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Existence and uniqueness of solutions to wave equations with nonlinear degenerate damping and source terms

Barbu V., Lasiecka I., Rammaha A.

Control and Cybernetics

2005

Vol. 34, no 3

665-687

In this article we focus on the global well-posedness of the differential equation u [...] in Omega x(O, T), where j' denotes the derivative of a C1 convex and real valued function j. The interaction between degenerate damping and a source term constitutes the main challenge of the problem. Problems with non-degenerate damping (k = 0) have been studied in the literature (Georgiev and Todorova, 1994; Levine and Serrin, 1997; Vitillaro, 2003). Thus the degeneracy of monotonicity is the main novelty of this work. Depending on the level of interaction between the source and the damping we characterize the domain of the parameters p, m, k, n (see below) for which one obt ains existence, regularity or finite time blow up of solutions. More specifically, when p [is less than or equal to] m + k global existence of generalized solutions in H1 x L2 is proved. For p > m + k, solutions blow up in a finite time. Higher energy solutions are studied as well. For H2 x H1 initial data we obtain both local and global solutions with the same regularity. Higher energy solutions are also proved to be unique.

Thermodynamic aspects of variational principles for fluids with heat flow

Sieniutycz S.

Archives of Thermodynamics

2005

Vol. 26, no. 2

45-71

Processing some known results of nonequilibrium statisctical mechanics we focus on nonequilibrium corrections Δs to entropy s of the fluid in terms of the nonequilibrium density distribution function, f. To evaluate corrections Δe to the energy e or kinetic potential L we apply a relationship that links energy and entropy representations of thermodynamics. We also evaluate coefficients of wave model of heat conduction, such as: relaxation time, propagation speed and thermal inertia. With corrections to L we formulate a quadratic Lagrangian and a variational principle of Hamilton's (least action) type for a fluid with heat flux, or other random-type effect, in the field or Eulerian representation of fluid motion. We discuss canonical and generalized conservation laws and show that satisfaction of the second law of thermodynamics under the constraint of canonical conservation laws.