This paper gives the local existence of mild solutions to the Cauchy problem for the complex Ginzburg-Landau type equation $${\displaystyle \frac{\partial u}{\partial t}}-(\lambda +\mathrm{i}\alpha )\Delta u+(\kappa +\mathrm{i}\beta ){\left|u\right|}^{q-1}u-\gamma u=0$$ in ${ℝ}^N\times (0,\infty )$ with $L^p$-initial data $u_0$ in the subcritical case ($1\le q<1+2p/N$), where $u$ is a complex-valued unknown function, $\alpha$, $\beta$, $\gamma$, $\kappa \in ℝ$, $\lambda >0$, $p>1$, $\mathrm{i}=\sqrt{-1}$ and $N\in \mathbb{N}$. The proof is based on the $L^p$-$L^q$ estimates of the linear semigroup $\{exp(t(\lambda +\mathrm{i}\alpha )\Delta \left)\right\}$ and usual fixed-point argument.

We study the Cauchy problem for the non-Newtonian incompressible fluid with the viscous part of the stress tensor ${\tau }^V\left(𝕖\right)=\tau \left(𝕖\right)-2{\mu }_1\Delta 𝕖$, where the nonlinear function $\tau \left(𝕖\right)$ satisfies ${\tau }_{ij}\left(𝕖\right)e_{ij}\ge c{\left|𝕖\right|}^p$ or ${\tau }_{ij}\left(𝕖\right)e_{ij}\ge {c\left(\right|𝕖|}^2+{\left|𝕖\right|}^p)$. First, the model for the bipolar fluid is studied and existence, uniqueness and regularity of the weak solution is proved for $p>1$ for both models. Then, under vanishing higher viscosity ${\mu }_1$, the Cauchy problem for the monopolar fluid is considered. For the first model the existence of the weak solution is proved for $p>\frac{3n}{n+2}$, its uniqueness and...

In this work we introduce a new class of lowest order methods for diffusive problems on general meshes with only one unknown per element. The underlying idea is to construct an incomplete piecewise affine polynomial space with optimal approximation properties starting from values at cell centers. To do so we borrow ideas from multi-point finite volume methods, although we use them in a rather different context. The incomplete polynomial space replaces classical complete polynomial spaces in discrete...

In this work we introduce a new class of lowest order methods for diffusive problems on general meshes with only one unknown per element. The underlying idea is to construct an incomplete piecewise affine polynomial space with optimal approximation properties starting from values at cell centers. To do so we borrow ideas from multi-point finite volume methods, although we use them in a rather different context. The incomplete polynomial space replaces classical complete polynomial spaces...

Solutions to nonlinear Schrödinger equations may blow up in finite time. We study the influence of the introduction of a potential on this phenomenon. For a linear potential (Stark effect), the blow-up time remains unchanged, but the location of the collapse is altered. The main part of our study concerns isotropic quadratic potentials. We show that the usual (confining) harmonic potential may anticipate the blow-up time, and always does when the power of the nonlinearity is $L^2$-critical. On the other...

In this work we consider a solid body $\Omega \subset {ℝ}^3$ constituted by a nonhomogeneous elastoplastic material, submitted to a density of body forces $\lambda f$ and a density of forces $\lambda g$ acting on the boundary where the real $\lambda$ is the loading parameter. The problem is to determine, in the case of an unbounded convex of elasticity, the Limit load denoted by $\overline{\lambda }$ beyond which there is a break of the structure. The case of a bounded convex of elasticity is done in [El-Fekih and Hadhri, RAIRO: Modél. Math. Anal. Numér. 29 (1995)...

In this work we consider a solid body $\Omega \subset {ℝ}^3$ constituted by a nonhomogeneous elastoplastic material, submitted to a density of body forces $\lambda f$ and a density of forces $\lambda g$ acting on the boundary where the real $\lambda$ is the loading parameter. The problem is to determine, in the case of an unbounded convex of elasticity, the Limit load denoted by $\overline{\lambda }$ beyond which there is a break of the structure. The case of a bounded convex of elasticity is done in [El-Fekih and Hadhri, RAIRO: Modél. Math. Anal. Numér. 29 (1995)...

Nous montrons dans cette étude l’existence de configurations stationnaires où une bille tombe le long d’un plan incliné sans le toucher. Nous donnons également des propriétés qualitatives de ces configurations. En particulier, nous nous intéressons à l’orientation du plan par rapport à la verticale quand la masse de la bille est proche de celle d’un volume équivalent de liquide i.e., quand l’écoulement autour de la bille est lent.