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Livio Pizzocchero - One of the best experts on this subject based on the ideXlab platform.
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on the reynolds number expansion for the navier stokes equations
Nonlinear Analysis-theory Methods & Applications, 2014Co-Authors: Carlo Morosi, Livio PizzoccheroAbstract:Abstract In a previous paper of ours (Morosi and Pizzocchero (2012) [1] ) we have considered the incompressible Navier–Stokes (NS) equations on a d -dimensional torus T d , in the functional setting of the Sobolev spaces H Σ 0 n ( T d ) of divergence free, zero mean vector fields ( n > d / 2 + 1 ). In the cited work we have presented a general setting for the a posteriori analysis of approximate solutions of the NS Cauchy problem; given any approximate solution u a , this allows to infer a lower bound T c on the time of existence of the exact solution u and to construct a function R n such that ‖ u ( t ) − u a ( t ) ‖ n ⩽ R n ( t ) for all t ∈ [ 0 , T c ) . In certain cases it is T c = + ∞ , so global existence is granted for u . In the present paper the framework of Morosi and Pizzocchero (2012) [1] is applied using as an approximate solution an expansion u N ( t ) = ∑ j = 0 N R j u j ( t ) , where R is the Reynolds number. This allows, amongst else, to derive the global existence of u when R is below some critical value R ∗ (increasing with N in the examples that we analyze). After a general discussion about the Reynolds expansion and its a posteriori analysis, we consider the expansions of orders N = 1 , 2 , 5 in dimension d = 3 , with the Initial Datum of Behr, Necas and Wu (2001) [11] . Computations of order N = 5 yield a quantitative improvement of the results previously obtained for this Initial Datum in Morosi and Pizzocchero (2012) [1] , where a Galerkin approximate solution was employed in place of the Reynolds expansion.
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on power series solutions for the euler equation and the behr necas wu Initial Datum
Mathematical Modelling and Numerical Analysis, 2013Co-Authors: Carlo Morosi, Mario Pernici, Livio PizzoccheroAbstract:We consider the Euler equation for an incompressible fluid on a three dimensional torus, and the construction of its solution as a power series in time. We point out some general facts on this subject, from convergence issues for the power series to the role of symmetries of the Initial Datum. We then turn the attention to a paper by Behr, Necas and Wu, ESAIM: M2AN 35 (2001) 229–238; here, the authors chose a very simple Fourier polynomial as an Initial Datum for the Euler equation and analyzed the power series in time for the solution, determining the first 35 terms by computer algebra. Their calculations suggested for the series a finite convergence radius τ 3 in the H 3 Sobolev space, with 0.32 3 35 (2001) 229–238, using again computer algebra; the order has been increased from 35 to 52, using the symmetries of the Initial Datum to speed up computations. As for τ 3 , our results agree with the original computations of E. Behr, J. Necas and H. Wu, ESAIM: M2AN 35 (2001) 229–238 (yielding in fact to conjecture that 0.32 3 3 is not at all an indication of a possible blow-up. (b) There is a strong indication that the solution of the Euler equation does not blow up at a time close to τ 3 . In fact, the solution is likely to exist, at least, up to a time θ 3 > 0.47. (c) There is a weak indication, based on Pade analysis, that the solution might blow up at a later time.
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on power series solutions for the euler equation and the behr necas wu Initial Datum
arXiv: Analysis of PDEs, 2012Co-Authors: Carlo Morosi, Mario Pernici, Livio PizzoccheroAbstract:We consider the Euler equation for an incompressible fluid on a three dimensional torus, and the construction of its solution as a power series in time. We point out some general facts on this subject, from convergence issues for the power series to the role of symmetries of the Initial Datum. We then turn the attention to a paper by Behr, Necas and Wu in ESAIM: M2AN 35 (2001) 229-238; here, the authors chose a very simple Fourier polynomial as an Initial Datum for the Euler equation and analyzed the power series in time for the solution, determining the first 35 terms by computer algebra. Their calculations suggested for the series a finite convergence radius \tau_3 in the H^3 Sobolev space, with 0.32 0.47. (c) Pade' analysis gives a rather weak indication that the solution might blow up at a later time.
