We study the existence, uniqueness, and regularity of weak solutions for a viscous Hamilton-Jacobi equation of the form: $u_t-\Delta u=a|\nabla u|^p, $ $p\in(0,\infty)$ and $a\in{{\bf R}}$, $a\neq 0$, with Dirichlet boundary condition and irregular initial data $\mu_0$. The cases of initial data $\mu_0$ a bounded Radon measure, or a function in the Lebesgue space $L^q, 1\leq q < \infty$ are investigated.

To compute the flow around an obstacle, it is now quite classical to add in the equations a penalization term on this obstacle. From a computational point of view, this method gives very accurate results and avoids the use of an unstructured mesh to discretize the equations in complex geometry. The aim of this paper is to give a mathematical explanation of such good results. This study is performed for the incompressible Navier-Stokes equations.

In this paper, we study a fluid--rigid-body interaction problem. The motion of the fluid is modeled by the Navier-Stokes equations, written in an unknown bounded domain depending on the displacement of the rigid body. Our main result yields existence and uniqueness of strong solutions. In the two-dimensional case, the solutions are global provided that the rigid body does not touch the boundary. In the three-dimensional case, we obtain local-in-time existence and global existence for small data. Moreover, we prove an asymptotic stability result.