The Experts below are selected from a list of 51045 Experts worldwide ranked by ideXlab platform
Sergey A Denisov - One of the best experts on this subject based on the ideXlab platform.
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on the growth of the support of positive vorticity for 2d Euler Equation in an infinite cylinder
Communications in Mathematical Physics, 2019Co-Authors: Kyudong Choi, Sergey A DenisovAbstract:We consider the incompressible 2D Euler Equation in an infinite cylinder $${\mathbb{R} \times \mathbb{T}}$$ in the case when the initial vorticity is non-negative, bounded, and compactly supported. We study d(t), the diameter of the support of vorticity, and prove that it allows the following bound: $${d(t) \leqslant Ct^{1/3}{\rm log}^{2}t}$$ when $${t \to \infty}$$ .
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2d Euler Equation on the strip stability of a rectangular patch
Communications in Partial Differential Equations, 2017Co-Authors: Jennifer Beichman, Sergey A DenisovAbstract:ABSTRACTWe consider the 2D Euler Equation of incompressible fluids on a strip ℝ×𝕋 and prove the stability of the rectangular stationary state χ|x|
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2d Euler Equation on the strip stability of a rectangular patch
arXiv: Analysis of PDEs, 2016Co-Authors: Jennifer Beichman, Sergey A DenisovAbstract:We consider the 2D Euler Equation of incompressible fluids on a strip and prove the stability of the rectangular stationary state.
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double exponential growth of the vorticity gradient for the two dimensional Euler Equation
Proceedings of the American Mathematical Society, 2014Co-Authors: Sergey A DenisovAbstract:For the two-dimensional Euler Equation on the torus, we prove that the L∞–norm of the vorticity gradient can grow as double exponential over arbitrary long but finite time provided that at time zero it is already sufficiently large. The method is based on the perturbative analysis around the singular stationary solution studied by Bahouri and Chemin in [1]. Our result on the growth of the vorticity gradient is equivalent to the statement that the operator of Euler evolution is not bounded in the linear sense in Lipschitz norm for any time t > 0.
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double exponential growth of the vorticity gradient for the two dimensional Euler Equation
arXiv: Analysis of PDEs, 2012Co-Authors: Sergey A DenisovAbstract:For the two-dimensional Euler Equation on the torus, we prove that the uniform norm of the vorticity gradient can grow as double exponential over arbitrarily long but finite time provided that at time zero it is already sufficiently large. Our result is equivalent to the statement that the Euler evolutions is linearly unbounded in Lipschitz norm for any time t>0.
Michael Taylor - One of the best experts on this subject based on the ideXlab platform.
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Euler Equation on a rotating surface
Journal of Functional Analysis, 2016Co-Authors: Michael TaylorAbstract:Abstract We study 2D Euler Equations on a rotating surface, subject to the effect of the Coriolis force, with an emphasis on surfaces of revolution. We bring in conservation laws that yield long time estimates on solutions to the Euler Equation, and examine ways in which the solutions behave like zonal fields, building on previous works that have examined how such 2D Euler Equations can account for the observed band structure of rapidly rotating planets. Specific results include both an analysis of time averages of solutions and a study of stability of stationary zonal fields. The latter study includes both analytical and numerical work.
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Euler Equation on a Rotating Surface
arXiv: Analysis of PDEs, 2015Co-Authors: Michael Taylor, Jeremy L. MarzuolaAbstract:We study 2D Euler Equations on a rotating surface, subject to the effect of the Coriolis force, with an emphasis on surfaces of revolution. We bring in conservation laws that yield long time estimates on solutions to the Euler Equation, and examine ways in which the solutions behave like zonal fields, building on work of B.~Cheng and A.~Mahalov, examining how such 2D Euler Equations can account for the observed band structure of rapidly rotating planets. Specific results include both an analysis of time averages of solutions and a study of stability of stationary zonal fields. The latter study includes both analytical and numerical work.
Dong Li - One of the best experts on this subject based on the ideXlab platform.
