Fourier Analysis and Nonlinear Partial Differential Equations

Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël

doi:10.1007/978-3-642-16830-7

Cited by 2,202 publications

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“…It is just a consequence of Bony decomposition and of continuity results for the paraproduct and remainder operators, as stated in e.g. Theorem 2.52 of [2].…”

Section: Notation Besov Spaces and Basic Propertiesmentioning

confidence: 88%

Compressible Navier–Stokes system: Large solutions and incompressible limit

Danchin

Mucha

2017

Advances in Mathematics

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Abstract. Here we prove the existence of global in time regular solutions to the twodimensional compressible Navier-Stokes equations supplemented with arbitrary large initial velocity v0 and almost constant density ̺0 , for large volume (bulk) viscosity. The result is generalized to the higher dimensional case under the additional assumption that the strong solution of the classical incompressible Navier-Stokes equations supplemented with the divergence-free projection of v0, is global. The systems are examined in R d with d ≥ 2 , in the criticalḂ s 2,1 Besov spaces framework. IntroductionWe are concerned with the following compressible Navier-Stokes equations in the whole space R d :supplemented with initial data: ̺| t=0 = ̺ 0 and v| t=0 = v 0 .The pressure function P is given and assumed to be strictly increasing. The shear and volume viscosity coefficients λ and µ are constant (just for simplicity) and fulfill the standard strong parabolicity assumption:(1.2) µ > 0 and ν := λ + 2µ > 0.Starting with the pioneering work by Matsumura and Nishida [18,19] in the beginning of the eighties, a number of papers have been dedicated to the challenging issue of proving the global existence of strong solutions for (1.1) in different contexts (whole space or domains, dimension d = 2 or d ≥ 3, and so on). One may mention in particular the works by Zajaczkowski [26], Shibata [13], Danchin [4], Mucha [21,23,24] and, more recently, by Kotschote [14,15]. The common point between all those papers is that the initial velocity is assumed to be small, and that the initial density is close to a stable constant steady state.Our main goal is to prove the global existence of strong solutions to (1.1) for a class of large initial data. In the two-dimensional case, we establish that, indeed, for fixed shear viscosity µ and any initial velocity-field v 0 (with critical regularity), the solution to (1.1) is global if λ is sufficiently large, and ̺ 0 sufficiently close (in terms of λ) to some positive constant (say 1 for notational simplicity). This result will strongly rely on the fact that, at least formally, the limit velocity for λ → +∞ satisfies the incompressible Navier-Stokes equations:with V 0 being the Leray-Helmholtz projection of v 0 on divergence-free vector-fields. Then the initial data V 0 of (1.3) is inḂ 0 2,1 (R 2 ). Therefore, in light of the well-known embeddingḂ 0 2,1 (R 2 ) ⊂ L 2 (R 2 ), we are guaranteed that it generates a unique global solution V in the energy classthat satisfies the energy identity:Based on that fact, one may prove that the additional regularity of V 0 is preserved through the time evolution (see Theorem 4.1 in the Appendix), that is). Let us now state our main existence result for (1.1) in the two-dimensional setting.There exists a large constant C such that for V 0 = Pv 0 , the divergence-free part of the initial velocity, we setand if ν satisfieswhere Q stands for the projection operator on potential vector-fields, then there exists a unique global in time regular solution (̺, v) to (1.1) such tha...

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“…It is just a consequence of Bony decomposition and of continuity results for the paraproduct and remainder operators, as stated in e.g. Theorem 2.52 of [2].…”

Section: Notation Besov Spaces and Basic Propertiesmentioning

confidence: 88%

Compressible Navier–Stokes system: Large solutions and incompressible limit

Danchin

Mucha

2017

Advances in Mathematics

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“…Another key Fourier analysis tool employed in this work is the paraproduct decomposition, introduced by Bony [15] (see also [5]). Given suitable functions g 1 , g 2 we may define the paraproduct decomposition (in either (z, v) or just v),…”

Section: The Paraproduct Decompositionmentioning

confidence: 99%

Landau Damping: Paraproducts and Gevrey Regularity

2016

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We give a new, simpler, but also and most importantly more general and robust, proof of nonlinear Landau damping on T d in Gevrey− 1 s regularity (s > 1/3) which matches the regularity requirement predicted by the formal analysis of Mouhot and Villani [67]. Our proof combines in a novel way ideas from the original proof of Landau damping Mouhot and Villani [67] and the proof of inviscid damping in 2D Euler Bedrossian and Masmoudi [10]. As in Bedrossian and Masmoudi [10], we use paraproduct decompositions and controlled regularity loss along time to replace the Newton iteration scheme of Mouhot and Villani [67]. We perform time-response estimates adapted from Mouhot and Villani [67] to control the plasma echoes and couple them to energy estimates on the distribution function in the style of the work Bedrossian and Masmoudi [10]. We believe the work is an important step forward in developing a systematic theory of phase mixing in infinite dimensional Hamiltonian systems.

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“…Therefore, a classical result (see e.g. Lemma 2.2 in [3]) ensures that there exists a constant C = C(N ) such that for all q ∈ Z, r ∈ [1, +∞] and tempered distribution z we have (78) ∆ q (1 − ε 2 ∆) ± 1 2 z L r ≤ C ∆ q z L r for all ε ∈ [0, ε 0 ]. Combining these inequalities with (76), we deduce that…”

Section: It Follows That If S > 7/2 Then Inequalities (46) To (48) Ymentioning

confidence: 99%

“…In dimension N ≥ 2, in order to handle longer time scales, one may take advantage of the dispersive properties of system (3). As a matter of fact, the linearization about (0, 0) of the system (3) does not exactly yield the wave operator, as appearing in Theorem 2, but rather the ε-depending operator L ε (a, u) = ∂ t a + √ 2 divu, ∂ t u + √ 2 ∇a − √ 2 ε 2 ∇∆a , which possesses even better dispersive properties.…”

Section: Introductionmentioning

confidence: 99%

On the linear wave regime of the Gross-Pitaevskii equation

Béthuel

Danchin²,

Smets³

2010

JAMA

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Abstract. We study a long wave-length asymptotics for the Gross-Pitaevskii equation corresponding to perturbation of a constant state of modulus one. We exhibit lower bounds on the first occurence of possible zeros (vortices) and compare the solutions with the corresponding solutions to the linear wave equation or variants. The results rely on the use of the Madelung transform, which yields the hydrodynamical form of the Gross-Pitaevskii equation, as well as of an augmented system.

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Fourier Analysis and Nonlinear Partial Differential Equations

Cited by 2,202 publications

References 0 publications

Compressible Navier–Stokes system: Large solutions and incompressible limit

Compressible Navier–Stokes system: Large solutions and incompressible limit

Landau Damping: Paraproducts and Gevrey Regularity

On the linear wave regime of the Gross-Pitaevskii equation

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