Investigation of the Dissipative Solvability for Alpha-Maxwell Model

We study initial-boundary value problems for the Lagrangian averaged alpha model for the equations of motion for the corotational Maxwell and inviscid fluids in 2D and 3D. We show existence of (global in time) dissipative solutions to these problems.

The Maxwell model is one of the basic and classical models of a viscoelastic material. Its mechanical analogy is comprised of a spring and a dashpot connected in series. The multidimensional Maxwell models generate complicated systems of PDEs due the frame-indifference restrictions and consecutive involvement of objective derivatives. Very few mathematical results are known for the corotational Maxwell fluid equations. In particular, there is no global solvability theorem, even in 2D. Moreover, there is evidence of non-existence of smooth solutions. In these circumstances, if we want the problem to be solvable globally, a possible way out is to consider a kind of a generalized solution different from the standard hydrodynamical weak solution framework.

We are going to use the concept of dissipative solution due to Lions. It was suggested for the Euler equations of ideal fluid flow, which have only been proven to be globally weakly solvable on the torus (this is a very recent result, and that wild weak solutions are necessarily not dissipative solutions). Later, in addition to the Eulere quation, the existence of such solutions was established for Boltzmann’s equation, and for various models arising in magnetohydrodynamics, diffusion in polymers, and image restoration.

The objective of our paper is to introduce dissipative solutions for the corotational Maxwell-alpha problem, and to show their existence and basic properties. These solutions are always global in time.

Keywords: alpha model of hydrodynamic, Maxwell model, dissipative solution, existence theorem, topological approximation method.