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Documents Neustupa, Jiri 14 résultats

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We discuss some stability problems related to the Navier-Stokes-Fourier system describing the motion of a compressible, viscous, and heat conducting fluids. We introduce the concept of relative entropy/energy and present some applications that concern:
- Existence and conditional regularity of weak solutions;
- singular limits;
- existence and regularity for the inviscid system.

76N10 ; 35Q30

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y
Given initial data $(b_0, u_0)$ close enough to the equilibrium state $(e_3, 0)$, we prove that the 3-D incompressible MHD system without magnetic diffusion has a unique global solution $(b, u)$. Moreover, we prove that $(b(t) - e_3, u(t))$ decay to zero with rates in both $L^\infty$ and $L^2$ norm. (This is a joint work with Wen Deng).

35Q30 ; 76D03

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y
The purpose of this talk is to present two 1d congestion models: a soft congestion model with a singular pressure, and a hard congestion model in which the dynamic is different in the congested and non-congested zone (incompressible vs. compressible dynamic). The hard congested model is the limit of the soft one as the parameter within the singular presure vanishes.
For each model, we prove the existence of traveling waves, and we study their stability. This is a joint work with Charlotte Perrin.[-]
The purpose of this talk is to present two 1d congestion models: a soft congestion model with a singular pressure, and a hard congestion model in which the dynamic is different in the congested and non-congested zone (incompressible vs. compressible dynamic). The hard congested model is the limit of the soft one as the parameter within the singular presure vanishes.
For each model, we prove the existence of traveling waves, and we study their ...[+]

35B35 ; 35Q35 ; 35R35

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y

Geometric structures in 2D Navier-Stokes flows - Brandolese, Lorenzo (Auteur de la conférence) | CIRM H

Multi angle

Geometric structures naturally appear in fluid motions. One of the best known examples is Saturn's Hexagon, the huge cloud pattern at the level of Saturn's north pole, remarkable both for the regularity of its shape and its stability during the past decades. In this paper we will address the spontaneous formation of hexagonal structures in planar viscous flows, in the classical setting of Leray's solutions of the Navier–Stokes equations. Our analysis also makes evidence of the isotropic character of the energy density of the fluid for sufficiently localized 2D flows in the far field: it implies, in particular, that fluid particles of such flows are nowhere at rest at large distances.[-]
Geometric structures naturally appear in fluid motions. One of the best known examples is Saturn's Hexagon, the huge cloud pattern at the level of Saturn's north pole, remarkable both for the regularity of its shape and its stability during the past decades. In this paper we will address the spontaneous formation of hexagonal structures in planar viscous flows, in the classical setting of Leray's solutions of the Navier–Stokes equations. Our ...[+]

35Q30 ; 76D05

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In this talk, I will present recent results on solutions to a one-dimensional Euler system coupling compressible and incompressible phases. With this original fluid system we intend to model congestion (or saturation) phenomena in heterogeneous flows (mixtures, wave-structure interactions, collective motion, etc.). Here the compressible-incompressible model will be seen as the limit of a fully compressible Euler system endowed with a singular pressure law. The goal of the talk is to present theoretical results concerning this singular limit. This is a joint work with Roberta Bianchini.[-]
In this talk, I will present recent results on solutions to a one-dimensional Euler system coupling compressible and incompressible phases. With this original fluid system we intend to model congestion (or saturation) phenomena in heterogeneous flows (mixtures, wave-structure interactions, collective motion, etc.). Here the compressible-incompressible model will be seen as the limit of a fully compressible Euler system endowed with a singular ...[+]

35Q35 ; 35L87 ; 35L81

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The H-Principle and Turbulence - Székelyhidi, László (Auteur de la conférence) | CIRM H

Multi angle

It is well known since the pioneering work of Scheffer and Shnirelman that weak solutions of the incompressible Euler equations exhibit a wild behaviour, which is very different from that of classical solutions. Nevertheless, weak solutions in three space dimensions have been studied in connection with a long-standing conjecture of Lars Onsager from 1949 concerning anomalous dissipation and, more generally, because of their possible relevance to the K41 theory of turbulence.
In recent joint work with Camillo De Lellis we established a connection between the theory of weak solutions of the Euler equations and the Nash-Kuiper theorem on rough isometric immersions. Through this connection we interpret the wild behaviour of weak solutions of Euler as an instance of Gromov's h-principle.
In this lecture we explain this connection and outline recent progress towards Onsager's conjecture.[-]
It is well known since the pioneering work of Scheffer and Shnirelman that weak solutions of the incompressible Euler equations exhibit a wild behaviour, which is very different from that of classical solutions. Nevertheless, weak solutions in three space dimensions have been studied in connection with a long-standing conjecture of Lars Onsager from 1949 concerning anomalous dissipation and, more generally, because of their possible relevance to ...[+]

