Search results for " 35B25"

showing 5 items of 5 documents

A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary

2016

We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair $\boldsymbol\varepsilon = (\varepsilon_1, \varepsilon_2 )$ of positive parameters, we consider a perforated domain $\Omega_{\boldsymbol\varepsilon}$ obtained by making a small hole of size $\varepsilon_1 \varepsilon_2 $ in an open regular subset $\Omega$ of $\mathbb{R}^n$ at distance $\varepsilon_1$ from the boundary $\partial\Omega$. As $\varepsilon_1 \to 0$, the perforation shrinks to a point and, at the same time, approaches the boundary. When $\boldsymbol\varepsilon \to (0,0)$, the size of the hole shrinks at a faster rate than its approach to the boundary. We denote by $u_{\bolds…

Asymptotic analysisGeneral MathematicsBoundary (topology)Asymptotic expansion01 natural sciences35J25; 31B10; 45A05; 35B25; 35C20Mathematics - Analysis of PDEsSettore MAT/05 - Analisi MatematicaFOS: Mathematics[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]Mathematics (all)Mathematics - Numerical Analysis0101 mathematicsMathematicsDirichlet problemLaplace's equationDirichlet problemAnalytic continuationApplied Mathematics010102 general mathematicsMathematical analysisHigh Energy Physics::PhenomenologyReal analytic continuation in Banach spaceNumerical Analysis (math.NA)Physics::Classical Physics010101 applied mathematicsasymptotic analysisLaplace operatorPhysics::Space PhysicsAsymptotic expansion; Dirichlet problem; Laplace operator; Real analytic continuation in Banach space; Singularly perturbed perforated domain; Mathematics (all); Applied MathematicsAsymptotic expansionLaplace operator[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]Singularly perturbed perforated domainAnalytic functionAnalysis of PDEs (math.AP)Asymptotic expansion; Dirichlet problem; Laplace operator; Real analytic continuation in Banach space; Singularly perturbed perforated domain;
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A Derivation of the Vlasov-Stokes System for Aerosol Flows from the Kinetic Theory of Binary Gas Mixtures

2016

In this short paper, we formally derive the thin spray equation for a steady Stokes gas, i.e. the equation consists in a coupling between a kinetic (Vlasov type) equation for the dispersed phase and a (steady) Stokes equation for the gas. Our starting point is a system of Boltzmann equations for a binary gas mixture. The derivation follows the procedure already outlined in [Bernard-Desvillettes-Golse-Ricci, arXiv:1608.00422 [math.AP]] where the evolution of the gas is governed by the Navier-Stokes equation.

Binary numberKinetic energy01 natural sciencesBoltzmann equationPhysics::Fluid Dynamics35Q20 35B25 82C40 76T15 76D07symbols.namesakeMathematics - Analysis of PDEshydrodynamic limitPhase (matter)FOS: Mathematics[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP][PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph]sprays0101 mathematicsSettore MAT/07 - Fisica MatematicaVlasov-Stokes systemPhysicsNumerical Analysisgas mixture.010102 general mathematicsMSC Primary: 35Q20 35B25; Secondary: 82C40 76T15 76D07.Stokes flowBoltzmann equationAerosol010101 applied mathematicsClassical mechanicsModeling and SimulationBoltzmann constantKinetic theory of gasessymbolsVlasov-Stokes system Boltzmann equation Hydrodynamic limit Aerosols Sprays Gas mixtureaerosolsAnalysis of PDEs (math.AP)
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Local uniqueness of the solutions for a singularly perturbed nonlinear nonautonomous transmission problem

2020

Abstract We consider the Laplace equation in a domain of R n , n ≥ 3 , with a small inclusion of size ϵ . On the boundary of the inclusion we define a nonlinear nonautonomous transmission condition. For ϵ small enough one can prove that the problem has solutions. In this paper, we study the local uniqueness of such solutions.

Local uniqueness of the solutionsLaplace's equation020502 materialsApplied MathematicsNonlinear nonautonomous transmission problem010102 general mathematicsMathematical analysisA domainBoundary (topology)02 engineering and technology01 natural sciencesNonlinear systemMathematics - Analysis of PDEs35J25 31B10 35J65 35B25 35A020205 materials engineeringTransmission (telecommunications)Settore MAT/05 - Analisi MatematicaLocal uniqueness of the solutions; Nonlinear nonautonomous transmission problem; Singularly perturbed perforated domainFOS: MathematicsUniqueness0101 mathematicsSingularly perturbed perforated domainAnalysisMathematicsAnalysis of PDEs (math.AP)
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A DERIVATION OF THE VLASOV-NAVIER-STOKES MODEL FOR AEROSOL FLOWS FROM KINETIC THEORY

2016

This article proposes a derivation of the Vlasov-Navier-Stokes system for spray/aerosol flows. The distribution function of the dispersed phase is governed by a Vlasov-equation, while the velocity field of the propellant satisfies the Navier-Stokes equations for incompressible fluids. The dynamics of the dispersed phase and of the propellant are coupled through the drag force exerted by the propellant on the dispersed phase. We present a formal derivation of this model from a multiphase Boltzmann system for a binary gaseous mixture, involving the droplets/dust particles in the dispersed phase as one species, and the gas molecules as the other species. Under suitable assumptions on the colli…

MSC: 35Q20 35B25 (82C40 76T15 76D05)aerosolVlasov-Navier-Stokes systemGeneral Mathematics01 natural sciencesPhysics::Fluid DynamicsBoltzmann equationsymbols.namesakeMathematics - Analysis of PDEsThermal velocityPhase (matter)35Q20 35B25 (82C40 76T15 76D05)SpraysFOS: Mathematics[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]0101 mathematicsSettore MAT/07 - Fisica MatematicaPhysicsPropellantAerosolsGas mixtureApplied Mathematics010102 general mathematicsMechanicsMass ratioBoltzmann equationAerosol010101 applied mathematicsDistribution functionsprayBoltzmann constantsymbolsHydrodynamic limitAnalysis of PDEs (math.AP)
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Global representation and multiscale expansion for the Dirichlet problem in a domain with a small hole close to the boundary

2019

For each pair (Formula presented.) of positive parameters, we define a perforated domain (Formula presented.) by making a small hole of size (Formula presented.) in an open regular subset (Formula presented.) of (Formula presented.) ((Formula presented.)). The hole is situated at distance (Formula presented.) from the outer boundary (Formula presented.) of the domain. Thus, when (Formula presented.) both the size of the hole and its distance from (Formula presented.) tend to zero, but the size shrinks faster than the distance. Next, we consider a Dirichlet problem for the Laplace equation in the perforated domain (Formula presented.) and we denote its solution by (Formula presented.) Our ai…

multiscale asymptotic expansionmulti-scale asymptotic expansionBoundary (topology)01 natural sciences35J25; 31B10; 45A05; 35B25; 35C20Domain (mathematical analysis)Settore MAT/05 - Analisi MatematicaSituated[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]Dirichlet problem; Laplace operator; multiscale asymptotic expansion; real analytic continuation in Banach space; singularly perturbed perforated domainSmall hole[MATH]Mathematics [math]0101 mathematicsRepresentation (mathematics)MathematicsDirichlet problemDirichlet problemApplied Mathematics010102 general mathematicsMathematical analysisA domain010101 applied mathematicssingularly perturbed perforated domainLaplace operatorLaplace operatorAnalysisreal analytic continuation in Banach space
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