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.