A local unique solvability is established for viscous conservation laws when the initial data may grow to infinity with a natural order. It is also shown that such a classical solution can be extended to a global-in-time solution provided that the growth order of the initial data is less than the critical order.

We study the problem of the order of approximation in modular spaces for a family of nonlinear integral operators of the form $$ (T_wf)(s) = \displaystyle\int_H K_w (s- h_w(t), f(h_w(t))) d \mu_H (t),~~w>0,~ s \in G, $$ where $G$~and $H$~ are locally compact topological groups, $f:G{\mbox{$\rightarrow$}}{\mathbb R}$ is measurable, $\{ h_w \}_{w>0}$~is a family of homeomorphisms $h_w:H \rightarrow h_w(H) \subset G$ and $\{K_w\}_{w>0}$ is a family of kernel functions. The general setting of modular spaces allows us to obtain, in particular, the rate of approximation for the above operators in $L^p$ spaces and in Orlicz-type spaces. Furthermore, the above general class contains, as particular cases, some classical families of integral operators well known in approximation theory, such as the classical convolution integral operators, the Mellin convolution integral operators and the sampling-type operators in their nonlinear form. Our approach, in the framework of modular spaces, is mainly based on the introduction of a suitable Lipschitz class and of a condition on a family of measures which is linked with the modulars involved and which is always fulfilled in classical and Musielak-Orlicz spaces.

This paper is first concerned with the trace problem for the transport equation. We prove the existence and the uniqueness of the traces as well as the well posedness of the initial- and boundary-value problem for the transport equation for any $L^p$ data ($p\in ]1,+\infty]$). In a second part, we use our study of the trace problem to prove that any solution to the transport equation is, roughly speaking, continuous with respect to the spatial variable along the direction of the transport vector field with values in a suitable $L^q$ space in the other variables. We want to emphasize the fact that we do not need to suppose any time regularity on the vector field defining the transport. This point is crucial in view of applications to fluid mechanics for instance. One of the main tools in our study is the theory of renormalized solutions of Di Perna and Lions.

We generalize recent results by L.G. Softova concerning $ W_p^{2,1} (\Omega_T) $ estimates ($ 1 < p < \infty$) for second order parabolic operators with VMO coefficients and the boundary condition $$ \sum_{i=1}^n b_i(\xi,t) \partial_i u(\xi,t) = g(\xi,t) \ \ \text{ on $ \partial \Omega_T $} $$ in the nondegenerate case (see Remark 2.2 i). While Softova assumed $ [(\xi,t) \mapsto b_i(\xi,t)] \in Lip(\partial \Omega_T) $, we weaken this assumption to $ b_i \in C^{ \alpha, \alpha/2 } (\partial \Omega_T) $ for some $ \alpha > 1 -1/p $.

We show that the Cauchy problem for the quintic NLS on $\mathbf{R}$ is globally well posed in $H^s$ for $4/9<s\leq 1/2$. Since we work below the energy space we cannot immediately use the energy. Instead we use the "I-method" introduced by J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. This method allows us to define a modification of the energy functional that is "almost conserved" and thus can be used to iterate the local result.