1,720,970 research outputs found
Weak solutions of a strongly coupled degenerate parabolic system
No full textWe prove existence in arbitrary space dimension of nonnegative bounded weak solutions of both the Cauchy problem and an initial-boundary value problem for a degenerate parabolic system which arises in plasma physics
Analisi Matematica
No full textAnalisi Matematica e` un testo che:
* offre una gamma completa degli argomenti classici dei corsi di base di Analisi Matematica;
* accoglie le esigenze dei corsi dei nuovi ordinamenti didattici;
* e` scritto in modo accessibile per lo studente, senza rinunciare al rigore matematico;
* lascia un alto grado di liberta` al docente nell’impostazione delle lezioni.
Il testo contiene, oltre a ‘‘quasi’’ tutti gli argomenti standard dei ‘‘vecchi’’ corsi di Analisi Matematica 1 e 2, un’introduzione alle funzioni olomorfe, alle serie di Fourier, alle trasformate di Laplace e di Fourier e al concetto di stabilita’ per soluzioni di equazioni differenziali ordinarie. Il ‘‘quasi’’ si riferisce al concetto di convergenza uniforme, piu`adatto a un corso avanzato
A system of degenerate parabolic nonlinear PDE's: a new free boundary problem
No full textWe prove existence of solutions of a new free boundary problem described by a system of degenerate parabolic equations. The problem arises in petroleum engineering and concerns fluid flows in diatomite rocks. The unknown functions represent the pressure of the fluid and a damage parameter of the porous rock. These quantities are not necessarily continuous on the free boundary, which considerably complicates the mathematical analysis
Rotationally symmetric 1-harmonic maps from D^2 to S^2
No full textWe consider rotationally symmetric 1-harmonic maps from D^2 to S^2 subject to Dirichlet boundary conditions. We prove that the corresponding energy—a degenerate nonconvex functional with linear growth—admits a unique minimizer, and that the minimizer is smooth in the bulk and continuously differentiable up to the boundary. We also show that, in contrast with 2-harmonic maps, a range of boundary data exists such that the energy admits more than one smooth critical point: more precisely, we prove that the corresponding Euler–Lagrange equation admits a unique (up to scaling and symmetries) global solution, which turns out to be oscillating, and we characterize the minimizer and the smooth critical points of the energy as the monotone, respectively non-monotone, branches of such solution
Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility
No full textIn one space dimension, we prove existence of weak solutions for the Neumann problem for a degenerate parabolic system consisting of a fourth-order and a second-order equation with singular lower-order terms. This system arises in the description of phase separation and ordering in binary alloys
A waiting time phenomenon for thin film equations
Full text linkWe prove the occurrence of a waiting time phenomenon for solutions to fourth order degenerate parabolic differential equations which model the evolution of thin films of viscous fluids. In space dimension less or equal to three, we identify a general criterion on the growth of initial data near the free boundary which guarantees that for sufficiently small times the support of strong solutions locally does not increase. It turns out that this condition only depends on the smoothness of the diffusion coefficient in its point of degeneracy. Our approach combines a new version of Stampacchia's iteration lemma with weighted energy or entropy estimates. On account of numerical experiments, we conjecture that the
aforementioned growth criterion is optimal
Waiting time phenomena for degenerate parabolic equations - A unifying approach
No full textWe present a new approach to establish the occurrence of
waiting time phenomena for solutions to degenerate parabolic equations. Originally developed by the authors for the thin film equation, it may in fact be used for a broad class of degenerate parabolic equations and systems having divergence structure, such as doubly nonlinear second order equations of porous media type as well as higher order doubly nonlinear equations with variational structure
A system of degenerate parabolic nonlinear PDE's: a new free boundary problem
No full textWe prove existence of solutions of a new free boundary problem described by a system of degenerate parabolic equations. The problem arises in petroleum engineering and concerns fluid flows in diatomite rocks. The unknown functions represent the pressure of the fluid and a damage parameter of the porous rock. These quantities are not necessarily continuous on the free boundary, which considerably complicates the mathematical analysis
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