Cesar J Niche - One of the best experts on this subject based on the ideXlab platform.
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sharp decay estimates and asymptotic behaviour for 3d magneto micropolar fluids
arXiv: Analysis of PDEs, 2020Co-Authors: Cesar J Niche, Cilon PerusatoAbstract:We characterize the $L^2$ decay rate of solutions to the 3D magneto-micropolar system in terms of the decay character of the Initial Datum. Due to a linear damping term, the micro-rotational field has a faster decay rate. We also address the asymptotic behaviour of solutions by comparing them to solutions to the linear part. As a result of the linear damping, the difference between the micro-rotational field and its linear part also decays faster. As part of the proofs of these results, we prove estimates for the derivatives of solutions which might be of independent interest.
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decay characterization of solutions to navier stokes voigt equations in terms of the Initial Datum
Journal of Differential Equations, 2016Co-Authors: Cesar J NicheAbstract:Abstract The Navier–Stokes–Voigt equations are a regularization of the Navier–Stokes equations that share some of its asymptotic and statistical properties and have been used in direct numerical simulations of the latter. In this article we characterize the decay rate of solutions to the Navier–Stokes–Voigt equations in terms of the decay character of the Initial Datum and study the long time behavior of its solutions by comparing them to solutions to the linear part.
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decay characterization of solutions to the navier stokes voigt equations in terms of the Initial Datum
arXiv: Analysis of PDEs, 2015Co-Authors: Cesar J NicheAbstract:The Navier-Stokes-Voigt equations are a regularization of the Navier-Stokes equations that share some of its asymptotic and statistical properties and have been used in direct numerical simulations of the latter. In this article we characterize the decay rate of solutions to the Navier-Stokes-Voigt equations in terms of the decay character of the Initial Datum and study the long time behaviour of its solutions by comparing them to solutions to the linear part.
Luis Vega - One of the best experts on this subject based on the ideXlab platform.
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on the relationship between the one corner problem and the m corner problem for the vortex filament equation
arXiv: Numerical Analysis, 2017Co-Authors: Francisco De La Hoz, Luis VegaAbstract:In this paper, we give evidence that the evolution of the Vortex Filament Equation for a regular $M$-corner polygon as Initial Datum can be explained at infinitesimal times as the superposition of $M$ one-corner Initial data. Therefore, and due to periodicity, the evolution at later times can be understood as the nonlinear interaction of infinitely many filaments, one for each corner. This interaction turns out to be some kind of nonlinear Talbot effect. We also give very strong numerical evidence of the transfer of energy and linear momentum for the $M$-corner case.
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The Vortex Filament Equation as a Pseudorandom Generator
Acta Applicandae Mathematicae, 2015Co-Authors: Francisco De la hoz, Luis VegaAbstract:In this paper, we consider the evolution of the so-called vortex filament equation (VFE), $$\mathbf{X}_t = \mathbf{X}_s\wedge\mathbf{X}_{ss}, $$ taking a planar regular polygon of M sides as Initial Datum. We study VFE from a completely novel point of view: that of an evolution equation which yields a very good generator of pseudorandom numbers in a completely natural way. This essential randomness of VFE is in agreement with the randomness of the physical phenomena upon which it is based.
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the vortex filament equation as a pseudorandom generator
arXiv: Number Theory, 2013Co-Authors: Francisco De La Hoz, Luis VegaAbstract:In this paper, we consider the evolution of the so-called vortex filament equation (VFE), \begin{equation*} \mathbf X_t = \mathbf X_s\wedge\mathbf X_{ss}, \end{equation*} taking a planar regular polygon of $M$ sides as Initial Datum. We study VFE from a completely novel point of view: that of an evolution equation which yields a very good generator of pseudorandom numbers in a completely natural way. This essential randomness of VFE is in agreement with the randomness of the physical phenomena upon which it is based.
Clement Mouhot - One of the best experts on this subject based on the ideXlab platform.
