35].”
Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces (2013, background): “In [7] Chae proved local existence and uniqueness of solutions to d-dimensional Euler in critical Besov space (for velocity) Bd/p+1 p,1 (Rd) with 1 < p < ∞.”
Incompressible Viscous Flows in Borderline Besov Spaces (2008, methodology): “In the critical case, we mention that we have the same result when we replace the space H d 2 by the Besov space B 1+ d p p,1 , p ∈ (1,∞) which is contained in the class of Lipschitz functions (see [2, 5], for instance).”
Strong illposedness of the incompressible Euler equation in integer Cm spaces (2014, background): “ace B2/p+1 p,1 (R 2) with 1 <p<∞. Note that in 2D the regularity index s= 2/p+ 1 is critical (albeit at the expense of l1-summation in dyadic frequency blocks). For general dimension d≥ 2, Chae [9] obtained local wellposedness of Euler in critical Besov space Bd/p+1 p,1 (R d) with 1 <p<∞”