The global existence theorem of classical solutions for one-dimensional nonlinear thermoelasticity is proved for small and smooth initial data in the case of a bounded reference configuration for a homogeneous medium, considering the Neumann type boundary conditions: traction free and insulated. Moreover, the asymptotic behaviour of solutions is investigated.

We consider nondiagonal elliptic and parabolic systems of equations with quadratic nonlinearities in the gradient. We discuss a new description of regular points of solutions of such systems. For a class of strongly nonlinear parabolic systems, we estimate locally the Hölder norm of a solution. Instead of smallness of the oscillation, we assume local smallness of the Campanato seminorm of the solution under consideration. Theorems about quasireverse Hölder inequalities proved by the author are essentially...

In questo articolo si ottengono stime di Schauder di tipo nuovo per equazioni ellittiche infinito-dimensionali del secondo ordine con coefficienti Hölderiani a valori nello spazio degli operatori Hilbert-Schmidt. In particolare si mostra che la derivata seconda delle soluzioni è Hilbert-Schmidt.

It is proved that the solution to the initial value problem ${\partial }_{t}u=\partial {²}_{x}u+u²$, u(0,x) = 1/(1+x²), does not belong to the Gevrey class $G^s$ in time for 0 ≤ s < 1. The proof is based on an estimation of a double sum of products of binomial coefficients.

The purpose of this work is to study the class of non-negative continuous weak solutions of the non-linear evolution equation∂u/∂t = ∆φ(u),   x ∈ Rn, 0 &lt; t &lt; T ≤ +∞.

$\mathrm{\Omega }$ is a bounded open set of ${\mathbb{R}}^n$, of class $C^2$ and $T>0$. In the cylinder $Q=\mathrm{\Omega }\times \left(0,T\right)$ we consider non variational basic operator $a\left(H\left(u\right)\right)-\partial u/\partial t$ where $a\left(\xi \right)$ is a vector in ${\mathbb{R}}^N$, $N\ge 1$, which is continuous in $\xi$ and satisfies the condition (A). It is shown that $\mathrm{\forall }f\in L^2\left(Q\right)$ the Cauchy-Dirichlet problem $u\in W_0^{2,1}\left(Q\right)$, $a\left(H\left(u\right)\right)-\partial u/\partial t=f$ in $Q$, has a unique solution. It is further shown that if $u\in W_0^{2,1}\left(Q\right)$ is a solution of the basic system $a\left(H\left(u\right)\right)-\partial u/\partial t=0$ in $Q$, then $H\left(u\right)$ and $\partial u/\partial t$ belong to $H_{loc}^1\left(Q\right)$. From this the Hölder continuity in $Q$ of the vectors $u$ and $Du$ are deduced respectively when $n\le 4$ and $n=2$.

We prove almost optimal local well-posedness for the coupled Dirac–Klein–Gordon (DKG) system of equations in $1+3$ dimensions. The proof relies on the null structure of the system, combined with bilinear spacetime estimates of Klainerman–Machedon type. It has been known for some time that the Klein–Gordon part of the system has a null structure; here we uncover an additional null structure in the Dirac equation, which cannot be seen directly, but appears after a duality argument.

We consider an uncoupled, modular regularization algorithm for approximation of the Navier-Stokes equations. The method is: Step 1.1: Advance the NSE one time step, Step 1.1: Regularize to obtain the approximation at the new time level. Previous analysis of this approach has been for specific time stepping methods in Step 1.1 and simple stabilizations in Step 1.1. In this report we extend the mathematical support for uncoupled, modular stabilization to (i) the more complex and better performing...