We consider the 2D Euler equation with periodic boundary conditions in a family of Banach spaces based on the Fourier coefficients, and show that it is ill-posed in the sense that 'norm inflation' occurs. The proof is based on the observation that the evolution of certain perturbations of the 'Kolmogorov flow' given in velocity by
$U(x,y) = \left( {\begin{array}{*{20}{c}}{\cos \;y}\\0\end{array}} \right)$
can be well approximated by the linear Schrödinger equation, at least for a short period of time.
Citation: |
J. Bourgain
and D. Li
, Strong Ill-posedness of the incompressible Euler equation inborderline Sobolev spaces, Invent. Math., 20 (2015)
, 97-157.
doi: 10.1007/s00222-014-0548-6.![]() ![]() ![]() |
|
J. Bourgain
and D. Li
, Strong illposedness of the incompressible Euler equation in integer $C^m$ spaces, Geom. Funct. Anal., 25 (2015)
, 1-86.
doi: 10.1007/s00039-015-0311-1.![]() ![]() ![]() |
|
A. Cheskidov
and R. Shvydkoy
, Ill-posedness of the basic equations of fluid dynamics in Besov spaces, Proceedings of AMS, 138 (2010)
, 1059-1067.
doi: 10.1090/S0002-9939-09-10141-7.![]() ![]() ![]() |
|
D. Ebin
and J. Marsden
, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math., 92 (1970)
, 102-163.
doi: 10.2307/1970699.![]() ![]() ![]() |
|
T. Elgindi and N. Masmoudi, L∞ Ill-posedness for a class of equations arising in hydrodynamics, preprint, arXiv: 1405.2478v2.
![]() |
|
A. Grünrock
and S. Herr
, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM Journal on Mathematical Analysis, 39 (2008)
, 1890-1920.
doi: 10.1137/070689139.![]() ![]() ![]() |
|
T. Kato
, Nonstationary flows of viscous and ideal fluids in $\mathbb{R}^3$, J. Func. Anal., 9 (1972)
, 296-305.
doi: 10.1016/0022-1236(72)90003-1.![]() ![]() ![]() |
|
T. Kato
, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975)
, 181-205.
doi: 10.1007/BF00280740.![]() ![]() ![]() |
|
J. Mattingly
and Ya. Sinai
, An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equations, Commun. Contemp. Math., 1 (1999)
, 497-516.
doi: 10.1142/S0219199799000183.![]() ![]() ![]() |