Catálogo de publicaciones - libros

In literature there is no mathematical proof of the experimentally trivial stability of the rest state for a layer of compressible fluid heated from above. In the case of layer heated from below it is known that the system shows a threshold in the temperature gradient below which the fluid is not sensible to the imposed difference of temperature. Only semi-empirical justifications are available for this phenomenon, see [6].

Neglecting the thermal conductivity, we are able to prove that for a layer of compressible fluid between two rigid planes kept at constant temperature, the rest state is linearly stable for every values of the parameters involved in two cases: the layer is heated from above; the layer is heated from below and the imposed gradient of temperature is less than a precise quantity, namely , where is the gravity constant, and is the specific heat at constant pressure, known as .

We consider the equation= Δ() with 2 ≤ < on a compact Riemannian manifold. We prove that any solution () approaches its (time-independent) mean with the quantitative boundfor any ∊ [2, +∞] and > 0 and the exponents , are shown to be the only possible for a bound of such type. The proof is based upon the connection between logarithmic Sobolev inequalities and decay properties of nonlinear semigroups.

The 3D incompressible Euler equations with initial data characterized by uniformly large vorticity are investigated. We prove existence on long time intervals of regular solutions to the 3D incompressible Euler equations for a class of large initial data in bounded cylindrical domains. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach is based on fast singular oscillating limits, nonlinear averaging and cancellation of oscillations in the nonlinear interactions for the vorticity field. With nonlinear averaging methods in the context of almost periodic functions, resonance conditions and a nonstandard small divisor problem, we obtain fully 3D limit resonant Euler equations. We establish the global regularity of the latter without any restriction on the size of 3D initial data and bootstrap this into the regularity on arbitrary large time intervals of the solutions of 3D Euler equations with weakly aligned uniformly large vorticity at = 0.

We present a mathematical model for the phototransduction cascade, taking into account the spatial localization of the different reaction processes. The geometric complexity of the problem (set in the rod outer segment) is simplified by a process of homogenization and concentration of capacity.

We formulate su cient conditions for regularity of a so-called suitable weak solution (; ) in a sub-domain of the time-space cylinder by means of requirements on one of the eigenvalues or on the eigenvectors of the rate of deformation tensor.

In literature there is no mathematical proof of the experimentally trivial stability of the rest state for a layer of compressible fluid heated from above. In the case of layer heated from below it is known that the system shows a threshold in the temperature gradient below which the fluid is not sensible to the imposed difference of temperature. Only semi-empirical justifications are available for this phenomenon, see [6].

Neglecting the thermal conductivity, we are able to prove that for a layer of compressible fluid between two rigid planes kept at constant temperature, the rest state is linearly stable for every values of the parameters involved in two cases: the layer is heated from above; the layer is heated from below and the imposed gradient of temperature is less than a precise quantity, namely , where is the gravity constant, and is the specific heat at constant pressure, known as .

We review the geometry of diffusion processes of differential forms on smooth compact manifolds, as a basis for the random representations of the kinematic dynamo equations on these manifolds. We realize these representations in terms of sequences of ordinary (for almost all times) differential equations. We construct the random symplectic geometry and the random Hamiltonian structure for these equations, and derive a new class of Poincaré-Cartan invariants of magnetohydrodynamics. We obtain a random Liouville invariant. We work out in detail the case of .

In mathematical models of incompressible flow problems, quasi-Lipschitz conditions present a useful link between a class of singular integrals and systems of ordinary differential equations. Such a condition, established in suitable form for the first-order derivatives of Newtonian potentials in (Section 2) gives the main tool for the proof (in Sections 3–6) of the existence of a unique classical solution to Cauchy’s problem of Helmholtz’s vorticity transport equation with partial discretization in for each bounded time interval. The solution depends continuously on its initial value and, in addition, fulfills a discretized form of Cauchy’s vorticity equation.

We study the Cauchy-Dirichlet problem for the degenerate parabolic equationwith the parameters ∊, > 1, > 0, satisfying the condition + ≥ 2. The problem domain is the exterior of the cylinder bounded by a simple-connected surface , supp is an annular domain. We show that the velocity of the outer interface Γ = ∂ {supp ()} is given by the formulawhere II() is a solution of the degenerate elliptic equation,depending on as a parameter. It is proved that the solution and its interface Γ preserve their initial regularity with respect to the space variables, and that they are real analytic functions of time . We also show that the regularity of the velocity v is better than it was at the initial instant. For the space dimensions = 1, 2, 3, these results were established in [8]. We propose a modification of the method of [8] that makes it applicable to equations with an arbitrary number of independent variables.

We describe some estimates for solutions of nonlinear discrete schemes, which are analogues of fundamental estimates of Krylov and Safonov for linear elliptic partial differential equations and the resultant Schauder estimates for nonlinear elliptic equations of Evans, Krylov and Safonov.