Self-similar solutions of the p-Laplace heat equation: the case when p>2

We study the self-similar solutions of the equation $$u_{t}-div(| \nabla u| ^{p-2}\nabla u)=0,$$ in $\mathbb{R}^{N},$ when $p>2.$ We make a complete study of the existence and possible uniqueness of solutions of the form $$u(x,t)=(\pm t)^{-\alpha/\beta}w((\pm t)^{-1/\beta}| x|)$$ of any sign, regular or singular at $x=0.$ Among them we find solutions with an expanding compact support or a shrinking hole (for $t>0),$ or a spreading compact support or a focussing hole (for $t<0).$ When $t<0,$ we show the existence of positive solutions oscillating around the particular solution $U(x,t)=C_{N,p}(| x| ^{p}/(-t))^{1/(p-2)}.

A new dynamical approach of Emden-Fowler equations and systems

We give a new approach on general systems of the form (G){[c]{c}% -\Delta_{p}u=\operatorname{div}(|\nabla u| ^{p-2}\nabla u)=\epsilon_{1}|x| ^{a}u^{s}v^{\delta}, -\Delta_{q}v=\operatorname{div}(|\nabla v|^{q-2}\nabla u)=\epsilon_{2}|x|^{b}u^{\mu}v^{m}, where $Q,p,q,\delta,\mu,s,m,$ $a,b$ are real parameters, $Q,p,q\neq1,$ and $\epsilon_{1}=\pm1,$ $\epsilon_{2}=\pm1.$ In the radial case we reduce the problem to a quadratic system of order 4, of Kolmogorov type. Then we obtain new local and global existence or nonexistence results. In the case $\epsilon_{1}=\epsilon_{2}=1,$ we also describe the behaviour of the ground states in two cases where the system is variational. We give an important result on existence of ground states for a nonvariational system with $p=q=2$ and $s=m>0.$ In the nonradial case we solve a conjecture of nonexistence of ground states for the system with $p=q=2$ and $\delta=m+1$ and $\mu=s+1.$Comment: 43 page

Evolution equations of p-Laplace type with absorption or source terms and measure data

Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$, and $Q=\Omega \times(0,T).$ We consider problems\textit{ }of the type % \left\{ \begin{array} [c]{l}% {u_{t}}-{\Delta_{p}}u\pm\mathcal{G}(u)=\mu\qquad\text{in }Q,\\ {u}=0\qquad\text{on }\partial\Omega\times(0,T),\\ u(0)=u_{0}\qquad\text{in }\Omega, \end{array} \right. where ${\Delta_{p}}$ is the $p$-Laplacian, $\mu$ is a bounded Radon measure, $u_{0}\in L^{1}(\Omega),$ and $\pm\mathcal{G}(u)$ is an absorption or a source term$.$ In the model case $\mathcal{G}(u)=\pm\left\vert u\right\vert ^{q-1}u$ $(q>p-1),$ or $\mathcal{G}$ has an exponential type. We prove the existence of renormalized solutions for any measure $\mu$ in the subcritical case, and give sufficient conditions for existence in the general case, when $\mu$ is good in time and satisfies suitable capacitary conditions.Comment: arXiv admin note: substantial text overlap with arXiv:1310.525

Stability properties for quasilinear parabolic equations with measure data

Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$, and $Q=\Omega \times(0,T).$ We study problems of the model type \left\{ \begin{array} [c]{l}% {u_{t}}-{\Delta_{p}}u=\mu\qquad\text{in }Q,\\ {u}=0\qquad\text{on }\partial\Omega\times(0,T),\\ u(0)=u_{0}\qquad\text{in }\Omega, \end{array} \right. where $p>1$, $\mu\in\mathcal{M}_{b}(Q)$ and $u_{0}\in L^{1}(\Omega).$ Our main result is a \textit{stability theorem }extending the results of Dal Maso, Murat, Orsina, Prignet, for the elliptic case, valid for quasilinear operators $u\longmapsto\mathcal{A}(u)=$div$(A(x,t,\nabla u))$\textit{. }Comment: arXiv admin note: substantial text overlap with arXiv:1310.525

Pointwise estimates and existence of solutions of porous medium and pp-Laplace evolution equations with absorption and measure data

Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}(N\geq 2)$. We obtain a necessary and a sufficient condition, expressed in terms of capacities, for existence of a solution to the porous medium equation with absorption \begin{equation*} \left\{ \begin{array}{l} {u_{t}}-{\Delta }(|u|^{m-1}u)+|u|^{q-1}u=\mu ~ \text{in }\Omega \times (0,T), \\ {u}=0~~~\text{on }\partial \Omega \times (0,T), \\ u(0)=\sigma , \end{array} \right. \end{equation*} where $\sigma$ and $\mu$ are bounded Radon measures, $q>\max (m,1)$, $m>\frac{N-2}{N}$. We also obtain a sufficient condition for existence of a solution to the $p$-Laplace evolution equation \begin{equation*} \left\{ \begin{array}{l} {u_{t}}-{\Delta _{p}}u+|u|^{q-1}u=\mu ~~\text{in }\Omega \times (0,T), \\ {u}=0 ~ \text{on }\partial \Omega \times (0,T), \\ u(0)=\sigma . \end{array} \right. \end{equation*} where $q>p-1$ and $p>2$

