The Maxwell-Klein-Gordon system in temporal gauge is unconditionally globally well-posed in energy space, especially uniqueness holds in the natural solution space. This improves earlier results where uniqueness was only shown in a suitable subspace. It is also locally well-posed for large data below energy space.

Here, we show new apriori estimates for the nonnegative solutions of the equation \[ u_{t}-\Delta u+|\nabla u|^{q}=0, \] in $Q_{\Omega,T}=\Omega\times\left( 0,T\right) ,$ $T\leqq\infty,$ where $q > 1,$ and $\Omega=\mathbb{R}^{N},$ or $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N}$ and $u=0$ on $\partial\Omega\times\left( 0,T\right) .$ In case $\Omega=\mathbb{R}^{N},$ we show that any solution $u\in C^{2,1}(Q_{\mathbb{R}^{N},T})$ of equation $(1.1)$ in $Q_{\mathbb{R}^{N}% ,T}$ (in particular, any weak solution if $q\leqq2),$ without condition as $\left\vert x\right\vert \rightarrow\infty,$ satisfies the universal estimate \[ \left\vert \nabla u(x,t)\right\vert ^{q}\leqq\frac{1}{q-1}\frac{u(x,t)}% {t}\qquad\text{in }Q_{\mathbb{R}^{N},T}. \] Moreover, we prove that the growth of the solutions is limited in space and time: \[ u(x,t)\leqq C(t+t^{-1/(q-1})(1+\left\vert x\right\vert ^{q^{\prime}}% )\qquad\text{in }Q_{\mathbb{R}^{N},T}, \] where $C$ depends on the initial data. We also give existence properties of solutions in $Q_{\Omega,T},$ for initial data locally integrable or unbounded measures. We give a nonuniqueness result in case $q > 2.$ Finally, we show that besides the local regularizing effect of the heat equation, $u$ satisfies a second effect of type $L_{loc}^{R}% $-$L_{loc}^{\infty},$ due to the gradient term.

Given a class of nonautonomous elliptic operators $\mathcal A(t)$ with unbounded coefficients, defined in $\overline{I \times \Omega}$ (where $I$ is a right-halfline or $I=\mathbb R$ and $\Omega\subset \mathbb R^d$ is possibly unbounded), we prove existence and uniqueness of the evolution operator associated to $\mathcal A(t)$ in the space of bounded and continuous functions, under Dirichlet and first order, non tangential homogeneous boundary conditions. Some qualitative properties of the solutions, the compactness of the evolution operator and some uniform gradient estimates are then proved.

We study the nonhomogeneous Dirichlet problem for the stationary Stokes equations on exterior smooth domains $\Omega$ in $\mathbb R^n , n \ge 3$. Our main result is the existence and uniqueness of very weak solutions in the Lorentz space $L^{p,q}(\Omega )^n$, where $(p,q)$ satisfies either $(p,q)=(n/(n-2),\infty)$ or $n/(n-2) < p < \infty , 1 \le q \le \infty$. This is deduced by a duality argument from our new solvability results on strong solutions in homogeneous Sobolev-Lorentz spaces. Homogeneous Sobolev-Lorentz spaces are also studied in quite details: particularly, we establish basic interpolation and density results, which are not only essential to our results for the Stokes equations but also themselves of independent interest.

We consider the system of $N$ partial differential equations $$ \sum\limits_{i=1}^{n} D_i (a_i^\alpha (x,Du(x))) = 0, \quad x \in \Omega, \quad \alpha \in \{1,...,N\}, $$ and the boundary condition $$ u(x) = u_*(x), \quad x \in \partial\Omega. $$ We show that higher integrability of the boundary datum $u_*$ forces solutions $u$ to have higher integrability as well, provided we assume suitable ellipticity and growth conditions on $a_i^\alpha$.

We shall establish the existence and non existence of solitons (travelling waves of finite energy) for a Benney-Luke equation of higher order, which includes models for long water waves with small amplitude. Following a variational approach, solitons are characterized as critical points of the action functional. Existence of solitons follows by the Concentration-Compactness principle by P.-L. Lions, applied to an appropriated minimization problem. It is also shown that solitons are smooth.