2. Wellposedness and sharp enstropy bound of quasilinear systems

Problem 2.1.
[Bartosz Protas] Prove sharp estimates of the $H ^1$ norm of solutions to the 1D periodic viscous Burger’s equation. It is proven the enstrophy $\mathcal E (t) := \left\{\partial _x u (t)}\right\ _{L ^2} ^2$ obeys \begin{align*} \mathcal E’ (t) \le C \mathcal E (t) ^\frac53. \end{align*} Prove the following bound which comes from numerical evidences: \begin{align*} \mathcal E (t) \le C \mathcal E (0) ^\frac32. \end{align*} 
Problem 2.2.
[Adam Larios] Prove the global existence of the 2D Burger’s equation with hyperviscosity. \begin{align*} \partial _t u + u \cdot \nabla u = \nu \Delta ^2 u \end{align*} 
Problem 2.3.
[Wojciech Ozanski] Prove the global existence of strong solutions to the 1D surface growth model. \begin{align*} \partial _t u + u _{xxxx} + \partial _{xx} (u _x ^2) = 0 \end{align*} Show the illposedness of the inviscid version of this model.
Cite this as: AimPL: Criticality and stochasticity in quasilinear fluid systems, available at http://aimpl.org/stochasticfluid.