On the connection between two quasilinear elliptic problems with source terms of order 0 or 1

We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of $\mathbb{R}^{N}.$ The first one, of the form $$-\Delta_{p}u=\beta(u)| \nabla u| ^{p}+\lambda f(x)+\alpha,$$ involves a source gradient term with natural growth, where $\beta$ is nonnegative, $\lambda>0,f(x)\geqq0$, and $\alpha$ is a nonnegative measure. The second one, of the form $$-\Delta_{p}v=\lambda f(x)(1+g(v))^{p-1}+\mu,$$ presents a source term of order $0,$where $g$ is nondecreasing, and $\mu$ is a nonnegative measure. Here $\beta$ and $g$ can present an asymptote. The correlation gives new results of existence, nonexistence, regularity and multiplicity of the solutions for the two problems, without or with measures. New informations on the extremal solutions are given when $g$ is superlinear

Initial trace of solutions of Hamilton-Jacobi parabolic equation with absorption

Here we study the initial trace problem 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 $q>0,$ and $\Omega=\mathbb{R}^{N},$ or $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N}$ and $u=0$ on $\partial\Omega\times\left( 0,T\right) .$ We can define the trace at $t=0$ as a nonnegative Borel measure $(\mathcal{S} ,u\_{0}),$ where $S$ is the closed set where it is infinite, and $u\_{0}$ is a Radon measure on $\Omega\backslash\mathcal{S}.$ We show that the trace is a Radon measure when $q\leqq1.$ For $q\in(1,(N+2)/(N+1)$ and any given Borel measure, we show the existence of a minimal solution, and a maximal one on conditions on $u\_{0}.$ When $\mathcal{S}$ $=\overline{\omega}\cap\Omega$ and $\omega$ is an open subset of $\Omega,$ the existence extends to any $q\leqq2$ when $u\_{0}\in L\_{loc}^{1}(\Omega)$ and any $q>1$ when $u\_{0}=0$. In particular there exists a self-similar nonradial solution with trace $(\mathbb{R}^{N+},0),$ with a growth rate of order $\left\vert x\right\vert ^{q^{\prime}}$ as $\left\vert x\right\vert \rightarrow\infty$ for fixed $t.$ Moreover we show that the solutions with trace $(\overline{\omega},0)$ in $Q\_{\mathbb{R}^{N},T}$ may present near $t=0$ a growth rate of order $t^{-1/(q-1)}$ in $\omega$ and of order $t^{-(2-q)/(q-1)}$ on $\partial \omega.

Separable solutions of some quasilinear equations with source reaction

We study the existence of singular separable solutions to a class of quasilinear equations with reaction term. In the 2-dim case, we use a dynamical system approach to construct our solutions.Comment: 34 page

THE INFLUENCE OF THERMAL LOADING ON THE LEAK TIGHTNESS BEHAVIOUR OF HORIZONTALLY SPLIT CENTRIFUGAL COMPRESSORS

LectureFor the purpose of maintenance requirements or for some applications in the chemical industry (for instance chlorine) the centrifugal compressor must be designed with a horizontally (axially) split casing. Beyond the decision about the type of manufacturing (cast or welded) or about the material selection one of the utmost issues regarding the design of the compressor lies in the leak tightness of the flanges. The design of the compressor casing needs detailed checking for tightness. In order to ensure a proper design with respect to integrity of stress and tightness under test and operating conditions several FE Analyses and resulting criteria have been developed by the OEMs. Furthermore in order to demonstrate the leak tightness the casing is subjected to a hydrostatic test prior to the assembly of the inner parts and rotor. According the API specification a hydrostatic pressure of at least 1.5 times maximum design pressure is applied. This standard procedure is usually considered to be sufficient for demonstrating the casing integrity, the tightness in operation and to check the accuracy of the FE analysis. However some applications require the use of several sections (consisting of some stages) inside one casing. According to the stage configuration hot and cold casing sections might be close to each other. In operation the compressor casing is subjected not only to pressure but also to thermal loading. These potential high temperature gradients can considerably influence the compressor behaviour regarding its tightness. The conventional hydrostatic test can even be less critical than at some particular operating conditions of the compressor on site. This paper describes some experiences of the author’s company with this type of compressors and the performed calculations. Different configurations are analysed and compared between each others. The paper shows the steps for the optimization of casings in order to develop an appropriate split line flange design supported by FE calculations. Some examples of different casing concepts are shown and discussed. At last the paper highlights some decisive issues which influence the tightness of compressor like: - Design of inner casing - Arrangement of sections - Geometry of flange - Design and arrangement of bolts