Existence to solutions of a kinetic aerosol model

Description

Niveau: Supérieur, Doctorat, Bac+8
Existence to solutions of a kinetic aerosol model ? Pierre-Emmanuel Jabin†, Christian Klingenberg‡ Abstract Consider small particles of varying size being transported by a fluid. If we allow these particels to coalesce their evolution may be described by the coagulation model ft + p m .?xf = Q(f) . Here f denotes the particle density f(t, x,m, p) of particles with mass m ? R+, momentum p ? R3, at time t > 0 and position x ? R3. For a general class of collion operators Q we prove existence of solutions. Under some natural restriction on the initial data we have existence without blowup of the solution. Contents 1 Introduction 2 2 The main theorem and the L∞ bound 4 2.1 Beginning of the proof of Lemma 2.2 . . . . . . . . . . . . . . 6 2.1.1 The contribution from F 1 . . . . . . . . . . . . . . . . 7 2.1.2 The contribution from F 2 . . . . . . . . . . . . . . . . 7 2.2 Conclusion of the proof of Lemma 2.2 . . . . . . . . . . . . . . 9 ?Mathematics Subject Classifications: 35L60, 82C22, 82C40. Key words: Coagulation process, collisional kinetic theory, space dependence.

• weak solution


• operator than


• existence results


• without leading


• particle density


• gets very


• restrictive collision


• particles

Sujets

• University of Würzburg
• Weak solution
• Particle density
• Particle

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 11
• Langue: English

Extrait

Existencetosolutionsofakineticaerosol model ∗ Pierre-EmmanuelJabin, † ChristianKlingenberg ‡ Abstract Considersmallparticlesofvaryingsizebeingtransportedbya fluid.Ifweallowtheseparticelstocoalescetheirevolutionmaybe describedbythecoagulationmodel p f t +. r x f=Q(f). m Herefdenotestheparticledensityf(t,x,m,p)ofparticleswithmass m ∈ R + ,momentump ∈ R 3 ,attimet>0andpositionx ∈ R 3 .For ageneralclassofcollionoperatorsQweproveexistenceofsolutions. Undersomenaturalrestrictionontheinitialdatawehaveexistence withoutblowupofthesolution. Contents 1Introduction2 2ThemaintheoremandtheL ∞ bound4 2.1BeginningoftheproofofLemma2.2..............6 2.1.1ThecontributionfromF 1 ................7 2.1.2ThecontributionfromF 2 ................7 2.2ConclusionoftheproofofLemma2.2..............9 ∗ MathematicsSubjectClassifications:35L60,82C22,82C40. Keywords:Coagulationprocess,collisionalkinetictheory,spacedependence. † ENS,DMA,45Rued’Ulm,75230Paris,Cedex05,France ‡ AppliedMathematics,Wu¨rzburgUniversity,AmHubland,97074Wu¨rzburg,Germany 1