We consider a free boundary problem for the Navier-Stokes equation in $\mathbb R^n$ ($n \geqq 2$). We prove a local in time unique existence theorem for any initial data and a global in time unique existence theorem for some small initial data. The problem we consider in this paper was already treated by V.~Solonnikov [15]. But, recently in [10] we proved an $L_p$-$L_q$ maximal regularity theorem for the Stokes equation with Neumann boundary condition which is a linearized version of the free boundary problem for the Navier-Stokes equation treated in this paper. Our proof is based on this theorem. Therefore our solution is obtained in the space $W^{2,1}_{q,p}$ ($2 < p < \infty$ and $n < q < \infty$) while a solution in [15] is in $W^{2,1}_q = W^{2,1}_{q,q}$ ($n < q < \infty$). Moreover, our proof of the global in time existence theorem is much simpler than [15], because in [10] we established a maximal regularity theorem on the whole time interval $(0, \infty)$ with exponential stability. The results obtained in this paper were already announced in Shibata-Shimizu [11].

A collisionless plasma is described by the Vlasov-Poisson system. This system is considered in one space dimension in the case when there is one species of ions. Using the fact that the time evolution preserves a certain partial order of points in phase space, two theorems on large-time asymptotics are established.

The non-local parabolic equation \[ v_t=\Delta v+\frac{\lambda e^v}{\int_\Omega e^v}\quad\mbox{in $\Omega\times (0,T)$} \] associated with Dirichlet boundary and initial conditions is considered here. This equation is a simplified version of the full chemotaxis system. Let $\lambda^*$ be such that the corresponding steady-state problem has no solutions for $\lambda>\lambda^*$, then it is expected that blow-up should occur in this case. In fact, for $\lambda>\lambda^*$ and any bounded domain $\Omega\subset {\bf R}^2$ it is proven, using Trudinger-Moser's inequality, that $\int_{\Omega} e^{v(x,t)}dx\to \infty$ as $t\to T_{max}\leq \infty.$ Moreover, in this case, some properties of the blow-up set are provided. For the two-dimensional radially symmetric problem, i.e. when $\Omega=B(0,1),$ where it is known that $\lambda^*=8\,\pi,$ we prove that $v$ blows up in finite time $T^* < \infty$ for $\lambda>8\,\pi$ and this blow-up occurs only at the origin $r=0$ (single-point blow-up, mass concentration at the origin).

For a certain range of the eigenvalue parameter we prove a new multiple and sign-changing solutions theorem. The novelties of our paper are twofold. First, unlike recent papers in the field we do not assume jumping nonlinearities and allow a rather general growth condition on the nonlinearity involved. Second, our approach strongly relies on a combined use of variational and topological arguments (e.g. critical points, mountain--pass theorem, second deformation lemma, variational characterization of the first and second eigenvalue of the p-Laplacian) on the one hand, and comparison principles for nonlinear differential inequalities, in particular, the existence of extremal constant-sign solutions, on the other hand.

We prove global well-posedness for the Gross-Pitaevskii equation on the plane for classes of initial data having nonzero topological degree at infinity and therefore infinite Ginzburg-Landau energy. These classes allow us to consider arbitrary configurations of vortices as initial data. Our work follows recent results of Patrick Gérard [9] and Clément Gallo [4], where the finite energy regime is treated.

We study approximation problems by means of nonlinear convolution integral operators for functions belonging to $BV_{\varphi}$-spaces, i.e., functions with bounded $\varphi$-variation in the sense of Musielak-Orlicz. In particular, we obtain estimates and convergence results with respect to $\varphi$-variation. Introducing suitable Lipschitz classes that take into account the $\varphi$-variational functional, the problem of the rate of approximation is also considered.