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stability convergence to self similarity and elastic limit for the boltzmann equation for inelastic hard spheres
arXiv: Analysis of PDEs, 2007Co-Authors: Stephane Mischler, Clement MouhotAbstract:We consider the spatially homogeneous Boltzmann equation for {\em inelastic hard spheres}, in the framework of so-called {\em constant normal restitution coefficients} $\alpha \in [0,1]$. In the physical regime of a small inelasticity (that is $\alpha \in [\alpha_*,1)$ for some constructive $\alpha_*>0$) we prove uniqueness of the self-similar profile for given values of the restitution coefficient $\alpha \in [\alpha_*,1)$, the mass and the momentum; therefore we deduce the uniqueness of the self-similar solution (up to a time translation). Moreover, if the Initial Datum lies in $L^1_3$, and under some smallness condition on $(1-\alpha_*)$ depending on the mass, energy and $L^1_3$ norm of this Initial Datum, we prove time asymptotic convergence (with polynomial rate) of the solution towards the self-similar solution (the so-called {\em homogeneous cooling state}). These uniqueness, stability and convergence results are expressed in the self-similar variables and then translate into corresponding results for the original Boltzmann equation. The proofs are based on the identification of a suitable elastic limit rescaling, and the construction of a smooth path of self-similar profiles connecting to a particular Maxwellian equilibrium in the elastic limit, together with tools from perturbative theory of linear operators. Some universal quantities, such as the "quasi-elastic self-similar temperature" and the rate of convergence towards self-similarity at first order in terms of $(1-\alpha)$, are obtained from our study. These results provide a positive answer and a mathematical proof of the Ernst-Brito conjecture [16] in the case of inelastic hard spheres with small inelasticity.
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regularity theory for the spatially homogeneous boltzmann equation with cut off
arXiv: Analysis of PDEs, 2006Co-Authors: Clement Mouhot, Cedric VillaniAbstract:We develop the regularity theory of the spatially homogeneous Boltzmann equation with cut-off and hard potentials (for instance, hard spheres), by (i) revisiting the Lp-theory to obtain constructive bounds, (ii) establishing propagation of smoothness and singularities, (iii) obtaining estimates about the decay of the sin- gularities of the Initial Datum. Our proofs are based on a detailed study of the "regularity of the gain operator". An application to the long-time behavior is presented.
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cooling process for inelastic boltzmann equations for hard spheres part i the cauchy problem
Journal of Statistical Physics, 2006Co-Authors: Stephane Mischler, Clement Mouhot, Rodriguez M RicardAbstract:We develop the Cauchy theory of the spatially homogeneous inelastic Boltzmann equation for hard spheres, for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and the relative velocity as well as the sticky particles model. We prove (local in time) non-concentration estimates in Orlicz spaces, from which we deduce weak stability and existence theorem. Strong stability together with uniqueness and instantaneous appearance of exponential moments are proved under additional smoothness assumption on the Initial Datum, for a restricted class of collision rates. Concerning the long-time behaviour, we give conditions for the cooling process to occur or not in finite time.
Vera Mikyoung Hur - One of the best experts on this subject based on the ideXlab platform.
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wave breaking in the whitham equation
arXiv: Analysis of PDEs, 2015Co-Authors: Vera Mikyoung HurAbstract:We prove wave breaking --- bounded solutions with unbounded derivatives --- in the nonlinear nonlocal equation which combines the dispersion relation of water waves and a nonlinearity of the shallow water equations, provided that the slope of the Initial Datum is sufficiently negative, whereby we solve a Whitham's conjecture. We extend the result to equations of Korteweg-de Vries type for a range of fractional dispersion.
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wave breaking for the whitham equation with fractional dispersion
Nonlinearity, 2014Co-Authors: Vera Mikyoung Hur, Lizheng TaoAbstract:We show wave breaking for the Whitham equation in a range of fractional dispersion, i.e. the solution remains bounded but its slope becomes unbounded in finite time, provided that the Initial Datum is sufficiently steep.