Mohammad A. Rammaha

The Navier-Stokes equations on the rotating 2- D sphere: Gevrey regularity and asymptotic degrees of freedom

Abstract. In this article we prove a Gevrey class global regularity to the Navier-Stokes equations on the rotating two dimensional sphere, S2 - a fundamental model that arises naturally in large scale atmospheric dynamics. As a result one concludes the exponential convergence of the spectral Galerkin numerical method, based on spherical harmonic functions. Moreover, we provide an upper bound for the number of asymptotic degrees of freedom for this system.

Global existence and nonexistence for nonlinear wave equations with damping and source terms

We consider an initial-boundary value problem for a nonlinear wave equation in one space dimension. The nonlinearity features the damping term |u| m-1 u t and a source term of the form |u| p-1 u, with m, p > 1. We show that whenever m > p, then local weak solutions are global. On the other hand, we prove that whenever p > m and the initial energy is negative, then local weak solutions cannot be global, regardless of the size of the initial data.

Gevrey Regularity for Nonlinear Analytic Parabolic Equations on the Sphere

The regularity of solutions to a large class of analytic nonlinear parabolic equations on the two-dimensional sphere is considered. In particular, it is shown that these solutions belong to a certain Gevrey class of functions, which is a subset of the set of real analytic functions. As a consequence it can be shown that the Galerkin schemes, based on the spherical harmonics, converge exponentially fast to the exact solutions, as the number of modes involved in the approximation tends to infinity. Furthermore, in the case that the underlying evolution equation has a global attractor, then this global attractor is contained in the space of spatially real analytic functions whose radii of analyticity are bounded uniformly from below.

Systems of nonlinear wave equations with damping and supercritical boundary and interior sources

utt −∆u + g1(ut) = f1(u, v) vtt −∆v + g2(vt) = f2(u, v), in a bounded domain Ω ⊂ R with Robin and Dirichlet boundary conditions on u and v respectively. The nonlinearities f1(u, v) and f2(u, v) are with supercritical exponents representing strong sources, while g1(ut) and g2(vt) act as damping. In addition, the boundary condition also contains a nonlinear source and a damping term. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data.

Blow-up of solutions to systems of nonlinear wave equations with supercritical sources

In this article, we focus on the life span of solutions to the following system of nonlinear wave equations: in a bounded domain Ω ⊂ ℝ n with Robin and Dirichlét boundary conditions on u and v, respectively. The nonlinearities f 1(u, v) and f 2(u, v) represent strong sources of supercritical order, while g 1(u t ) and g 2(v t ) represent interior damping. The nonlinear boundary condition on u, namely ∂ν u + u + g(u t ) = h(u) on Γ, also features h(u), a boundary source, and g(u t ), a boundary damping. Under some restrictions on the parameters, we prove that every weak solution to system above blows up in finite time, provided the initial energy is negative.

Wave equations with super-critical interior and boundary nonlinearities

Abstract: This article presents a unified overview of the latest, to date, results on boundary value problems for wave equations with super-critical nonlinear sources on both the interior and the boundary of a bounded domain @W@?R^n. The presented theorems include Hadamard local wellposedness, global existence, blow-up and non-existence theorems, as well as estimates on the uniform energy dissipation rates for the appropriate classes of solutions.

Critically and degenerately damped systems of nonlinear wave equations with source terms

This article is concerned with the global well-posedness of the critically and degenerately damped system of nonlinear wave equations in a bounded domain Ω ⊂ ℝ n , n = 1, 2, 3, with Dirichlét boundary conditions. The nonlinearities f 1(u, v) and f 2(u, v) act as a strong source in the system. Under some restriction on the parameters in the system we obtain several results concerning the existence of local solutions, global solutions, uniqueness and the blow up in finite time.

Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources

This paper investigates a quasilinear wave equation with Kelvin-Voigt damping, utt − Δpu − Δut = f(u), in a bounded domain Ω ⊂ ℝ3 and subject to Dirichlet boundary conditions. The operator Δp, 2 < p < 3, denotes the classical p-Laplacian. The nonlinear term f(u) is a source feedback that is allowed to have a supercritical exponent, in the sense that the associated Nemytskii operator is not locally Lipschitz from W01,p(Ω) into L2(Ω). Under suitable assumptions on the parameters, we prove existence of local weak solutions, which can be extended globally provided the damping term dominates the source in an appropriate sense. Moreover, a blow-up result is proved for solutions with negative initial total energy.

Well-Posedness and Stability of a Mindlin–Timoshenko Plate Model with Damping and Sources

This note gives a concise summary of results concerning the well-posedness and long-time behavior of (Reissner)–Mindlin–Timoshenko plate equations as presented in Pei et al. (Local and global well-posedness for semilinear Reissner–Mindlin–Timoshenko plate equations, 2013 and Global well-posedness and stability of semilinear Mindlin–Timoshenko system, 2013). The main feature of the considered model is the interplay between nonlinear viscous interior damping and nonlinear source terms. The results include Hadamard local well-posedness, global existence, blow-up theorems, as well as estimates on the uniform energy decay rates.