Search results for "Blow–up time"

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Singularity formation for Prandtl’s equations

2009

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 distanc…

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 timePhysica D: Nonlinear Phenomena
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Singularities for Prandtl's equations.

2006

We used a mixed spectral/finite-difference numerical method to investigate the possibility of a finite time blow-up of the solutions of Prandtl's equations for the case of the impulsively started cylinder. Our toll is the complex singularity tracking method. We show that a cubic root singularity seems to develop, in a time that can be made arbitrarily short, from a class of data uniformely bounded in H^1.

Prandtl’s equations Separation Spectral methods Complex singularities Blow–up time Regularizing viscosity.Settore MAT/07 - Fisica Matematica
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