Pietro Galimberti (Università di Modena e Reggio Emilia) Title: A perturbation of the Cahn-Hilliard equation with non-degenerate mobility Abstract: We study a perturbation of the Cahn-Hilliard equation with non-degenerate mobility and nonlinear terms of logarithmic type. We extend the results known in the constant mobility case, proving the existence, regularity and uniqueness of weak and strong solutions. Moreover we show that weak solutions enjoy the so-called separation property from the pure phases, also in three space dimensions. We finally prove, in dimension two, the convergence to the Cahn–Hilliard equation, on finite time intervals. |
Daniele Nassisi (Università di Modena e Reggio Emilia) Title: Results on the long term behaviour of a perturbed Cahn-Hilliard equation with logarithmic potential Abstract: In this poster, our aim is to investigate the long term behaviour of a perturbed Cahn-Hilliard equation on a bounded domain \( \Omega \subset \mathbb R^d, \ d = 2, 3 \) given by $$ \begin{cases} \varphi_t = \Delta \mu – \varepsilon \mu_t, & \text{in } \Omega \times (0, +\infty), \\ \mu = -\Delta \varphi + \Psi'(\varphi), & \text{in } \Omega \times (0, +\infty), \\ \partial_{\mathbf{n}}\varphi = \partial_{\mathbf{n}}\mu = 0 & \text{on } \partial\Omega \times (0, +\infty), \\ \varphi(0) = \varphi_0 & \text{in } \Omega, \end{cases} \qquad \textrm{(1)} $$ where \( \varphi(x, t) \in (-1, 1) \) is the order parameter, \( \mu(x, t) \) is the chemical potential, \(\varepsilon > 0 \) is a small parameter, and \( \Psi \) is the thermodynamically relevant logarithimic double-well potential defined by $$ \Psi(s) = \frac{\theta}{2} \left((1 + s)\ln(1 + s) + (1 – s)\ln(1 – s) \right) – \frac{\theta_0}{2}s^2, $$ where \( 0 < \theta < \theta_0 \) are respectively the temperature and the critical temperature of the system. In particular, we establish the existence of a closed semigroup of operators \( S(t) \) generated by the strong solutions of \( \textrm{(1)} \) on a suitable phase space and its dissipative nature. As a consequence, we obtain new long term regularities for the solutions on the interval \( [0, +\infty) \), including the uniform (in time) strict separation from the pure phases holding in \((\sigma, +\infty), \ \forall \sigma > 0 \). To conclude we prove the existence of the universal attractor of \( S(t)\). |
Greta Ricchi (Università di Modena e Reggio Emilia) Title: Weak solutions for a Cahn-Hilliard equation with a non-local incompressibility constraint Abstract: We consider a Cahn-Hilliard equation variant proposed by Weinan E and Palffy-Muhoray, which describes the dynamics of phase separation of an incompressible mixture. The equation includes a degenerate mobility and the Flory-Huggins potential, as in the classical Cahn-Hilliard model, together with an additional convective term involving the Leray projection of the solution itself. The additional key term arises from a non-local incompressibility constraint, which leads to a faster energy dissipation. In particular, we demonstrate the existence of global-in-time weak solutions for the initial-boundary value problem associated to the Cahn-Hilliard equation with a non-local incompressibility constraint. The system is endowed with homogeneous Neumann boundary conditions for the concentration and the chemical potential. Joint work with Stefania Gatti and Andrea Giorgini. |
Francesca Toni (Università di Modena e Reggio Emilia) Title: Annealed spherical model on random graphs Abstract: While classical spin models, such as the Ising and Potts systems, have long been analysed on regular and deterministic lattices, only recently have they been systematically investigated on sparse, heterogeneous networks. Originally introduced by Berlin and Kac in 1952, the spherical model replaces discrete \(+1\), \(-1\) spins on a \(d\)-dimensional lattice with real‐valued variables subject to a spherical constraint. Thanks to its quadratic Hamiltonian, diagonalization reduces the partition function to a constrained Gaussian integral, yielding an exact second‐order phase transition for \(d>2\) and mean‐field critical exponents above the upper critical dimension \(d_c=4\). In this project, we plan to investigate the spherical model on a random graph. In particular, we focus on the annealed setup and study the phase transition on inhomogeneous random graphs: edges are independent Bernoulli variables, and spins remain continuous under the spherical constraint. The main goal is to identify the critical temperature and the critical exponents. Joint work in collaboration with C. Giardinà. |
Xu Yan (Osaka University) Title: The compactness of Trudinger-Moser type functionals with variable exponents for domains in \(\mathbb R^N\) Abstract: We consider the compactness property of several Trudinger-Moser type functionals with variable exponents, and we establish various nearly optimal conditions on the variable exponents which assure the compactness or the noncompactness of the functionals. We treat the problem both on bounded domains and the entire domain. The results are crucial in the view of the associated maximization problems. |
