Related papers: $L^1$-Theory for Hele-Shaw flow with linear drift
We study the existence and long-time asymptotics of weak solutions to a system of two nonlinear drift-diffusion equations that has a gradient flow structure in the Wasserstein distance. The two equations are coupled through a…
We show a connection between the linear trace Li-Yau-Hamilton inequality for the Kaehler-Ricci flow and the monotonicity formula for the positive currents. As an application of the linear trace Li-Yau-Hamilton stated in this paper and the…
We consider in this paper a spray constituted of an incompressible viscous gas and of small droplets which can breakup. This spray is modeled by the coupling (through a drag force term) of the incom- pressible Navier-Stokes equation and of…
We deal with the uniqueness of distributional solutions to the continuity equation with a Sobolev vector field and with the property of being a Lagrangian solution, that means transported by a flow of the associated ordinary differential…
This paper is concerned with entropy solutions of scalar conservation laws of the form $\partial_{t}u+\diver f=0$ in $\mathbb{R}^d\times(0,\infty)$. The flux $f=f(x,u)$ depends explicitly on the spatial variable $x$. Using an extension of…
We study the Willmore flow for graphs over a bounded domain in $\mathbb{R}^2$ with Dirichlet (clamped) boundary conditions, a still little-studied setting that also serves as a prototype for higher-order flows with fixed boundary data. We…
We derive, through the deterministic homogenization theory in thin domains, a new model consisting of Hele-Shaw equation with memory coupled with the convective Cahn-Hilliard equation. The obtained system, which models in particular tumor…
Topology changes in multi-phase fluid flows are difficult to model within a traditional sharp interface theory. Diffuse interface models turn out to be an attractive alternative to model two-phase flows. Based on a…
In this paper, we study the continuity in initial data of a classical reaction-diffusion equation with arbitrary $p>2$ order nonlinearity and in any space dimension $N\geq 1$. It is proved that the weak solutions can be $(L^2, L^\gamma\cap…
In this note, we discuss the quantum Hele-Shaw flow, a random measure process in the complex plane introduced by the physicists P.Wiegmann, A. Zabrodin, et al. This process arises in the theory of electronic droplets confined to a plane…
We formulate a relativistic hydrodynamic theory for fluids with spin and intrinsic dilation charges. Using an entropy-current analysis, we derive constitutive relations featuring a bulk viscosity and a dilation conductivity governing the…
The recently developed theory of Lagrangian flows for transport equations with low regularity coefficients enables to consider non BV vector fields. We apply this theory to prove existence and stability of global Lagrangian solutions to the…
An useful approximation for the displacement of two immiscible fluids in a porous medium is the Hele-Shaw model. We consider several liquids with different constant viscosities, inserted between the displacing fluids. The linear stability…
The porous medium type reaction-diffusion equation and the Hele-Shaw problem are two free boundary problems linked through the incompressible (Hele-Shaw) limit. We investigate and compare the sharp power concavities of the pressures on…
We introduce a novel primal-dual flow for affine constrained convex optimization problems. As a modification of the standard saddle-point system, our primal-dual flow is proved to possess the exponential decay property, in terms of a…
This paper investigates the connection between the chemotaxis--Navier--Stokes system with porous medium type nonlinear diffusion and the Hele--Shaw problem in $\mathbb{R}^d$ ($d\geq2$). First, we prove the global-in-time existence of weak…
With the Lattice Boltzmann method (using the BGK approximation) we investigate the dynamics of Hele-Shaw flow under conditions corresponding to various experimental systems. We discuss the onset of the instability (dispersion relation), the…
We study the global existence of solutions to a one-dimensional drift-diffusion equation with logistic term, generalizing the classical parabolic-elliptic Keller-Segel aggregation equation arising in mathematical biology. In particular, we…
We study the long time behavior of solutions of the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition or monostable type. We prove a one-to-one relation between the…
We develop a renormalization group for weak Harris-marginal disorder in otherwise strongly interacting quantum critical theories, focusing on systems which have emergent conformal invariance. Using conformal perturbation theory, we argue…