Renormalized solutions of some transport equations with partially W1,1 velocities and applications

By C. Le Bris, Pierre-Louis Lions. 2004

78 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics, Finance

View Paper: https://www.semanticscholar.org/paper/b57947c3d3d127cc78c2ee8f45e260b1e9798503

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Citations in the corpus

A stochastic Lagrangian representation of the three‐dimensional incompressible Navier‐Stokes equations (2005, methodology): “This can be seen from the techniques used by LeBris, Lions [ 18 ] or those used in Section 3. Notice immediately that u = E˜u and ˜u0 = u0. The circulation of the stochastic velocity is conserved by the stochastic flow.”
The Flow Associated to Weakly Differentiable Vector Fields: Recent Results and Open Problems (2011, methodology): “An important fact is that the approach is completely different from the one of [48]: the main idea is to estimate the difference quotient of bt in terms of maximal functions of the modulus of ∇bt.”
Differential equations with singular fields (2010, background): “ divergence and in W1,1. The crucial concept there is the one of renormalized solutions, namely weak solutions us.t. φ(u) is also a solution to the same transport equation. This was extended in [29], [27] and [26] ; also see [8] where the connection between well posedness for the transport equation ”
Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields (2009, background): “The main theorem in [73] is the following:”
Renormalized Solutions for Stochastic Transport Equations and the Regularization by Bilinear Multiplicative Noise (2010, background): “The notion of renormalized solutions has been investigated further by several authors, see for instance [3, 8, 9, 12, 18, 19, 21] and many others.”
Stochastic flows of SDEs with irregular coefficients and stochastic transport equations (2009, background): “’ result to the stochastic case so that σcan be non-constant. The motivation of such extension partly comes from the study of some modeling equations of polymeric ﬂuids (see Le Bris and Lions’ papers [18, 19]). We also mention that Figalli [12] has already developed a stochastic counterpart for DiPe”