Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in and space dimensions
A system of quasilinear parabolic equations modelling chemotaxis and taking into account the volume filling effect is studied under no-flux boundary conditions. The resulting system is non-uniformly parabolic. A Lyapunov functional for the system is constructed. The proof of existence and uniqueness of regular global-in-time solutions is given in cases when either the Lyapunov functional is bounded from below or chemotactic forces are suitably weakened. In the first case solutions are uniformly...
We consider a class of possibly degenerate second order elliptic operators on ℝⁿ. This class includes hypoelliptic Ornstein-Uhlenbeck type operators having an additional first order term with unbounded coefficients. We establish global Schauder estimates in Hölder spaces both for elliptic equations and for parabolic Cauchy problems involving . The Hölder spaces in question are defined with respect to a possibly non-Euclidean metric related to the operator . Schauder estimates are deduced by sharp...
We show the solvability of a nonlinear degenerate parabolic system of two equations describing the displacement of one compressible fluid by another, completely miscible with the first, in a one-dimensional porous medium, neglecting the molecular diffusion. We use the technique of renormalised solutions for parabolic equations in the derivation of a priori estimates for viscosity type solutions. We pass to the limit, as the molecular diffusion coefficient tends to 0, on the parabolic system, owing...
In this work we prove both local and global Harnack estimates for weak supersolutions to second order nonlinear degenerate parabolic partial differential equations in divergence form. We reduce the proof to an analysis of so-called hot and cold alternatives, and use the expansion of positivity together with a parabolic type of covering argument. Our proof uses only the properties of weak supersolutions. In particular, no comparison to weak solutions is needed.
If is a strongly continuous and contractive semigroup on a complex Banach space , then , , generates a holomorphic semigroup on . This was proved by K. Yosida in [7]. Using similar techniques, we present a class of Bernstein functions such that for all , the operator generates a holomorphic semigroup.
In this work we consider a diffusion problem in a periodic composite having three phases: matrix, fibers and interphase. The heat conductivities of the medium vary periodically with a period of size ( and ) in the transverse directions of the fibers. In addition, we assume that the conductivity of the interphase material and the anisotropy contrast of the material in the fibers are of the same order (the so-called double-porosity type scaling) while the matrix material has a conductivity of...
Questo articolo considera una successione di equazioni differenziali a derivate parziali non lineari in forma di divergenza del tipo in un dominio limitato dello spazio -dimensionale; e sono matrici con coefficenti limitati, e è invertibile e la sua matrice inversa ha anche coefficenti limitati. La non linearità è dovuta alla funzione ; la condizione di crescita, la monotonicità e le ipotesi di coercitività sono modellate sul -Laplaciano, , ed assicurano l'esistenza di una soluzione...