1,977 research outputs found
A uniqueness result for the continuity equation in two dimensions
No full textWe characterize the autonomous, divergence-free vector fields on the plane such that the Cauchy problem for the continuity equation admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential associated to .
As a corollary we obtain uniqueness under the assumption that the curl of is a measure. This result can be extended to certain non- autonomous vector fields with bounded divergence
Renormalized solutions to the continuity equation with an integrable damping term
No full textWe consider the continuity equation with a nonsmooth vector field and a damping term. In their fundamental paper, DiPerna and Lions (Invent Math 98:511–547, 1989) proved that, when the damping term is bounded in space and time, the equation is well posed in the class of distributional solutions and the solution is transported by suitable characteristics of the vector field. In this paper, we prove existence and uniqueness of renormalized solutions in the case of an integrable damping term, employing a new logarithmic estimate inspired by analogous ideas of Ambrosio et al. (Rendiconti del Seminario Fisico Matematico di Padova 114:29–50, 2005), Crippa and De Lellis (J Reine Angew Math 616:15–46, 2008) in the Lagrangian case
Flows of singular vector fields and applications to fluid and kinetic equations
No full textSeveral physical phenomena arising in fluid dynamics and kinetic equations can be modeled by nonlinear transport PDE. Such quantities are the vorticity of a fluid, or the density of a collection of particles advected by a velocity field which is highly irregular. The theory of characteristics provides a link between this PDE and the ODE dX/dt=b(t,X(t,x)), where b is the velocity field. When b has Sobolev or BV regularity and bounded divergence, the theory of DiPerna-Lions and Ambrosio gives a good notion of solution to the ordinary differential equation using the concept of regular Lagrangian flow. Extending the results of Crippa-DeLellis, and more recently Bouchut-Crippa, we study Lagrangian flows associated to velocity fields with anisotropic regularity: those with gradient given by the singular integral of an L^1 function in some directions, and the singular integral of a measure in others. We exploit an anisotropic version of the previous arguments and estimate the difference quotients in this context, thereby gaining quantitative estimates in terms of the given regularity bounds. One then recovers well-posedness for the ordinary differential equation. This answers positively the question of existence of Lagrangian solutions to the Vlasov Poisson and Euler equations with L^1 data
Nuclear myosin VI enhances RNA polymerase II-dependent transcription
No full textAbstract not availableSarah Vreugde, Carmelo Ferrai, Annarita Miluzio, Ehud Hauben, Pier Carlo Marchisio, Massimo P. Crippa, Mario Bussi, Stefano Biff
Renormalized solutions of the 2D Euler equations
No full textIn this paper we prove that solutions of the 2D Euler equations in vorticity formulation obtained via vanishing viscosity approximation are renormalized
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