sala posiedzeń Rady Wydziału, godz. 10:30–12:00
Franka Baaske (Friedrich-Schiller-University of Jena, Germany)
Generalized heat equations in supercritical function spaces
Summary. We deal with a~generalized heat equation
\begin{align}
\frac{\partial}{\partial t} u(x,t) + (-\Delta_x)^{\alpha} u(x,t)&= f(x,t), \quad \text{in }\mathbb{R}^n\times (0, T),\label{gheat} \\
u(\cdot,0)&=u_0(x), \quad \text{in }\mathbb{R}^n,\label{gheatawp}
\end{align}
where \( 0< T \leq \infty\), \(2\leq n\in\mathbb{N}\), \(\alpha\in\mathbb{N}\) and \(u(x,t)\) in the above equation is a~scalar function. The case \(\alpha=1\) corresponds to the classical heat equation. In order to apply the results to the generalized Navier-Stokes equations we choose \(f=Du^2=\sum\limits_{j=1}^{n}\frac{\partial}{\partial x_i} u^2\).
We assume for the initial data \(u_0\in A_{p,q}^{\sigma}\) with \(A\in\{B,F\}\), that \(\sigma>\frac{n}{p}\), \(1\leq p,q\leq\infty\), i.e. that spaces are multiplication algebras. Later we lower this assumption to \(\frac{n}{p}-2\alpha+1<\sigma<\frac{n}{p}\). Then these spaces cover all supercritical cases for the initial data. We show the existence and uniqness of solutions \(u\) belonging to some spaces \( L_{2\alpha v}((0,T),\frac{a}{2\alpha} A^{\sigma}_{p,q} (\mathbb{R}^n))\), where \begin{align*} \|u| L_{2\alpha v}((0,T),\frac{a}{2\alpha} A^{\sigma}_{p,q} (\mathbb{R}^n))\| = \left(\int\limits_0^T t^{av} \|u(\cdot,t)| A^{\sigma}_{p,q}(\mathbb{R}^n)\|^{2\alpha v} \text{d}t\right)^{1/2\alpha v}<\infty. \end{align*}
Therese Mieth (Friedrich Schiller University Jena, Germany)
Approximation numbers via bracketing
Summary. Let \(B\) be the unit ball in \(\mathbb{R}^n, m\in\mathbb{N}\) and \(1\leq p<\infty\). We define the weighted Sobolev space \(E^m_{p,\sigma}(B)\) as the completion of \(C^m_0(B)=\{f\in C^m(B): \operatorname{supp} f \text{ compact}\}\) with respect to the norm
\begin{equation*}
\Vert f\ \vert E^m_{p,\sigma}(B) \Vert := \bigg(\int_B|x|^{mp}\ (1+|\log|x||)^{\sigma p}\sum_{|\alpha|=m}|D^\alpha f (x)|^pd x\bigg)^{1/p}.
\end{equation*}
Then, if \( \sigma>0,\) the embedding
\begin{equation*}
\operatorname{id}: E^m_{p,\sigma}(B)\rightarrow L_p(B)
\end{equation*}
is compact. In case of Hilbert spaces, \(p=2\), Triebel obtained in~\cite{Tri12} sharp results for the corresponding approximation numbers
\begin{equation*}\label{eq1}
a_k(\operatorname{id})\sim
\begin{cases}
\ k^{-\frac{m}{n}}\hspace{1cm}& \text{, if } \sigma>\frac{m}{n}\\
\ k^{-\frac{m}{n}}(\log k)^{\frac{m}{n}}\hspace{1cm}& \text{, if } \sigma=\frac{m}{n}\\
\ k^{-\sigma}\hspace{1cm}& \text{, if } 0<\sigma<\frac{m}{n}.
\end{cases}
\end{equation*}
Therefore the Courant-Weyl method of Dirichlet-Neumann-bracketing was used. This technique is not available for \(p\neq2\), but a~partial analogue was established by Evans and Harris for Sobolev spaces \(W^1_p(\Omega)\) on a~wide class of domains. We want to transfer this idea and extend the results to the general case of Banach spaces \(1\leq p<\infty\).