On the Uniqueness of the Classical Solution of the Initial-boundary Value Problem for a Viscoelastic Maxwell Medium Flow with Zero Velocities at the Boundary
A system of equations with a rotational Jaumann derivative describing the motion of a viscoelastic incompressible Maxwell medium in a bounded region of three-dimensional space in the Stokes approximation (without convective terms in both the equation of motion and the rheological relation) is under consideration. The choice of the objective derivative of Jaumann type in the rheological ratio is due to the presence of an energy identity, the validity of which cannot be proved for the cases of using the upper and lower convective derivatives as objective derivatives. The uniqueness of the classical solution of the initial-boundary value problem with the adherence condition on the boundary of the flow domain in the class of sufficiently smooth functions under the assumption that the system has four distinct real sound characteristics is proved. The proof is based on the application of integral estimates and the use of the hyperbolicity condition of the system. For the sake of clarity, the calculations are carried out for the two-dimensional case, however, nowhere does it use the properties of two-dimensionality. Differences from the three-dimensional case are purely technical in nature. It is noted that the presence of convective terms in the equations of motion does not prevent the proof of uniqueness. Uniqueness is initially proved for a small interval of time, then in the standard way, the statement is extended to an interval of arbitrary length.
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