35Q31 ; 76B03

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2y

The Onsager Theorem - De Lellis, Camillo (Auteur de la conférence) | CIRM H

Post-edited

In the fifties John Nash astonished the geometers with his celebrated isometric embedding theorems. A folkloristic explanation of his first theorem is that you should be able to put any piece of paper in your pocket without crumpling or folding it, no matter how large it is.
Ten years ago László Székelyhidi and I discovered unexpected similarities with the behavior of some classical equations in fluid dynamics. Our remark sparked a series of discoveries and works which have gone in several directions. Among them the most notable is the recent proof of Phil Isett of a long-standing conjecture of Lars Onsager in the theory of turbulent flows. In a joint work with László, Tristan Buckmaster and Vlad Vicol we improve Isett's theorem to show the existence of dissipative solutions of the incompressible Euler equations below the Onsager's threshold.[-]
In the fifties John Nash astonished the geometers with his celebrated isometric embedding theorems. A folkloristic explanation of his first theorem is that you should be able to put any piece of paper in your pocket without crumpling or folding it, no matter how large it is.
Ten years ago László Székelyhidi and I discovered unexpected similarities with the behavior of some classical equations in fluid dynamics. Our remark sparked a series of ...[+]

35Q31 ; 35D30 ; 76B03

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We first summarize the derivation of viscoelastic (rate-type) fluids with stress diffusion that generates the models that are compatible with the second law of thermodynamics and where no approximation/reduction takes place. The approach is based on the concept of natural configuration that splits the total response between the current and initial configuration into the purely elastic and dissipative part. Then we restrict ourselves to the class of fluids where elastic response is purely spherical. For such class of fluids we then provide a mathematical theory that, in particular, includes the long-time and large-data existence of weak solution for suitable initial and boundary value problems. This is a joint work with Miroslav Bulicek, Vit Prusa and Endre Suli.[-]
We first summarize the derivation of viscoelastic (rate-type) fluids with stress diffusion that generates the models that are compatible with the second law of thermodynamics and where no approximation/reduction takes place. The approach is based on the concept of natural configuration that splits the total response between the current and initial configuration into the purely elastic and dissipative part. Then we restrict ourselves to the class ...[+]

76A10 ; 80A10 ; 35D30 ; 35Q35

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An asymptotic regime for the Vlasov-Poisson system - Miot, Evelyne (Auteur de la conférence) | CIRM H

Multi angle

We investigate the gyrokinetic limit for the two-dimensional Vlasov-Poisson system in a regime studied by F. Golse and L. Saint-Raymond. First we establish the convergence towards the Euler equation under several assumptions on the energy and on the norms of the initial data. Then we provide a first analysis of the asymptotics for a Vlasov-Poisson system describing the interaction of a bounded density with a moving point charge.

82D10 ; 82C40 ; 35Q35 ; 35Q83 ; 35Q31

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Consider the motion of a viscous incompressible fluid in a 3D exterior domain $D$ when a rigid body $\mathbb R^3\setminus D$ moves with prescribed time-dependent translational and angular velocities. For the linearized non-autonomous system, $L^q$-$L^r$ smoothing action near $t=s$ as well as generation of the evolution operator $\{T(t,s)\}_{t\geq s\geq 0}$ was shown by Hansel and Rhandi [1] under reasonable conditions. In this presentation we develop the $L^q$-$L^r$ decay estimates of the evolution operator $T(t,s)$ as $(t-s)\to\infty$ and then apply them to the Navier-Stokes initial value problem.[-]
Consider the motion of a viscous incompressible fluid in a 3D exterior domain $D$ when a rigid body $\mathbb R^3\setminus D$ moves with prescribed time-dependent translational and angular velocities. For the linearized non-autonomous system, $L^q$-$L^r$ smoothing action near $t=s$ as well as generation of the evolution operator $\{T(t,s)\}_{t\geq s\geq 0}$ was shown by Hansel and Rhandi [1] under reasonable conditions. In this presentation we ...[+]

35Q30 ; 76D05 ; 76D07

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