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Strong ill-posedness of the incompressible Euler Equation in borderline Sobolev spaces
Inventiones mathematicae, 2015Co-Authors: Jean Bourgain, Dong LiAbstract:For the $$d$$ d -dimensional incompressible Euler Equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space $$H^s(\mathbb R^d)$$ H s ( R d ) , $$s>s_c:=d/2+1$$ s > s c : = d / 2 + 1 . The borderline case $$s=s_c$$ s = s c was a folklore open problem. In this paper we consider the physical dimension $$d=2$$ d = 2 and show that if we perturb any given smooth initial data in $$H^{s_c}$$ H s c norm, then the corresponding solution can have infinite $$H^{s_c}$$ H s c norm instantaneously at $$t>0$$ t > 0 . In a companion paper [ 1 ] we settle the 3D and more general cases. The constructed solutions are unique and even $$C^{\infty }$$ C ∞ -smooth in some cases. To prove these results we introduce a new strategy: large Lagrangian deformation induces critical norm inflation . As an application we also settle several closely related open problems.
Jean Bourgain - One of the best experts on this subject based on the ideXlab platform.
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Strong ill-posedness of the incompressible Euler Equation in borderline Sobolev spaces
Inventiones mathematicae, 2015Co-Authors: Jean Bourgain, Dong LiAbstract:For the $$d$$ d -dimensional incompressible Euler Equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space $$H^s(\mathbb R^d)$$ H s ( R d ) , $$s>s_c:=d/2+1$$ s > s c : = d / 2 + 1 . The borderline case $$s=s_c$$ s = s c was a folklore open problem. In this paper we consider the physical dimension $$d=2$$ d = 2 and show that if we perturb any given smooth initial data in $$H^{s_c}$$ H s c norm, then the corresponding solution can have infinite $$H^{s_c}$$ H s c norm instantaneously at $$t>0$$ t > 0 . In a companion paper [ 1 ] we settle the 3D and more general cases. The constructed solutions are unique and even $$C^{\infty }$$ C ∞ -smooth in some cases. To prove these results we introduce a new strategy: large Lagrangian deformation induces critical norm inflation . As an application we also settle several closely related open problems.
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strong ill posedness of the incompressible Euler Equation in borderline sobolev spaces
arXiv: Analysis of PDEs, 2013Co-Authors: Jean BourgainAbstract:For the $d$-dimensional incompressible Euler Equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space $H^s(\mathbb R^d)$, $s>s_c:=d/2+1$. The borderline case $s=s_c$ was a folklore open problem. In this paper we consider the physical dimensions $d=2,3$ and show that if we perturb any given smooth initial data in $H^{s_c}$ norm, then the corresponding solution can have infinite $H^{s_c}$ norm instantaneously at $t>0$. The constructed solutions are unique and even $C^{\infty}$-smooth in some cases. To prove these results we introduce a new strategy: large Lagrangian deformation induces critical norm inflation. As an application we also settle several closely related open problems.
Pierre-henri Chavanis - One of the best experts on this subject based on the ideXlab platform.
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Euler-like modelling of dense granular flows: application to a rotating drum
European Physical Journal B: Condensed Matter and Complex Systems, 2009Co-Authors: Daniel Bonamy, Pierre-henri Chavanis, Pierre-philippe Cortet, François Daviaud, Bérengère Dubrulle, Mathieu RenoufAbstract:General conservation Equations are derived for 2D dense granular flows from the Euler Equation within the Boussinesq approximation. In steady flows, the 2D fields of granular temperature, vorticity and stream function are shown to be encoded in two scalar functions only. We checked such prediction on steady surface flows in a rotating drum simulated through the Non-Smooth Contact Dynamics method. This result is non trivial because granular flows are dissipative and therefore not necessarily compatible with Euler Equation. Finally, we briefly discuss some possible ways to predict theoretically these two functions using statistical mechanics.
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Quasilinear theory of the 2D Euler Equation
Physical review letters, 2000Co-Authors: Pierre-henri ChavanisAbstract:We develop a quasilinear theory of the 2D Euler Equation and derive an integrodifferential Equation for the evolution of the coarse-grained vorticity omega;(r,t). This Equation respects all of the invariance properties of the Euler Equation and conserves angular momentum in a circular domain and linear impulse in a channel. We show under which hypothesis we can derive an H theorem for the Fermi-Dirac entropy and make the connection with statistical theories of 2D turbulence.