Attractors for nonlinear reaction-diffusion systems in unbounded domains via the method of short trajectories

We consider a nonlinear reaction-diffusion equation settled on the whole euclidean space. We prove the well-posedness of the corresponding Cauchy problem in a general functional setting, namely, when the initial datum is uniformly locally bounded in L^2. Then we adapt the short trajectory method to establish the existence of the global attractor and, if the space dimension is at most 3, we also find an upper bound of its Kolmogorov's entropy

Strong solutions and attractor dimension for 2D NSE with dynamic boundary conditions

We consider incompressible Navier-Stokes equations in a bounded 2D domain, complete with the so-called dynamic slip boundary conditions. Assuming that the data are regular, we show that weak solutions are strong. As an application, we provide an explicit upper bound of the fractal dimension of the global attractor in terms of the physical parameters. These estimates comply with analogous results in the case of Dirichlet boundary condition

On LpL^p semigroup to Stokes equation with dynamic boundary condition in the half-space

We consider evolutionary Stokes system, coupled with the so-called dynamic boundary condition, in the simple geometry of $d$-dimensional half-space. Using the Fourier transform, we obtain an explicit formula for the resolvent. Maximal regularity estimates and existence of analytic semigroup in the $L^p$-setting are then deduced using classical multiplier theorems

Remarks on the uniqueness of second order ODEs\rm ODEs

summary:We are concerned with the uniqueness problem for solutions to the second order ODE of the form $x''+f(x,t)=0$, subject to appropriate initial conditions, under the sole assumption that $f$ is non-decreasing with respect to $x$, for each $t$ fixed. We show that there is non-uniqueness in general; on the other hand, several types of reasonable additional assumptions make the problem uniquely solvable. The interest in this problem comes, among other, from the study of oscillations of lumped parameter systems with implicit constitutive relations

Mechanical oscillators with dampers defined by implicit constitutive relations

summary:We study the vibrations of lumped parameter systems, the spring being defined by the classical linear constitutive relationship between the spring force and the elongation while the dashpot is described by a general implicit relationship between the damping force and the velocity. We prove global existence of solutions for the governing equations, and discuss conditions that the implicit relation satisfies that are sufficient for the uniqueness of solutions. We also present some counterexamples to the uniqueness when these conditions are not met

On the sign of Colombeau functions and applications to conservation laws

summary:A generalized concept of sign is introduced in the context of Colombeau algebras. It extends the sign of the point-value in the case of sufficiently regular functions. This concept of generalized sign is then used to characterize the entropy condition for discontinuous solutions of scalar conservation laws