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Existence to solutions of a kinetic aerosol model

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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


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ExistencetosolutionsofakineticaerosolmodelPierre-EmmanuelJabin,ChristianKlingenbergAbstractConsidersmallparticlesofvaryingsizebeingtransportedbyafluid.Ifweallowtheseparticelstocoalescetheirevolutionmaybedescribedbythecoagulationmodelpft+.rxf=Q(f).mHerefdenotestheparticledensityf(t,x,m,p)ofparticleswithmassmR+,momentumpR3,attimet>0andpositionxR3.ForageneralclassofcollionoperatorsQweproveexistenceofsolutions.Undersomenaturalrestrictionontheinitialdatawehaveexistencewithoutblowupofthesolution.Contents1Introduction22ThemaintheoremandtheLbound42.1BeginningoftheproofofLemma2.2..............62.1.1ThecontributionfromF1................72.1.2ThecontributionfromF2................72.2ConclusionoftheproofofLemma2.2..............9MathematicsSubjectClassifications:35L60,82C22,82C40.Keywords:Coagulationprocess,collisionalkinetictheory,spacedependence.ENS,DMA,45Rued’Ulm,75230Paris,Cedex05,FranceAppliedMathematics,Wu¨rzburgUniversity,AmHubland,97074Wu¨rzburg,Germany1