Related papers: Compactness for nonlinear transport equations
We prove existence of solutions to a nonlinear transport equation in the plane, for which the velocity field is obtained as the convolution of the classical Cauchy Kernel with the unknown. Even though the initial datum is bounded and…
We provide an informal overview on the theory of transport equations with non smooth velocity fields, and on some applications of this theory to the well-posedness of hyperbolic systems of conservation laws.
Given $k:\mathbb{R}^n\setminus\{0\} \to \mathbb{R}^n$ a kernel of class $C^2$ and homogeneous of degree $1-n$, we prove existence and uniqueness of H\"older regular solutions for some non-linear transport equations with velocity fields…
This work gathers new results concerning the semi-geostrophic equations: existence and stability of measure valued solutions, existence and uniqueness of solutions under certain continuity conditions for the density, convergence to the…
We prove quantitative regularity estimates for the solutions to non-linear continuity equations and their discretized numerical approximations on Cartesian grids when advected by a rough force field. This allow us to recover the known…
We investigate the presence of localized solutions in models described by a single real scalar field with generalized dynamics. The study offers a method to solve very intricate nonlinear ordinary differential equations, and we illustrate…
We consider linear and nonlinear transport equations with irregular velocity fields, motivated by models coming from mean field games. The velocity fields are assumed to increase in each coordinate, and the divergence therefore fails to be…
We prove the well-posedness of entropy solutions for a wide class of nonlocal transport equations with nonlinear mobility in one spatial dimension. The solution is obtained as the limit of approximations constructed via a deterministic…
A kinetic equation which combines the quasiparticle drift of Landau's equation with a dissipation governed by a nonlocal and noninstant scattering integral in the spirit of Snider's equation for gases is derived. Consequent balance…
The conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces are derived. For discrete models the conserved charges are constructed explicitly. The applications of the general…
We investigate the presence of static solutions in models described by real scalar field in two-dimensional spacetime. After taking advantage of a procedure introduced sometime ago, we solve intricate nonlinear ordinary differential…
We consider a quasilinear KdV equation that admits compactly supported traveling wave solutions (compactons). This model is one of the most straightforward instances of degenerate dispersion, a phenomenon that appears in a variety of…
We prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds. The proof is based on a refined bubbling analysis, for which the main estimates are given in integral form. Our result is…
We provide closed form solutions for an equation which describes the transport of turbulent kinetic energy in the framework of a turbulence model with a single equation.
We prove that the solution map for a family of non-linear transport equations in $\mathbb{R}^n$, with a velocity field given by the convolution of the density with a kernel that is smooth away from the origin and homogeneous of degree…
We study a nonlinear transport equation defined on an oriented network where the velocity field depends not only on the state variable, but also on the solution itself. We prove existence, uniqueness and continuous dependence results for…
We prove global existence, uniqueness and $\L1$ stability of solutions to general systems of nonlocal conservation laws modeling multiclass vehicular traffic. Each class follows its own speed law and has specific effects on the other…
In this paper, we introduce a class of nonlinear optimisation problems. Under mild assumptions, we obtain the existence of potential functions and show that the potential function is a generalised solution of a Monge-Amp\`ere type equation.…
We introduce a model for nonlinear viscoelastic solids where traveling shear waves with compact support are possible. We obtain an exact compact solution. We also derive a new Burger's type evolution equation associated with the introduced…
Conservation laws are key theoretical and practical tools for understanding, characterizing, and modeling nonlinear dynamical systems. However, for many complex systems, the corresponding conserved quantities are difficult to identify,…