Ill-posedness in B^s_p,∞ of the Euler equations: Non-continuous dependence

2025/09/16 by Jinlu Li, Li, Jinlu, Yanghai Yu +1
Mathematics · Economics, Econometrics and Finance · #Navier-Stokes equation solutions #advanced mathematical theories #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2509.12619

Abstract

In this paper, we solve an open problem left in the monographs \cite[Bahouri-Chemin-Danchin, (2011)]BCD. Precisely speaking, it was obtained in \cite[Theorem 7.1 on pp293, (2011)]BCD the existence and uniqueness of B^s_p,∞ solution for the Euler equations. We furthermore prove that the solution map of the Euler equation is not continuous in the Besov spaces from B^s_p,∞ to L_T^∞ B^s_p,∞ for s>1+d/p with 1≤ p≤ ∞ and in the Hölder spaces from C^k,α to L_T^∞ C^k,α with k∈ ℕ⁺ and α∈(0,1), which later covers particularly the ill-posedness of C^1,α solution in \cite[Trans. Amer. Math. Soc., (2018)]MYtams. Beyond purely technical aspects on the choice of initial data, a remarkable novelty of the proof is the construction of an approximate solution to the Burgers equation.

Related

• Ill-posedness for the Euler equations in Besov spaces
• Local ill-posedness of the Euler equations in B¹_∞,1
• Continuity of the solution map of the Euler equations in Hölder spaces and weak norm inflation in Besov spaces
• Non-uniform dependence for Euler equations in Besov spaces
• The failure of Hölder regularity of solutions for the Euler-Poincaré equations in Besov spaces