Related papers: Transport and continuity equations with (very) rou…
We study the statistical properties of stochastic evolution equations driven by space-only noise, either additive or multiplicative. While forward problems, such as existence, uniqueness, and regularity of the solution, for such equations…
We prove well-posedness and rough path stability of a class of linear and semi-linear rough PDE's on $\mathbb{R}^d$ using the variational approach. This includes well-posedness of (possibly degenerate) linear rough PDE's in…
We study an integro-differential equation that describes the slow erosion of granular flow. The equation is a first order non-linear conservation law where the flux function includes an integral term. We show that there exist unique…
Distribution functions of many static transport equations are found using the Maximum Entropy Principle. The equations of constraint which contain the relevant dynamical information are simply the low-lying moments of the distributions.…
Recent advances in time series, where deterministic and stochastic modelings as well as the storage and analysis of big data are useless, permit a new approach to short-term traffic flow forecasting and to its reliability, i.e., to the…
The displacement and deviation vectors in spaces (manifolds), the tangent bundle of which is endowed with a transport along paths, are introduced. In case these spaces are equipped with a linear connection, the deviation equations (between…
The Kolmogorov equation associated to a stochastic 2D Euler equations with transport type noise and random initial conditions is studied by a direct approach, based on Fourier analysis, Galerkin approximation and Wiener chaos methods. The…
We study a stochastically-driven standard map. The addition of a noise term destroys the invariant manifolds that organize the phase space which allows for more widespread transport than in the noiseless case. Using appropriately defined…
The conditions under which, the continuity equation can be substituted by an ordinary non differential equation, will be discussed. Since continuity equation is a fundamental equation, this result will be applicable in a vast area of…
We study a two-dimensional incompressible vorticity equation on the torus driven by transport-type fractional Brownian noise with Hurst parameter $H \in (1/2,1)$. The model captures persistent, long-range correlated forcing consistent with…
We prove the existence of local stable, unstable, and center manifolds for stochastic semiflows induced by rough differential equations driven by rough paths valued stochastic processes around random fixed points of the equation. Examples…
We address the description of solutes flow with trapping processes in porous media. Starting from a small-scale model for tracer particles trajectories, we derive the corresponding governing equations for the concentration of the mobile and…
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 analyze common lifts of stochastic processes to rough paths/rough drivers-valued processes and give sufficient conditions for the cocycle property to hold for these lifts. We show that random rough differential equations driven by such…
We give an overview of the existence and regularity results for curvature flows and how these flows can be used to solve some problems in geometry and physics.
The generalized transport equations for a consistent description of kinetic and hydrodynamic processes in dense gases and liquids are considered. The inner structure of the generalized transport kernels for these equations is established.…
Turbulence is omnipresent in Nature and technology, governing the transport of heat, mass, and momentum on multiple scales. For real-world applications of wall-bounded turbulence, the underlying surfaces are virtually always rough; yet…
Many real life queueing networks have finite buffers with overflows. To understand the behavior of such networks, we consider traffic equations that generalize the traffic equations of classic open queueing networks where some nodes are…
We use the vorticity transportation equation as the start point--with the help of stream function for two-dimensional planar incompressible flows--to obtain exact solutions that characterize evolution and dynamics of the flows. These…
The purpose of this article is to solve rough differential equations with the theory of regularity structures. These new tools recently developed by Martin Hairer for solving semi-linear partial differential stochastic equations were…