Global solutions of the equations of elastodynamics for incompressible materials.
In this note we present a proof of existence of global in time regular (unique) solutions to the Navier-Stokes equations in an arbitrary three dimensional domain with a general boundary condition. The only restriction is that the L₂-norm of the initial datum is required to be sufficiently small. The magnitude of the rest of the norm is not restricted. Our considerations show the essential role played by the energy bound in proving global in time results for the Navier-Stokes equations.
Global solutions of semilinear parabolic equations are studied in the case when some weak a priori estimate for solutions of the problem under consideration is already known. The focus is on the rapid growth of the nonlinear term for which existence of the semigroup and certain dynamic properties of the considered system can be justified. Examples including the famous Cahn-Hilliard equation are finally discussed.
We study the chemotaxis system with singular sensitivity and logistic-type source: , under the non-flux boundary conditions in a smooth bounded domain , , and . It is shown with that the system possesses a global generalized solution for which is bounded when is suitably small related to and the initial datum is properly small, and a global bounded classical solution for .
In this study, we consider the Takagi-Sugeno (T-S) fuzzy model to examine the global asymptotic stability of Clifford-valued neural networks with time-varying delays and impulses. In order to achieve the global asymptotic stability criteria, we design a general network model that includes quaternion-, complex-, and real-valued networks as special cases. First, we decompose the -dimensional Clifford-valued neural network into -dimensional real-valued counterparts in order to solve the noncommutativity...