We study by computer simulations the complex solutions of the two-dimensional Burgers equations in the whole plane in absence of external forces. For such model the existence of singularities, corresponding to a divergence of the total energy at a finite time, is proved by Li and Sinai [“

Singularities of complex-valued solutions of the two-dimensional Burgers system
,” J. Math. Phys.51, 015205 (2010)] for a large class of initial data. The simulations show that the blow-up takes place in a very short time, of the order of 10−5 time units. Moreover near the blow-up time the support of the solution in Fourier space moves out to infinity along a straight line. In x-space the solutions are concentrated in a finite region, with large space derivatives, as one would expect for physical phenomena such as tornadoes.

1.
R.
Temam
, “
Navier-Stokes equations: Theory and numerical analysis
,” in
Studies in Mathematics and its Applications
2 (
North-Holland
,
Amsterdam
,
1979
).
2.
J.
Leray
, “
Sur le mouvement d'un liquide visqueux emplissant l'espace
,”
Acta Math.
63
,
193
248
(
1934
).
3.
Th. Y.
Hou
, “
Blow-up or no blow-up? A unified computational and analytic approach to 3D incompressible Euler and Navier-Stokes equations
,”
Acta Numerica
18
,
277
346
(
2009
).
4.
D.
Li
and
Ya. G.
Sinai
, “
Blowups of complex solutions of the 3D Navier-Stokes system and renormalization group method
,”
J. Eur. Math. Soc.
10
,
267
313
(
2008
).
5.
D.
Li
and
Ya. G.
Sinai
, “
Singularities of complex-valued solutions of the two-dimensional Burgers system
,”
J. Math. Phys.
51
,
015205
(
2010
).
6.
D.
Li
and
Ya. G.
Sinai
, “
Blowups of complex-valued solutions for some hydrodynamic models
,”
Regular Chaotic Dyn.
15
,
521
531
(
2010
).
7.
P.
Polacik
and
V.
Sverak
, “
Zeroes of complex caloric functions and singularities of complex viscous Burgers equations
,”
J. Reine Angew. Math.
616
,
205
215
(
2008
).
8.
D.
Li
and
Ya. G.
Sinai
, “
Complex singularities of solutions of some 1D hydrodynamic model
,”
Physica D
237
,
1945
1950
(
2008
).
9.
L. M.
Burgers
, “
The formation of vortex sheets in a simplified type of turbulent motion
,”
Nederl. Akad. Wetensch., Proc.
53
,
122133
(
1950
).
10.
D.
Li
and
Ya. G.
Sinai
, “
Asymptotic behavior of generalized convolution
,”
Regular Chaotic Dyn.
14
,
248
262
(
2009
).
11.
See http://www.fftw.org for information about the subroutine library FFTW.
12.
M. D.
Arnol'd
and
A. V.
Khokhlov
, “
Modeling a blow-up solution of tornado type for the complex version of the three-dimensional Navier-Stokes equation
,”
Usp. Mat. Nauk
64
,
171
172
(
2009
)
M. D.
Arnol'd
and
A. V.
Khokhlov
, [
Russ. Math. Suveys
64
,
1133
1135
(
2009
)].
You do not currently have access to this content.