In this article, we present several new results for Palais--Smale sequences. Consequently, we unify the Vitali convergence theorem and many main concepts in the variational methods by Lions, Lien--Tzeng--Wang, del Pino--Felmer and Chabrowski.

We consider Euler--Stokes splitting approximation of the Navier--Stokes equations with no--slip boundary condition. This consists in alternate solving of the Euler equations with tangential boundary condition and Stokes equations with no-slip boundary condition on small time intervals of the same length $k$. In a previous paper, J.T. Beale and C. Greengard proved the convergence of this approximation scheme in $L^p$ norm as $k$ tends to zero, for smooth solutions of the Navier--Stokes equations. Here we show how a certain simplification in their arguments improves their main result in the following way: the convergence holds without any additional regularity assumption on the solution of the Navier--Stokes equations.

We consider higher-order variational integrals of the type $$ \mathcal{F}(u,\Omega )=\int_{\Omega }f(x,u,D^{[k]}u,D^{k+1}u)\ dx$$ and study relaxation and lower semicontinuity properties of such functionals. In particular, under a bound of the type $$ 0\leq f(x,u,z_{1},z_{2},\ldots,z_{k+1}) \leq L(1+| z_{k+1}|^{q}) $$ the following relaxed energies are studied: \begin{eqnarray*} \mathcal{F}^{q,p}(u,\Omega) & = & \inf_{\{u_n\}} \Big \{\liminf_n \int_{\Omega}f(x,u_n,D^{[k]}u_n ,D^{k+1}u_n)\ dx \ : \\ & & u_n\in W^{k+1 ,q}(\Omega ;\Bbb R^d),\ u_n \rightharpoonup u \mbox{ in }W^{k+1 ,p}(\Omega ;\Bbb R^d) \Big \} \\ \mathcal{F}^{q,p}_{{loc}}(u,\Omega) & = & \inf_{\{u_n\}} \Big \{\liminf_n \int_{\Omega}f(x,u_n,D^{[k]}u_n ,D^{k+1}u_n)\ dx \ : \\ & & u_n\in W^{k+1 ,q}_{{loc}}(\Omega ;\Bbb R^d),\ u_n \rightharpoonup u \mbox{ in }W^{k+1 ,p}(\Omega ;\Bbb R^d) \Big \} \end{eqnarray*} with $\frac{q}{p} < \frac{Nk}{Nk-1}.$

We prove some improved Strichartz estimates for null forms, some of which were conjectured recently in [7]. The results follow from combining the usual Strichartz estimates with div-curl lemma techniques rather than space-time Fourier transform.

Let $L$ be a second-order elliptic partial differential operator of non-divergence form acting on ${\bf R^n}$ with bounded coefficients. We show that for each $1 < p_0 <2, L$ has a bounded $H_{\infty}$-functional calculus on $L^p({\bf R^n})$ for $p_0 <p <\infty$ if the $BMO$ norm of the coefficients is sufficiently small.

One considers the damped semilinear viscoelastic wave equation $$u_{tt}-\Delta u+\alpha u+f(u)+\int_0^tg(t-\tau )\Delta u(\tau )\, d\tau +h(u_t)=0\,\,\,\hbox{in}\,\,\,\Omega\times (0,\infty ),$$ where $\Omega$ is any bounded or finite measure domain of ${\bf R}^ n$, $\alpha\geq 0$ and $f,h$ are power like functions. The existence of global regular and weak solutions is proved by means of the Faedo-Galerkin method and uniform decay rates of the energy are obtained following the perturbed energy method by assuming that $g$ decays exponentially.

We consider the nonlinear Schrödinger equation with critical power $iu_t=-\Delta u-|u|^{4/N}u$, $t\geq 0$ and $x\in\mathbb T^N$ (the space-periodic case) in $H^1$. We consider a blow-up solution with minimal mass. We obtain in this context an optimal lower bound for the blow-up rate (that is, for $ | {\nabla u(t)}| _{L^2}$), and we observe that this lower bound equals the blow-up rate (which is explicitly known) of the minimal blow-up solutions in $\mathbb R^N$.