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Toan Nguyen's blogTue, 05 Jul 2016 07:16:26 +0000hourly1http://wordpress.com/Comment on Math 505, Mathematical Fluid Mechanics: Notes 1 by Math 505, Mathematical Fluid Mechanics: Notes 2 | Snapshots in Mathematics !
https://nttoan81.wordpress.com/2016/01/11/math-505-mathematical-fluid-mechanics-notes-1/#comment-35
Tue, 05 Jul 2016 07:16:26 +0000http://nttoan81.wordpress.com/?p=165#comment-35[…] Math 505, Mathematical Fluid Mechanics: Notes 1 Instabilities in the mean field limit […]
]]>Comment on Madelung version of Schrödinger: a link between classical and quantum mechanics by cnpde
https://nttoan81.wordpress.com/2015/08/04/madelung-version-of-schrodinger-a-link-between-classical-and-quantum-mechanics/#comment-26
Tue, 11 Aug 2015 15:10:19 +0000http://nttoan81.wordpress.com/?p=150#comment-26http://arxiv.org/abs/1503.06894 http://arxiv.org/abs/1501.06803
I remember there is also another paper by Antonelli and Marcati dealing with the nonviscid case. They proved the global weak solution. However, I am not sure whether the regularity can be improved further by dispersive effect.
]]>Comment on Madelung version of Schrödinger: a link between classical and quantum mechanics by nttoan81
https://nttoan81.wordpress.com/2015/08/04/madelung-version-of-schrodinger-a-link-between-classical-and-quantum-mechanics/#comment-25
Wed, 05 Aug 2015 06:43:36 +0000http://nttoan81.wordpress.com/?p=150#comment-25do you have their reference? thanks!
]]>Comment on Madelung version of Schrödinger: a link between classical and quantum mechanics by cnpde
https://nttoan81.wordpress.com/2015/08/04/madelung-version-of-schrodinger-a-link-between-classical-and-quantum-mechanics/#comment-24
Tue, 04 Aug 2015 21:55:04 +0000http://nttoan81.wordpress.com/?p=150#comment-24Interesting! I remember recently, Vasseur and Yu also use similar quantum potential to prove the global weak solution of compressible NS equation with variable density coefficient.
]]>Comment on Math 597F, Notes 4: Prandtl boundary layer theory by Math 597F, Notes 5: A few examples of 2D boundary layers | Snapshots in Mathematics !
https://nttoan81.wordpress.com/2015/01/12/math-597f-notes-4-prandtl-boundary-layer-theory/#comment-23
Sat, 07 Mar 2015 15:27:44 +0000http://nttoan81.wordpress.com/?p=92#comment-23[…] us give a few examples of boundary layer solutions to the Prandtl problem, derived in the last lecture. In 2D, we recall the Prandtl layer […]
]]>Comment on Math 597F, Notes 2: Inviscid limit problem: absence of a boundary by Math 597F, Notes 3: Inviscid limit in the presence of a boundary | Snapshots in Mathematics !
https://nttoan81.wordpress.com/2015/01/07/math-597f-notes-2-inviscid-limit-problem-absence-of-a-boundary/#comment-22
Sat, 07 Mar 2015 15:26:47 +0000http://nttoan81.wordpress.com/?p=30#comment-22[…] be a smooth domain in that has a nonempty boundary. Consider again the usual NS equations (see equation (1) in the last lecture), accompanied with the classical zero boundary […]
]]>Comment on MATH 597F: Overview of a new graduate topics course: Boundary Layers in Fluid Dynamics by Math 597F, Notes 1: Euler and Navier-Stokes equations | Snapshots in Mathematics !
https://nttoan81.wordpress.com/2015/01/07/math-597f-overview-of-a-new-topics-course-boundary-layers-in-fluid-dynamics/#comment-21
Sat, 07 Mar 2015 15:17:31 +0000http://nttoan81.wordpress.com/?p=33#comment-21[…] is the first lecture of my Math 597F topics course. In this lecture, I will derive the partial differential equations, known as Euler and […]
]]>Comment on Math 597F, Notes 4: Prandtl boundary layer theory by Math 597F, Notes 5: A few examples of 2D boundary layers | Snapshots in Mathematics !
https://nttoan81.wordpress.com/2015/01/12/math-597f-notes-4-prandtl-boundary-layer-theory/#comment-20
Thu, 05 Feb 2015 00:28:48 +0000http://nttoan81.wordpress.com/?p=92#comment-20[…] « Math 597F, Notes 4: Prandtl boundary layer theory […]
]]>Comment on Math 597F, Notes 3: Inviscid limit in the presence of a boundary by nttoan81
https://nttoan81.wordpress.com/2015/01/08/notes-3-inviscid-limit-in-the-presence-of-a-boundary/#comment-17
Tue, 20 Jan 2015 16:40:28 +0000http://nttoan81.wordpress.com/?p=61#comment-17The point is that the smoothness is needed for the initial data (not just the smoothness of a boundary). In fact, even in the flat boundary, Sammartino and Caflisch proved less than what’s mentioned in the blog: they were able to prove the asymptotic expansion of Navier-Stokes = Euler, plus a Prandtl layer, plus small perturbation, where all data for Euler, Prandtl, and perturbation are in the analytic function space. In particular, the result implies the inviscid limit. This type of results is sometimes referred to as “inviscid limit with well-prepared initial data”). Since there is a loss of derivative in the estimates for Prandtl layers, their use of analytic function space was very crucial to control this loss of derivatives. I don’t think such a result is extended to a curved boundary, for various reasons, which I could only think of: (1) Prandtl equation now depends on the curvature of the boundary, essentially the equation is different near each region on the boundary, and (2) one might need to work with micro-local analysis, as now if you’d straight the boundary, the Laplace operator becomes a second order operator with variable coefficients…., and there might be others more serious that I didn’t know!
]]>Comment on Math 597F, Notes 3: Inviscid limit in the presence of a boundary by cnpde
https://nttoan81.wordpress.com/2015/01/08/notes-3-inviscid-limit-in-the-presence-of-a-boundary/#comment-16
Tue, 20 Jan 2015 02:11:52 +0000http://nttoan81.wordpress.com/?p=61#comment-16Consider an unbounded domain with a simple curve as boundary. if we assume the boundary is analytic, can we just adapt Sammartino and Caflisch’s proof to this curved boundary case?
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