1,485 research outputs found
Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach
Five types of blow-up patterns that can occur for the 4th-order semilinear
parabolic equation of reaction-diffusion type
u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1,
\quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For
the semilinear heat equation , various blow-up patterns
were under scrutiny since 1980s, while the case of higher-order diffusion was
studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure
Conditional and unconditional nonlinear stability in fluid dynamics
In this thesis we examine some of the interesting aspects of stability for some convection problems. Specifically, the first part of the thesis deals with the BĂ©nard problem for various Non-Newtonian fluids, whereas the second part develops a stability analysis for convection in a porous medium. The work on stability for viscoelastic fluids includes nonlinear stability analyses for the second grade fluid, the generalised second grade fluid, the fluid of dipolar type and the fluid of third grade. It is worth remarking that throughout the work the viscosity is supposed to be any given function of temperature, with the first derivative bounded above by a positive constant. The connection between the two parts of the thesis is made through the method used to approach the nonlinear stability analysis, namely the energy method. It is shown in the introductory chapter how this method works and what are its advantages over the linear analysis. Nonlinear stability results established in both Part I and Part II are the best one can get for the considered physical situations. Different choices of energy have been considered in order to achieve conditional or unconditional nonlinear stability results
Mathematics for 2d Interfaces
We present here a survey of recent results concerning the mathematical
analysis of instabilities of the interface between two incompressible, non
viscous, fluids of constant density and vorticity concentrated on the
interface. This configuration includes the so-called Kelvin-Helmholtz (the two
densities are equal), Rayleigh-Taylor (two different, nonzero, densities) and
the water waves (one of the densities is zero) problems. After a brief review
of results concerning strong and weak solutions of the Euler equation, we
derive interface equations (such as the Birkhoff-Rott equation) that describe
the motion of the interface. A linear analysis allows us to exhibit the main
features of these equations (such as ellipticity properties); the consequences
for the full, non linear, equations are then described. In particular, the
solutions of the Kelvin-Helmholtz and Rayleigh-Taylor problems are necessarily
analytic if they are above a certain threshold of regularity (a consequence is
the illposedness of the initial value problem in a non analytic framework). We
also say a few words on the phenomena that may occur below this regularity
threshold. Finally, special attention is given to the water waves problem,
which is much more stable than the Kelvin-Helmholtz and Rayleigh-Taylor
configurations. Most of the results presented here are in 2d (the interface has
dimension one), but we give a brief description of similarities and differences
in the 3d case.Comment: Survey. To appear in Panorama et Synth\`ese
Spatial pattern formation induced by Gaussian white noise
The ability of Gaussian noise to induce ordered states in dynamical systems
is here presented in an overview of the main stochastic mechanisms able to
generate spatial patterns. These mechanisms involve: (i) a deterministic local
dynamics term, accounting for the local rate of variation of the field
variable, (ii) a noise component (additive or multiplicative) accounting for
the unavoidable environmental disturbances, and (iii) a linear spatial coupling
component, which provides spatial coherence and takes into account diffusion
mechanisms. We investigate these dynamics using analytical tools, such as
mean-field theory, linear stability analysis and structure function analysis,
and use numerical simulations to confirm these analytical results.Comment: 11 pages, 8 figure
Kinetic theory of jet dynamics in the stochastic barotropic and 2D Navier-Stokes equations
We discuss the dynamics of zonal (or unidirectional) jets for barotropic
flows forced by Gaussian stochastic fields with white in time correlation
functions. This problem contains the stochastic dynamics of 2D Navier-Stokes
equation as a special case. We consider the limit of weak forces and
dissipation, when there is a time scale separation between the inertial time
scale (fast) and the spin-up or spin-down time (large) needed to reach an
average energy balance. In this limit, we show that an adiabatic reduction (or
stochastic averaging) of the dynamics can be performed. We then obtain a
kinetic equation that describes the slow evolution of zonal jets over a very
long time scale, where the effect of non-zonal turbulence has been integrated
out. The main theoretical difficulty, achieved in this work, is to analyze the
stationary distribution of a Lyapunov equation that describes quasi-Gaussian
fluctuations around each zonal jet, in the inertial limit. This is necessary to
prove that there is no ultraviolet divergence at leading order in such a way
that the asymptotic expansion is self-consistent. We obtain at leading order a
Fokker--Planck equation, associated to a stochastic kinetic equation, that
describes the slow jet dynamics. Its deterministic part is related to well
known phenomenological theories (for instance Stochastic Structural Stability
Theory) and to quasi-linear approximations, whereas the stochastic part allows
to go beyond the computation of the most probable zonal jet. We argue that the
effect of the stochastic part may be of huge importance when, as for instance
in the proximity of phase transitions, more than one attractor of the dynamics
is present
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