6533b839fe1ef96bd12a5b10
RESEARCH PRODUCT
Singularity formation for Prandtl’s equations
Marco SammartinoFrancesco GarganoVincenzo Sciaccasubject
Complex singularitiePrandtl numberFOS: Physical sciencesRegularizing viscositySeparationPhysics::Fluid Dynamicssymbols.namesakeViscosityMathematics - Analysis of PDEsSingularityFOS: MathematicsUniform boundednessSpectral methodSettore MAT/07 - Fisica MatematicaMathematical PhysicsMathematicsMathematical analysisStatistical and Nonlinear PhysicsMathematical Physics (math-ph)Condensed Matter PhysicsPrandtl–Glauert transformationPrandtl’s equationsymbolsGravitational singularitySpectral methodComplex planeAnalysis of PDEs (math.AP)Blow–up timedescription
Abstract We consider Prandtl’s equations for an impulsively started disk and follow the process of the formation of the singularity in the complex plane using the singularity tracking method. We classify Van Dommelen and Shen’s singularity as a cubic root singularity. We introduce a class of initial data, uniformly bounded in H 1 , which have a dipole singularity in the complex plane. These data lead to a solution blow-up whose time can be made arbitrarily short within the class. This is numerical evidence of the ill-posedness of the Prandtl equations in H 1 . The presence of a small viscosity in the streamwise direction changes the behavior of the singularities. They stabilize at a distance from the real axis which depends on the amount of viscosity. We show that the Van Dommelen and Shen singularity and the singularity predicted by E and Engquist in [W. E, B. Engquist, Blowup of the solutions to the unsteady Prandtl’s equations, Comm. Pure Appl. Math. 50 (1997) 1287–1293.] have different complex structures.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2009-10-01 | Physica D: Nonlinear Phenomena |