This paper is devoted to the study of a stationary problem consisting of the Boussinesq approximation of the Navier–Stokes equations and two convection–diffusion equations for the temperature and concentration, respectively. The equations are considered in 3D and a velocity–pressure formulation of the Navier–Stokes equations is used. The problem is complicated by nonstandard boundary conditions for velocity on the liquid–gas interface where tangential surface forces proportional to surface gradients of temperature and concentration (Marangoni effect) and zero normal component of the velocity are assumed. The velocity field is coupled through this boundary condition and through the buoyancy term in the Navier–Stokes equations with both the temperature and concentration fields. In this paper a weak formulation of the problem is stated and the existence of a weak solution is proved. For small data, the uniqueness of the solution is established.

Petr Knobloch

Lutz Tobiska

Analysis of a mathematical model related to Czochralski crystal growth

E. N. Dancer

K. Y. Lam

S. Yan

The effect of the graph topology on the existence of multipeak
solutions for nonlinear Schrödinger equations

Given an <mml:math alttext="$x_0\in R^n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>&#8712;</mml:mo><mml:msup><mml:mi>R</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math> we study the infinite horizon problem of minimizing the expression <mml:math alttext="$\int_0^Tf(t,x(t),x'(t))dt$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mo>&#8747;</mml:mo><mml:mn>0</mml:mn><mml:mi>T</mml:mi></mml:msubsup><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mo>&#8242;</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:math> as <mml:math alttext="$T$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math> grows to infinity where <mml:math alttext="$x:[0,\infty)\to R^n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>x</mml:mi><mml:mo>:</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8594;</mml:mo><mml:msup><mml:mi>R</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math> satisfies the initial condition <mml:math alttext="$x(0) = x_0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math>. We analyse the existence and the properties of approximate solutions for every prescribed initial value <mml:math alttext="$x_0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math>. We also establish that for every bounded set <mml:math alttext="$E\subset R^n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>E</mml:mi><mml:mo>&#8834;</mml:mo><mml:msup><mml:mi>R</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math> the <mml:math alttext="$C([0,T])$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> norms of approximate solutions <mml:math alttext="$x:[0,T]\rightarrow R^n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>x</mml:mi><mml:mo>:</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>&#8594;</mml:mo><mml:msup><mml:mi>R</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math> for the minimization problem on an interval <mml:math alttext="$[0,T]$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math> with <mml:math alttext="$x(0),x(T)\in E$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>T</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8712;</mml:mo><mml:mi>E</mml:mi></mml:mrow></mml:math> are bounded by some constant which does not depend on <mml:math alttext="$T$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math>.

Alexander J. Zaslavski

Existence and uniform boundedness of optimal solutions of
variational problems

In this part of our paper we present several new theorems concerning the existence of common fixed points of asymptotically regular uniformly lipschitzian semigroups.

Monika  Budzy&#324;ska

Tadeusz  Kuczumow

Simeon  Reich

Uniform asymptotic normal structure, the uniform semi-Opial
property, and fixed points of asymptotically regular uniformly lipschitzian semigroups. Part II

It is well known that the identity is an operator with the following property: if the operator, initially defined on an <mml:math alttext="$n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-dimensional Banach space <mml:math alttext="$V$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>V</mml:mi></mml:math>, can be extended to any Banach space with norm <mml:math alttext="$1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn></mml:math>, then <mml:math alttext="$V$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>V</mml:mi></mml:math> is isometric to <mml:math alttext="$\ell^{(n)}_\infty$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>&#8467;</mml:mi><mml:mi>&#8734;</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow></mml:math>. We show that the set of all such operators consists precisely of those with spectrum lying in the unit circle. This result answers a question raised in [5] for complex spaces.

B. L. Chalmers

B.  Shekhtman

Spectral properties of operators that characterize <svg style="vertical-align:-3.3907pt" height="20.950001" version="1.1" viewBox="0 0 21.9375 20.950001" width="21.9375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,16.675)"><path id="x2113" d="M363 114l21 -20q-62 -106 -161 -106q-61 0 -94 40t-33 138v56l-54 -40l-14 27l68 57v89q0 112 15.5 166t49.5 91q43 46 96 46q44 0 71 -32.5t27 -88.5q0 -41 -17.5 -81.5t-49.5 -76t-55.5 -57t-55.5 -46.5v-87q0 -138 81 -138q60 0 105 63zM177 421v-101q117 105 117 217&#xA;q0 38 -15 60.5t-41 22.5q-34 0 -47.5 -46.5t-13.5 -152.5z" /></g><g transform="matrix(.012,-0,0,-.012,7.05,20.75)"><path id="x221E" d="M983 225q0 -112 -67 -174.5t-150 -62.5q-91 0 -154.5 43.5t-113.5 129.5q-49 -85 -104 -129t-138 -44q-98 0 -158.5 66t-60.5 154q0 59 21 106.5t54.5 75.5t70.5 43t73 15q90 0 152.5 -43.5t112.5 -128.5q48 84 104.5 128t140.5 44q93 0 155 -65t62 -158zM478 196&#xA;q-27 49 -47 80t-50 67t-64 54t-73 18q-48 0 -81.5 -47t-33.5 -128q0 -96 37.5 -157.5t99.5 -61.5q68 0 117.5 47t94.5 128zM889 204q0 91 -35.5 151t-99.5 60q-68 0 -119 -47t-95 -127q27 -49 47 -80.5t50 -67.5t65 -54t74 -18q113 0 113 183z" /></g><g transform="matrix(.012,-0,0,-.012,7.05,8.513)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.012,-0,0,-.012,11.167,8.513)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g><g transform="matrix(.012,-0,0,-.012,17.14,8.513)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g></svg> 

