This paper is concerned with optimal lower bounds of decay rates for solutions to the Navier-Stokes equations in ${ℝ}^n$. Necessary and sufficient conditions are given such that the corresponding Navier-Stokes solutions are shown to satisfy the algebraic bound $$\parallel u\left(t\right)\parallel \ge {(t+1)}^{-\frac{n+4}{2}}.$$

Using interpolation techniques we prove an optimal regularity theorem for the convolution $u\left(t\right)={\int }_0^{t}T\left(t-s\right)f\left(s\right)ds$, where $T\left(t\right)$ is a strongly continuous semigroup in general Banach space. In the case of abstract parabolic problems – that is, when $T\left(t\right)$ is an analytic semigroup – it lets us recover in a unified way previous regularity results. It may be applied also to some non analytic semigroups, such as the realization of the Ornstein-Uhlenbeck semigroup in $L^p\left({\mathbb{R}}^n\right)$, $1<p<\mathrm{\infty }$, in which case it yields new optimal regularity results in fractional...

Sharp $L^p$ estimates are proven for oscillatory integrals with phase functions Φ(x,y), (x,y) ∈ X × Y, under the assumption that the canonical relation $C_{\Phi }$ projects to T*X and T*Y with fold singularities.

The limit behavior of a periodic assembly of a finite number of elasto-plastic phases is investigated as the period becomes vanishingly small. A limit quasi-static evolution is derived through two-scale convergence techniques. It can be thermodynamically viewed as an elasto-plastic model, albeit with an infinite number of internal variables.

Sufficient conditions for the problem $$\frac{{\partial }^{2}u}{\partial x\partial y}=P_0(x,y)u+P_1(x,y)\frac{\partial u}{\partial x}+P_2(x,y)\frac{\partial u}{\partial y}+q(x,y),u(x+{\omega }_1,y)=u(x,y),u(x,y+{\omega }_2)=u(x,y)$$ to have the Fredholm property and to be uniquely solvable are established, where ${\omega }_1$ and ${\omega }_2$ are positive constants and $P_j:R^2\to R^{n\times n}$$(j=0,1...$

In this paper we consider a chain of strings with fixed end points coupled with nearest neighbour interaction potential of exponential type, i.e.$$\left\{\begin{array}{cc}{\varphi }_{tt}^i-{\varphi }_{xx}^i=exp({\varphi }^{i+1}-{\varphi }^i)-exp({\varphi }^i-\varphi i-1)\hfill & 0\&lt;x\&lt;\pi ,t\in ℝ,i\in \mathbb{Z}\left(TC\right)\hfill \\ {\varphi }^i(0,t)={\varphi }^i(\pi ,t)=0\hfill & \forall t,i.\hfill \end{array}\right.$$We consider the case of “closed chains” i.e. ${\varphi }^{i+N}={\varphi }^i\forall i\in \mathbb{Z}$ and some $N\in \mathbb{N}$ and look for solutions which are peirodic in time. The existence of periodic solutions for the dual problem is proved in Orlicz space setting.

In this paper we consider a chain of strings with fixed end points coupled with nearest neighbour interaction potential of exponential type, i.e.$$\left\{\begin{array}{c}{\varphi }_{tt}^i-{\varphi }_{xx}^i=exp({\varphi }^{i+1}-{\varphi }^i)-exp({\varphi }^i-\varphi i-1)0<x<\pi ,t\in \mathrm{I}\mathrm{R},\mathrm{i}\in \mathrm{Z}\mathrm{Z}\left(\mathrm{TC}\right){\varphi }^{\mathrm{i}}(0,\mathrm{t})={\varphi }^{\mathrm{i}}(\pi ,\mathrm{t})=0\forall \mathrm{t},\mathrm{i}.\hfill \end{array}\right.$$ We consider the case of “closed chains" i.e.${\varphi }^{i+N}={\varphi }^i\forall i\in ZZ$ and some $N\in IN$ and look for solutions which are peirodic in time. The existence of periodic solutions for the dual problem is proved in Orlicz space setting.

the existence of an $\omega$-periodic solution of the equation $\frac{{\partial }^{2}u}{\partial t^2}+\alpha \frac{{\partial }^{4}u}{\partial x^4}+\gamma \frac{{\partial }^{5}u}{\partial x^4\partial t}-\tilde{\gamma }\frac{{\partial }^{3}u}{\partial x^2\partial t}+\delta \frac{\partial u}{\partial t}-\left[\beta +\aleph {\int }_0^n{\left(\frac{\partial u}{\partial x}\right)}^2(·,\xi )d\xi +\sigma {\int }_0^n\frac{{\partial }^{2}u}{\partial x\partial t}(·,\xi )\frac{\partial u}{\partial x}(·,\xi )d\xi \right]\frac{{\partial }^{2}u}{\partial x^2}=f$ sarisfying the boundary conditions $u(t,0)=u(t,\pi )=\frac{{\partial }^{2}u}{\partial x^2}\left(t,0\right)=\frac{{\partial }^{2}u}{\partial x^2}\left(t,\pi \right)=0$ is proved for every $\omega$-periodic function $f\in C\left(\left[0,\omega \right],L_2\right)$.

The paper deals with the existence of time-periodic solutions to the beam equation, in which terms expressing torsion and damping are also considered. The existence of periodic solutions is proved in the cas of time-periodic outer forces by means of an apriori estimate and the Fourier method.