We consider a phase field system based on the Maxwell Cattaneo heat conduction law, with a logarithmic nonlinearity, associated with Dirichlet boundary conditions. In particular, we prove, in one and two space dimensions, the existence of a solution which is strictly separated from the singularities of the nonlinear term and that the problem possesses a finite-dimensional global attractor in terms of exponential attractors.
A special two-sided condition for the incremental magnetic reluctivity is introduced which guarantees the unique existence of both the weak and the approximate solutions of the nonlinear stationary magnetic field distributed on a region composed of different media, as well as a certain estimate of the error between the two solutions. The condition, being discussed from the physical as well as the mathematical point of view, can be easily verified and is fulfilled for various magnetic reluctivity...
In this paper, we prove that the regularity property, in the sense of Gehring-Giaquinta-Modica, holds for weak solutions to non-stationary Stokes type equations. For the construction of solutions, Rothe's scheme is adopted by way of introducing variational functionals and of making use of their minimizers. Local estimates are carried out for time-discrete approximate solutions to achieve the higher integrability. These estimates for gradients do not depend on approximation.
We derive a global differential inequality for solutions of a free boundary problem for a viscous compressible heat concluding capillary fluid. The inequality is essential in proving the global existence of solutions.
We propose a Diphasic Low Mach Number (DLMN) system for the modelling of diphasic flows without phase change at low Mach number, system which is an extension of the system proposed by Majda in [Center of Pure and Applied Mathematics, Berkeley, report No. 112] and [Combust. Sci. Tech. 42 (1985) 185–205] for low Mach number combustion problems. This system is written for a priori any equations of state. Under minimal thermodynamic hypothesis which are satisfied by a large class of generalized van...
We propose a Diphasic Low Mach Number (DLMN) system for the modelling of diphasic flows without phase change at low Mach number, system which is an extension of the system proposed by Majda in [Center of Pure and Applied Mathematics, Berkeley, report No. 112] and [Combust. Sci. Tech.42 (1985) 185–205] for low Mach number combustion problems. This system is written for a priori any equations of state. Under minimal thermodynamic hypothesis which are satisfied by a large class of generalized van...
The problem of finding the summational collision invariants for the Boltzmann equation leads to a functional equation related to the Cauchy equation. The solution of this equation is known under different assumptions on its unknown . Most proofs assume that the equation is pointwise satisfied, while the result needed in kinetic theory concerns the solutions of the equation when the latter is satisfied almost everywhere. The only results of this kind appear to be due to the authors of the present...