1,721,028 research outputs found
Hopf bifurcations and global nonlinear L2-energy stability in thermal MHD
No full textThe transfer of heat and mass by convection in a fluid horizontal layer L, heated from below – in the past as nowadays – has attracted the attention of many scientists since it can be driven by different factors and has a very relevant influence on the behaviour of many phenomena of the real world concerning meteorology, sun and stars physics, oceanography, heat insulation, air and water pollution, ... (see [1–12, 25–29] and references therein). When L is filled by a plasma and is embedded in a transverse constant magnetic field, in the non-relativistic scheme of magneto-hydrodynamic (MHD), in the early of 1950 ([1–2]), the Nobel Prize laureate (1983) S. Chandrasekhar – in linear thermal MHD – obtained a relevant inhibition of convection by a magnetic field, verified experimentally [9] and appeared in 1961 in the celebrated monograph [3]. Successively, during the years, many efforts have been done in order to recover this relevant stabilizing effect in the nonlinear thermal MHD theory (see [6, 10, 11, 13, 16]). This goal has been reached partially in 1988 in [13] and totally in [16] but only under very severe restrictions on the initial data (of the order <10−6). Recently in [17], via a non standard approach, assuming the verticality of the gradient pressure perturbations, it has been totally recovered in the nonlinear thermal MHD theory, the linear inhibition of convection by magnetic field for any admissible initial data (Linearization Principle). In the present paper, we return to the problem and show that, in thermal MHD, the linear asymptotic stability implies the global exponential nonlinear L2−energy stability, without requiring the verticality of the perturbations to the pressure gradient. As concerns the onset of instabilities in the free-free case, since Pr≥Pm (with Pr,Pm Prandtl and Prandlt magnetic numbers), implies the onset of steady bifurcation for any value of the Chandrasekhar number Q2, we analyze the case $P_rHopf bifurcation number the threshold Qc that the Chandrasekhar number has to cross for the occurring of Hopf bifurcations, we obtain that Qc=1+PrPm−Prπ2. This formula – new in the existing literature – removes the difficulties (mentioned in page 184 of [3]) on finding a "simpler formula which gives Qc as function of Pr, Pm"
Hopf bifurcations in dynamical systems
No full textThe onset of instability in autonomous dynamical systems (ADS) of ordinary differential equations is investigated. Binary, ternary and quaternary ADS are taken into account. The stability frontier of the spectrum is analyzed. Conditions necessary and sufficient for the occurring of Hopf, Hopf–Steady, Double-Hopf and unsteady aperiodic bifurcations—in closed form—and conditions guaranteeing the absence of unsteady bifurcations via symmetrizability, are obtained. The continuous triopoly Cournot game of mathematical economy is taken into account and it is shown that the ternary ADS governing the Nash equilibrium stability, is symmetrizable. The onset of Hopf bifurcations in rotatory thermal hydrodynamics is studied and the Hopf bifurcation number (threshold that the Taylor number crosses at the onset of Hopf bifurcations) is obtained
On the dynamic of the nonlinear reaction diffusion equation u_t=∆F(u)+f(x,u,∇u) under Robin boundary data
No full textLet Omega be a bounded smooth domain in R^3. Under Robin boundary data, the nonlinear
reaction diffusion equation
u_t=∆F(u)+f(x,u,∇u), (x, t) in Omega x R^+, is studied. Existence, uniqueness, stability and longtime behaviour of solutions are analyzed. An application
to the dynamic of a population inhabiting a strongly heterogeneous environment is considered
Stability and absorbing set of parabolic chemotaxis model of Escherichia coli
No full textThis paper is devoted to model (1) for escherichia coli, introduced in [1]. Based on the experimental observations of Budrene and Berg [2, 3], Tyson and coworkers derived (1) with n cell density, c chemotrattactant concentration and s stimulant concentration. Our aim is to study the stability of constant meaning full solution and ultimately boundedness of the solutions. Precisely: (i) linear and nonlinear stability is proved by using a peculiar Lyapunov function, (ii) the ultimately boundedness of the solutions in the L2-norm is obtained, (iii) conditions guaranteeing the global stability are also obtained
Mathematical problems for miscible, incompressible fluids with Korteweg stresses
No full textGaldi, P.; Joseph, D.D.; Preziosi, L.; Rionero, S.. (1990). Mathematical problems for miscible, incompressible fluids with Korteweg stresses. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/1435
Stability properties of the solutions of a reaction diffusion equation with Robin boundary
No full textEquation u_t=∆F(u)+f(x,u,∇u) is considered under Robin boundary data in a bounded domains. Existence of equilibria and longtime behaviour of solutions, under appropriate assumptions on F anf f, are established
Proceedings, "WASCOM 2007" : 14th Conference on Waves and Stability in Continuous Media : Baia Samuele, Sicily, Italy ; 30 June - 7 July 2007
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