Compactness in the space ${L}^{p}(0,T;B)$, $B$ being a separable Banach space, has been deeply investigated by J.P. Aubin (1963), J.L. Lions (1961, 1969), J. Simon (1987), and, more recently, by J.M. Rakotoson and R. Temam (2001), who have provided various criteria for relative compactness, which turn out to be crucial tools in the existence proof of solutions to several abstract time dependent problems related to evolutionary PDEs. In the present paper, the problem is examined in view of Young measure theory: exploiting...

This note addresses the Cauchy problem for the gradient flow equation in a Hilbert space $H$
$$\left\{\begin{array}{c}{u}^{\text{'}}\left(t\right)+\partial \phi \left(u\left(t\right)\right)\ni 0\phantom{\rule{1.0em}{0ex}}\text{a.e.}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{0.166667em}{0ex}}(0,T),\hfill \\ u\left(0\right)={u}_{0},\hfill \end{array}\right.$$
where $\phi :H\to (-\infty ,+\infty \phantom{\rule{0.166667em}{0ex}}]$ is a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional and $\partial \phi $ is (a suitable limiting version of) its subdifferential. The interest for this kind of equations is motivated by a number of examples, which show that several mathematical models describing phase transitions phenomena and leading to systems of evolutionary PDEs...

This paper is devoted to the analysis of a one-dimensional model for phase transition phenomena in thermoviscoelastic materials. The corresponding parabolic-hyperbolic PDE system features a internal energy balance equation, governing the evolution of the absolute temperature $\vartheta $, an evolution equation for the phase change parameter $\chi $, including constraints on the phase variable, and a hyperbolic stress-strain relation for the displacement variable $\mathbf{u}$. The main novelty of the model is that the equations...

This paper addresses the Cauchy problem for the
gradient flow equation in a Hilbert space $\mathscr{H}$
$$\left\{\begin{array}{cc}{u}^{\text{'}}\left(t\right)+{\partial}_{\ell}\phi \left(u\left(t\right)\right)\ni f\left(t\right)\hfill & \mathit{\text{a.e.}}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}(0,T),u\left(0\right)={u}_{0},\hfill \end{array}\right.$$
where $\phi :\mathscr{H}\to (-\infty ,+\infty ]$ is a proper,
lower semicontinuous functional which is not supposed to be a
(smooth perturbation of a) convex functional and ${\partial}_{\ell}\phi $ is
(a suitable limiting version of) its subdifferential.
We will present some new existence results for the solutions of the
equation by exploiting a
technique, featuring some ideas from the theory of
and of .
Our analysis
is also motivated by some models...

In the nonconvex case, solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate-independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential that is a viscous regularization of a given rate-independent dissipation...

This paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of for gradient flows in metric spaces, see [5]), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract doubly nonlinear equations...

In the nonconvex case, solutions of rate-independent systems may develop jumps as a
function of time. To model such jumps, we adopt the philosophy that rate-independence
should be considered as limit of systems with smaller and smaller viscosity. For the
finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear
equations given in terms of a differentiable energy functional and a dissipation potential
that is a viscous regularization...

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