We obtain a parabolic analogue of the well-known maximum principle established by R. Finn for solutions of the minimal surface equation.

Vasyl V. Kurta

A parabolic analogue of Finn's maximum principle

We study nonlinear semigroups of holomorphic mappings on certain domains in complex Banach spaces. We examine, in particular, their differentiability and their representations by exponential and other product formulas. In addition, we also construct holomorphic retractions onto the stationary point sets of such semigroups.

Simeon Reich

David Shoikhet

Metric domains, holomorphic mappings and nonlinear semigroups

In this paper we will investigate the existence of multiple
solutions for the problem <mml:math display="block" alttext="$-\Delta_pu + g(x,u) =\lambda_1h(x)|u|^{p-2}u,\mathrm{in}\Omega,u\in H_0^{1,p}(\Omega)(P)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>P</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mo>&#8722;</mml:mo><mml:msub><mml:mi>&#916;</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>&#955;</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>u</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>in</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mi>&#937;</mml:mi><mml:mo>,</mml:mo><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mtext>&#8201;</mml:mtext><mml:mi>u</mml:mi><mml:mo>&#8712;</mml:mo><mml:msubsup><mml:mi>H</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#937;</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> where <mml:math alttext="$\Delta_{p}u =\mathrm{div}(|\nabla u|^{p-2}\nabla u)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>&#916;</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mtext>div</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mo>&#8711;</mml:mo><mml:mi>u</mml:mi></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>&#8711;</mml:mo><mml:mi>u</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> is the <mml:math alttext="$p$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-Laplacian operator, <mml:math alttext="$\Omega\subseteq\mathbb{R}^N$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>&#937;</mml:mi><mml:mo>&#10949;</mml:mo><mml:msup><mml:mi>&#8477;</mml:mi><mml:mi>N</mml:mi></mml:msup></mml:mrow></mml:math> is a bounded domain with smooth boundary, <mml:math alttext="$h$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>h</mml:mi></mml:math> and <mml:math alttext="$g$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>g</mml:mi></mml:math> are bounded functions, <mml:math alttext="$N\geq 1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi><mml:mo>&#8805;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> and <mml:math alttext="$1 &#60; p &#60; \infty$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#60;</mml:mo><mml:mi>p</mml:mi><mml:mo>&#60;</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow></mml:math>. Using the Mountain Pass Theorem and the Ekeland Variational Principle, we will show the existence of at least three solutions for (P).

C. O. Alves

P. C. Carri&#227;o

O. H. Miyagaki

Multiple solutions for a problem with resonance involving the
<svg style="vertical-align:-3.50804pt" height="13.3" version="1.1" viewBox="0 0 10.2 13.3" width="10.2" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,8.863)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g></svg>-Laplacian

The Czochralski method of the industrial production of a silicon single crystal consists of pulling up the single crystal from the silicon melt. The flow of the melt during the production is called the Czochralski flow. The mathematical description of the flow consists of a coupled system of six P.D.E. in cylindrical coordinates containing Navier-Stokes equations (with the stream function), heat convection-conduction equations, convection-diffusion equation for oxygen impurity and an equation describing magnetic field effect.This paper deals with the analysis of the system in the form used
for numerical simulation. The weak formulation is derived and the existence of the weak solution to the stationary and the evolution problem is proved.

Jan  Franc

Modelling of the Czochralski flow

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<p><em>Abstract and Applied Analysis</em> publishes research with an emphasis on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimisation theory, and control theory.</p>


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<p>Chief Editor, Dr Wong, is an associate professor at Nanyang Technological University, Singapore. Her research interests include differential equations, difference equations, integral equations, and numerical mathematics.</p>


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Abstract and Applied Analysis - Table of contents 1998

title

Abstract and Applied Analysis publishes research with an emphasis on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimisation theory, and control theory.

Abstract and Applied Analysis