Competing effects in fourth-order aggregation-diffusion equations (Q6591565)

The paper focuses on the analysis of global existence for gradient flow models involving Cahn-Hilliard equations of the form \N\[\N\partial_t \rho=-\operatorname{div} (\rho \nabla(\Delta \rho))-\chi \Delta \rho^m.\N\]\NTo do so, the authors analyse a 2-Wassertein gradient flow of a free energy, with the Dirichlet energy and the power-law internal energy as competing effects. Those models are not only related to the classical Cahn-Hilliard equation but also to the so-called thin film equation from lubrication theory.\N\NUsing classical variational methods they obtain interesting results on the existence of global solutions for such gradient flows, in terms of sharp conditions on the exponent \(m\) or the parameter \(\chi\) regarding a critical mass \(\chi_c\), involved in the systems.\N\NIn particular, for the free energy functional of the problems the authors prove some lower bound conditions and regularity results. Identifying the critical exponent for \(m\), i.e. \(m=m_c\), the infimum of the free energy functional is also achieved, for cases under the critical mass of the problem \(\chi_c\). Those results are extended to a wider range of exponents, finding the existence of minimisers for a 2-Wassertein gradient flow energy problem after performing a self-similar ansatz. These results characterise the energy minimisers for the self-similar profiles. Analysing the supercritical regime \(m>m_c\) the infimum of the free energy is not achieved and using similar computations as those performed for Keller-Segel models to compute the evolution of the second order moment, some finite blow-up results are discussed. The subcritical regime, as well as the long time behaviour of the problem is left open. It is interesting to note that thanks to the plot of some numerical solutions one can observe that for the subcritical exponent, solutions evolve towards a compactly supported steady state while the free energy stays bounded below. Moreover, for the critical exponent with subcritical mass the free energy is bounded below by zero, but in this case solutions tend to the self-similar profile. Finally, finite time blow-up occurs for critical exponent with supercritical mass and for supercritical exponent. In both cases, the free energy is unbounded from below.\N\NFurthermore, the existence of weak solutions is analysed applying the so-called JKO scheme of minimising movement, and direct methods of calculus of variations. Such as boundedness from below of the functional for a minimising sequence, lower semicontinuity and strong convergence of the minimising sequence. However, the strong convergence is not enough to pass to the limit in the Euler-Lagrange associated equation. To overcome such a difficulty, the authors use as an auxiliary problem the heat equation to obtain uniform bounds on the Hessian of the sequence. That argument helps the authors to ascertain the existence of weak solutions for the problems under analysis. Those arguments are also extended to coupled systems of Cahn-Hilliard form.