The course will review several aspects of discrete integration by parts methods. The ultimate goal is to construct finite difference approximations of the first order derivative that satisfy a similar integration by parts formula as in the continuous setting on a half-line. Basic questions (and partial answers) include existence, uniqueness and non-existence results. We shall connect the...
The course will review several aspects of discrete integration by parts methods. The ultimate goal is to construct finite difference approximations of the first order derivative that satisfy a similar integration by parts formula as in the continuous setting on a half-line. Basic questions (and partial answers) include existence, uniqueness and non-existence results. We shall connect the...
For hyperbolic systems of conservation laws in 1-D, fundamental questions about uniqueness and blow up of weak solutions still remain even for the apparently “simple” systems of two conserved quantities such as isentropic Euler and the p-system. Similarly, in the multi-dimensional case, a longstanding open question has been the uniqueness of weak solutions with initial data corresponding to...
The Euler-Korteweg equations are a modification of the Euler equations which include in the momentum equation a term modelling capillary forces. Mathematically, this supplementary term is of dispersive nature, and after a reformulation the system looks like a degenerate Schrödinger equation. We consider here the behaviour of smooth solutions when the capillary coefficient is very small. When...
In Hamiltonian systems, periodic waves often correspond to coherent structures: recurrent, robust patterns that persist over time. Notable examples include water waves, periodic sequences of light pulses in nonlinear optical fibers, and soliton trains in Bose-Einstein condensates. To date, nonlinear stability results for periodic standing or traveling waves in Hamiltonian systems have...
In Hamiltonian systems, periodic waves often correspond to coherent structures: recurrent, robust patterns that persist over time. Notable examples include water waves, periodic sequences of light pulses in nonlinear optical fibers, and soliton trains in Bose-Einstein condensates. To date, nonlinear stability results for periodic standing or traveling waves in Hamiltonian systems have...
We consider the one-dimensional diffusion approximation, non-equilibrium model of radiation hydrodynamics derived by Buet and Després (J. Quant. Spectrosc. Radiat. Transf. 85 (2004), no. 3-4, 385–418). This system describes a non-relativistic inviscid fluid subject to a radiative field under the non-equilibrium hypothesis, that is, when the temperature of the fluid is different from the...
For evident reasons, Cancer Biology is one of the most challenging topics of current medical research and understanding the mechanism behind its uncontrolled growth is a crucial issue. Among other explanations of the process, the Warburg effect posits that a pivotal role is played by the so-called aerobic glycolysis, i.e. the fact that, even in presence of oxygen, lactic acid fermentation can...
We consider the one-dimensional Euler-Poisson system equipped with the Boltzmann relation. We provide the exact asymptotic behavior of the peaked solitary wave solutions near the peak. This enables us to study the cold ion limit of the peaked solitary waves with the sharp range of Holder exponents. Furthermore, we provide numerical evidence for $C^1$ blow-up solutions to the pressureless...
Among weak solutions of Burgers' equation, a single strictly convex entropy is sufficient to characterize the sign of all entropy productions. In particular, if that entropy production vanishes, then the solution must be continuous. It turns out that this fact can be interpreted as a regularity result for a degenerate elliptic equation in the plane, and generalized to prove partial regularity...