**Models of hydrodynamics with memory and nonlocality and special solutions with self-origins of collapses**

**Abstract:**

The collapses problem has a long history since XVIII century. It is known that this problem is very complicated.
Collapses origin is also
closely connected to the considering the energy flows between different scales in turbulence.
One of the tools for considering such problems is
mathematical modelling, that is investigation of physical phenomena by models.
The common models for hydrodynamics are Navies-Stokes, Burgers, non-viscid
flows, Helmholtz for vortices and others [1]. But general problem still is open.
One way for searching solution of general problem consist in considering
more and more correct equations. Remark that in case of smallest space scales the
primary dynamical equations for particles is useful. But for intermediate
scales the hydrodynamics equations with memory and nonlocaliry accounting is intrinsic
(see reviews of literature in [2, 3]). One of the most known are
visco-elastic equations. Just as say Navies-Stokes equations, complete equations with
memory and nonlocality effects are very complex object for mathematical and numerical
investigations. This follows to the necessity of more
simple models. The examples of such classical models are Burgers equations, Lorenz equations,
Korteveg-de Vriz equations and many others. For equations
with memory effects earlier it had been proposed hyperbolic modification of Burgers equation and
low-dimensional counterpart for Lorenz equations. Some
results for such equations were described in [2, 3]. In one-dimensional case it was found the
fragmentation of solutions, origins of collapses (blow-up
regimes) in the case of high velocity of flow. So it is next natural step to investigate the
collapses and structures origin in 2D and 3D case. In the
report we propose model equations for 2D and 3D cases and results of numerical modelling collapses
origins in models with memory effects. The evidence of
such solutions poses new research problems of searching new types of solutions by symmetry methods.

References:

[1] Frish U. Turbulence. Cambridge Univ.Press. 1996.

[2] Makarenko A. Mathematical modeling of memory effects influence on fast hydrodynamic and heat conduction processes. Control and Cybernetics 25:
621- 630, 1996.

[3] Makarenko A., Moskalkov M., Levkov S. On blow- up solutions in turbulence. Phys. Letters A. A235: 391- 397, 1997.