Abstract:
Our talk will focus on transport equations with singular-integral forcing. By this we mean equations of the form:
$$\partial_t f+b\cdot\nabla_x f= R_x(f)$$
where $b$ is a given divergence-free vector field and $R_x$ is a singular integral operator such as the Hilbert transform. This type of transport equation shows up often in the study of fluid equations and can be seen as a prototype for many equations which have local and non-local forces at play. We are interested in solutions of such equations in $L^p$ spaces. Several recent results have been achieved for these types of equations depending upon whether the vector field $b$ is Lipschitz or not.
When $b$ is a Lipschitz function:
(1) The singular transport equation may be ill-posed on $L^\infty$ in the sense that bounded initial data may become unbounded immediately.
(2) The singular transport equation is well-posed in the class of functions of bounded mean oscillation (BMO)
(3) The singular transport equation exhibits "well-behaved" growth properties in $L^p.$
When $b$ is only taken to be bounded:
(1') The singular transport equation may be ill-posed in $L^\infty$ and BMO.
(2') The singular transport equation may have what we call "cascading solutions" starting from smooth initial data. These solutions belong to $L^p$ for all $p<\infty$ with $L^p$ norms growing on the order of $exp(p).$
We will discuss most of these results quickly and then focus on the constructions which lead to (2'). |