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Global solutions, structure of initial data and the Navier-Stokes equations

Piotr Bogusław Mucha (2008)

Banach Center Publications

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 via partial information and the Cahn-Hilliard equation

Jan Cholewa, Tomasz Dłotko (1996)

Banach Center Publications

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.

Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source

Xiangdong Zhao (2024)

Czechoslovak Mathematical Journal

We study the chemotaxis system with singular sensitivity and logistic-type source: u t = Δ u - χ ∇ · ( u ∇ v / v ) + r u - μ u k , 0 = Δ v - v + u under the non-flux boundary conditions in a smooth bounded domain Ω ⊂ ℝ n , χ , r , μ > 0 , k > 1 and n ≥ 1 . It is shown with k ∈ ( 1 , 2 ) that the system possesses a global generalized solution for n ≥ 2 which is bounded when χ > 0 is suitably small related to r > 0 and the initial datum is properly small, and a global bounded classical solution for n = 1 .

Global stability of Clifford-valued Takagi-Sugeno fuzzy neural networks with time-varying delays and impulses

Ramalingam Sriraman, Asha Nedunchezhiyan (2022)

Kybernetika

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 n -dimensional Clifford-valued neural network into 2 m n -dimensional real-valued counterparts in order to solve the noncommutativity...

Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

Thierry Gallay, Romain Joly (2009)

Annales scientifiques de l'École Normale Supérieure

We consider the damped wave equation α u t t + u t = u x x - V ' ( u ) on the whole real line, where V is a bistable potential. This equation has travelling front solutions of the form u ( x , t ) = h ( x - s t ) which describe a moving interface between two different steady states of the system, one of which being the global minimum of V . We show that, if the initial data are sufficiently close to the profile of a front for large | x | , the solution of the damped wave equation converges uniformly on ℝ to a travelling front as t → + ∞ . The proof of this global stability...

Currently displaying 361 – 380 of 484