Let $\Omega\subseteq\mathbb{R}^2$ be a domain and let $f\in W^{1,1}(\Omega,\mathbb{R}^2)$ be a homeomorphism (between $\Omega$ and $f(\Omega)$). Then there exists a sequence of smooth diffeomorphisms $f_k$ converging to $f$ in $W^{1,1}(\Omega,\mathbb{R}^2)$ and uniformly. This is a joint result with A. Pratelli.

# Partial Differential Equations Seminar

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This work is devoted to the study of the following boundary value problem for compressible Navier-Stokes equations\begin{align*}&\begin{aligned}[b] \partial_t(\varrho \mathbf{u})+\text{div}(\varrho \mathbf u\otimes\mathbf u) &+\nabla p(\rho)\\&= \text{div} \mathbb S(\mathbf u)+\varrho\, \mathbf f\quad\text{ in }\Omega\times (0,T),\end{aligned} \\[6pt]&\partial_t\varrho+\text{div}(\varrho \mathbf u)=0\quad\text{ in }\Omega\times (0,T), \\[6pt]&\begin{aligned}[c] &\mathbf u=0\quad\text{ on }\partial \Omega\times( 0,T), \\ &\mathbf u(x,0)=\mathbf u_0(x)\quad\text{ in } \Omega,\\&\varrho(x,0)= \varrho_0(x) \quad\text{ in } \Omega, \end{aligned}\end{align*} where $\Omega$ is a bounded domain in $\mathbb R^d$, $d=2,3$, $\varrho_0>0$, $\mathbf u_0$, $\mathbf f$ are given functions, $p(\varrho)=\varrho^\gamma$, $\mathbb S(\mathbf u)=\mu(\nabla\mathbf{u}+\nabla\mathbf{u}^\top)+\lambda \text{div } \mathbf{u}$, $\mu, \lambda$ are positive constants. We consider the endpoint cases $\gamma=3/2$, $d=3$ and $\gamma=1$, $d=2$, when the energy estimate does not guarantee the integrability of the kinetic energy density with an exponent greater than 1, which leads to the so-called concentration problem. In order to cope with this difficulty we develop new approach to the problem. Our method is based on the estimates of the Newton potential of $p(\varrho)$. We prove that the kinetic energy density in 3-dimensional problem with $\gamma=3/2$ is bounded in $L\log L^\alpha$ Orlitz space and obtain new estimates for the pressure function. In the case $d=2$ and $\gamma=1$ we prove the existence of the weak solution to the problem. We also discuss the structure of concentrations for rotationally-symmetric and stationary solutions.