In a joint work with Chenyin Qian (Zhejiang Normal University, Jinhua, China) and Ping Zhang (Academy of Mathematics & Systems Science, The Chinese Academy of Sciences, Beijing) we consider the 3D inhomogeneous Navier-Stokes equations \begin{align*} \partial_t\rho+\operatorname{div}(\rho u) & =0\\ \partial_t(\rho u)+\mbox{div}(\rho u\otimes u)-\mu\Delta u+\nabla\pi & = 0\\ \operatorname{div}u & = 0 \end{align*} in a bounded domain of \(\mathbb{R}^3\), complemented by a Dirichlet boundary condition for the velocity field \(u\) and initial values for \(u\) and the density \(\rho\). Under the assumption that the density term \(\rho-1\) is only bounded and small we prove the existence of a strong global-in-time solution \(u,\pi,\rho\). This solution is unique provided the initial velocity \(u_0\) has a slightly better regularity than needed for the existence result.

The analysis is performed in \(L^s(L^q)\)-spaces with weights in time, i.e., in other words, in Besov spaces of solenoidal vector fields on domains where techniques based on Littlewood-Paley decomposition are not available.

Let \(F\) be a smooth function. We consider the question "When \(f\) belongs to some function space, does \(F(f)\) belong to the same function space again?" The answers to this question for the Sobolev, Besov and Triebel-Lizorkin spaces are well known in virtue of the theory of paradifferential operators developed by Bony and Meyer. This talk is a trial to answer the same question for the modulation spaces. This is joint work with Tomoya Kato and Naohito Tomita (Osaka University).

In this talk, we consider the bidomain equations with FitzHugh-Nagumo nonlinearities arising in the study of electrophysiology. We show that the associated bidomain operator \(A\) admits a bounded \(H^\infty\)-calculus within the \(L^p\)-setting. This allows us to prove local as well as global well-posedness of this system in weak and strong settings for initial data in critical spaces. Moreover, we give stability results for spatially constant equilibria. This is joint work with Jan Prüss.

Consider the steady Navier-Stokes system in the frame attached to a rotating rigid body in 2D with constant angular velocity. It is known that the oscillation of the body leads to the resolution of the Stokes paradox. Given scale-critical Navier-Stokes flows, being assumed to be small, without specifying boundary conditions, we deduce their asymptotic representation at infinity in which the leading term is given by a self-similar Navier-Stokes flow. To be precise, it exhibits a circular profile and the coefficient is a sort of torque. This talk is based on a joint work with Mads Kyed.

We consider the Stokes IBVP: \[ \begin{array}{l} v_t+\nabla\pi=\Delta v\,,\; \nabla \cdot v=0\,,\mbox{ in }(0,T)\times\Omega\,,\\ v=0\mbox{ on }(0,T)\times\partial \Omega\,,\; v=v_0\mbox{ on }\{0\}\times\Omega\,. \end{array} \] We study estimates of the gradient of the solutions both in a bounded and in an exterior domain. More precisely, we look for estimates of the kind \(\|\nabla v(t)\|_q \leq g(t)\| \nabla v_0\|_p\,,\;q\geq p>1\,,\) for all \(t>0\) where function \(g\) is independent of \(v\).

Morrey space is the one of the powerful tool on analysis, especially, PDE. In 1980's-90's, it had been used by many researchers: One of the most important results are given by Kozono-Yamazaki (1995). However, interpolation theory of Morrey space is mysterious. In this talk, we shall treat its development and some results.

We discuss sparse bounds for operators involving time integral of the wave propagator. Our operator is regarded an intermediate one between maximal Riesz means and an operator appears in the local smoothing conjecture by Sogge. Main result is related to \(L^1\) bound for these operators. But the result is not sharp, the sharp weighted inequality can not be concluded.

In this talk, we consider the time-periodic problem for the Navier-Stokes equation in the whole space. We show that if the time-periodic external force is sufficiently small, then there exists a time-periodic solution \(\{ u,p \}\) of the Navier-Stokes equation such that \(| \nabla^j u(t,x)|=O(|x|^{1-n-j})\) and \(| \nabla^j p(t,x)|=O(|x|^{-n-j})\) (\(j=0,1,\ldots\)) uniformly in time as \(|x| \rightarrow \infty\). Our solution decays faster than the time-periodic Stokes fundamental solution. The proof is based on the representation formula of a solution via the Stokes fundamental solution and its properties.