1,720,978 research outputs found
A note on the slow convergence of solutions to conservation laws with mean curvature diffusions
No full textWe study the asymptotic behaviour of solutions to a scalar conservation law with a mean curvature's type diffusion, focusing our attention to the stability/metastability properties of the steady state. In particular, we show the existence of a unique steady state that slowly converges to its asymptotic configuration, with a speed rate which is exponentially small with respect to the viscosity parameter epsilon; the rigorous results are also validated by numerical simulations
Long time dynamics of layered solutions to the shallow water equations
No full textWe study the existence of a positive connection, i.e. a stationary solution connecting the boundary data, for the initial-boundary value problem for the viscous shallow water system in a bounded interval (-a"", a"") of the real line. Subsequently, we investigate the asymptotic behavior of the time-dependent solutions, showing that they first develop into a layered function and then they drift towards the steady state in an exponentially long time interval. The main tool of our analysis is given by the derivation of an ODE for the interface location
Slow dynamics in reaction–diffusion systems
No full textWe consider a system of reaction–diffusion equations in a bounded interval of the real line, with emphasis on the
metastable dynamics, whereby the time-dependent solution approaches the steady state in an asymptotically exponentially long
time interval as the viscosity coefficient ε > 0 goes to zero. To rigorous describe such behavior, we analyze the dynamics
of layered solutions localized far from the stable configurations of the system, and we derive an ODE for the position of the
internal interfaces
EXISTENCE AND UNIQUENESS OF A POSITIVE CONNECTION FOR THE SCALAR VISCOUS SHALLOW WATER SYSTEM IN A BOUNDED INTERVAL
No full textWe study the existence and the uniqueness of a positive connection, that is a stationary solution connecting the boundary data, for the initial-boundary value problem for the viscous shallow water system (Equation presented) in a bounded interval (-l, l) of the real line. We firstly consider the general case where the term of pressure P(u) satisfies P(0) = 0, P(+∞) = +∞, P'(u) and P"(u) > 0 ∀u 1. The viscous Saint-Venant system, corresponding to γ = 2, fits in the general framework
STABILITY PROPERTIES OF THE STEADY STATE FOR THE ISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY DEPENDENT VISCOSITY IN BOUNDED INTERVALS
No full textWe prove existence and asymptotic stability of the stationary solution for the compressible Navier-Stokes equations for isentropic gas dynamics with a density-dependent diffusion in a bounded interval. We present the necessary conditions to be imposed on the boundary data which ensure existence and uniqueness of the steady state, and we subsequently investigate its stability properties by means of the construction of a suitable Lyapunov functional for the system. The Saint-Venant system, modeling the dynamics of a shallow compressible fluid, fits into this general framework
METASTABLE DYNAMICS OF INTERFACES FOR THE HYPERBOLIC JIN-XIN SYSTEM
No full textWe consider the slow motion of internal shock layer exhibited by the solution to the initial boundary-value problem for the Jin-Xin system in one space dimension. In this contest, the time-dependent solution drifts towards its steady state in an exponentially long time interval. The derivation of an ODE for the shock layer position is the main tool of our analysis
Monotone wave fronts for (p, q)-Laplacian driven reaction-diffusion equations
No full textWe study the existence of monotone heteroclinic traveling waves for the 1-dimensional reaction-diffusion equation ut = (|ux| p − 2 ux + |ux| q − 2 ux)x + f(u), t ∈ R, x ∈ R, where the non-homogeneous operator appearing on the right-hand side is of (p, q)-Laplacian type. Here we assume that 2 ≤ q < p and f is a nonlinearity of Fisher type on [0, 1], namely f(0) = 0 = f(1) and f > 0 on ]0, 1[ . We give an estimate of the critical speed and we comment on the roles of p and q in the dynamics, providing some numerical simulations
Metastability for nonlinear parabolic equations with application to scalar viscous conservation laws
Full text linkThe aim of this paper is to contribute to the definition of a versatile language for metastability in the context of partial differential equations of evolutive type. A general framework suited for parabolic equations in one-dimensional bounded domains is proposed, based on choosing a family of approximate steady states {U-epsilon(mu;.)}(epsilon is an element of J) and on the spectral properties of the linearized operators at such states. The slow motion for solutions belonging to a cylindrical neighborhood of the family {U-epsilon} is analyzed by means of a system of an ODE for the parameter. =.(t), coupled with a PDE describing the evolution of the perturbation v := u -U-epsilon(xi;.). We state and prove a general result concerning the reduced system for the couple (xi v), called quasi-linearized system, obtained by disregarding the nonlinear term in v, and we show how such an approach suits to the prototypical example of scalar viscous conservation laws with Dirichlet boundary conditions in a bounded one-dimensional interval with convex flux
Transition from hyperbolicity to ellipticity for a p-system with viscosity
No full textIn this article we consider a viscous regularization of a p-system with
a Van der Waals pressure law, which presents both hyperbolic and elliptic
zones. Even if the purely hyperbolic Van der Walls system is strongly ill-posed,
we prove that the solutions of the regularized equation exist and experience
a transition from ellipticity to hyperbolicity, i.e. solutions issued
from initial data in the elliptic zone will enter the hyperbolic zone at
some time T>0, and viceversa
Heteroclinic traveling fronts for reaction-convection-diffusion equations with a saturating diffusive term
No full textThis paper deals with the existence of monotone heteroclinic traveling waves for some reaction-convection-diffusion equations with saturating (and possibly density-dependent) nonlinear diffusion, modeling physical situations where a saturation effect appears for large values of the gradient. An estimate for the critical speed—namely, the least speed for which a monotone heteroclinic traveling wave exists—is provided in the presence of different kinds of reaction terms (e.g., monostable and bistable ones). The dependence of the admissible speeds on a small real parameter breaking the diffusion is also briefly discussed, and some numerical simulations are also shown
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