Global existence in critical spaces for compressible Navier-Stokes equations

By Raphaël Danchin. 2000

621 citations recorded by the source corpus.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a09bdcf4771841b2198a621cf9c17ea4e23c4f64

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In Applied Mathematics

1. Zhifei Zhang
2. Ping Zhang
3. Jing Li
4. Boris Haspot
5. Zhouping Xin
6. Changxing Miao
7. F. Lin
8. Xianpeng Hu

In Control and Systems Engineering

1. Ping Zhang
2. Zhifei Zhang
3. Changxing Miao
4. Boris Haspot
5. Yonggeun Cho
6. Jing Li
7. F. Lin
8. Dehua Wang

In Mathematical Physics

1. Zhifei Zhang
2. Ping Zhang
3. Jing Li
4. Changxing Miao
5. Zhouping Xin
6. Xianpeng Hu
7. Boris Haspot
8. F. Lin

Citations in the corpus

• ON THE ILL-POSEDNESS OF THE COMPRESSIBLE NAVIER-STOKES EQUATIONS (2011, methodology): “Motived by Fujita-Kato’s result [12] on the incompressible Navier-Stokes equations, Danchin studied in a series of papers [7, 8, 9] the well-posedness for the compressible Navier-Stokes equations in the critical spaces.”
• On the uniqueness in critical spaces for compressible Navier-Stokes equations (2005, methodology): “In a series of papers (see [4], [5] and [6]), we studied whether Fujita and Kato’s approach is relevant for flows of compressible fluids.”
• Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations (2015, background): “[9] R. Danchin, Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density.”
• On the well-posedness of the full compressible Navier-Stokes system in critical Besov spaces (2014, methodology): “Let us recall that in the barotropic case, the critical Besov regularity was first considered by the latter author in a L2 type framework to obtain a global solution [6] for small perturbations of a stable constant state (ρ∗, 0) with ρ∗ > 0.”
• Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations (2008, background): “atsumura et al [35–37], who proved global existence of smooth solutions for data close to a non-vacuum equilibrium, and later by Hoff for discontinuous initial data [19], and more recently, by Danchin [11], who obtained existence and uniqueness of global solutions in a functional space invariant by t”
• A Global Existence Result for the Compressible Navier–Stokes Equations in the Critical Lp Framework (2010, methodology): “The following statement is a consequence of the results that have been proved in [ 7 ,11].”
• Global well‐posedness for compressible Navier‐Stokes equations with highly oscillating initial velocity (2009, background): “[15] R. Danchin, Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density.”
• Well-posedness of a 3D parabolic–hyperbolic Keller–Segel system in the Sobolev space framework (2012, neutral): “For various aspects of the smoothing properties, we refer the readers to see, for instance [6, 9] and the references therein.”
• Global Existence for the Multi-Dimensional Compressible Viscoelastic flows (2010, methodology): “Let us recall that using hybrid Besov spaces has been crucial for proving global well-posedness for compressible gases in critical spaces (see [4, 5]).”
• On the well-posedness of the incompressible density-dependent Euler equations in the Lp framework (2009, background): “timates in Besov spaces for the transport equation (T) ( ∂tf+v· ∇f= g, f|t=0 = f0. We shall often use the following result, the proof of which may be found in e.g. [2], Chap. 3, or in the appendix of [8]. Proposition 5. Let 1 ≤ p,r≤ ∞ and σ>0. Let f0 ∈ Bσ p,r, g∈ L1([0,T];Bσ ) and v be a time dep”
• A New Bernstein’s Inequality and the 2D Dissipative Quasi-Geostrophic Equation (2006, neutral): “ard limit argument, we can prove that the limit function θ(t,x) satisfies the equation (1.1) in the sense of distribution. 11 We still have to prove θ(t,x) belongs to C(R+;Bσ p,q). Our idea comes from [15]. We observe that ∂t∆jθ= −κΛ2α∆jθ−∆j∇· (uθ) (4.18) For fixed j, the right hand side of (4.18) bel”
• Optimal Time-decay Estimates for the Compressible Navier–Stokes Equations in the Critical Lp Framework (2016, methodology): “There, thanks to a smart modification of the method of [7], Inequality (1.”
• Equation de Navier-Stokes avec densité et viscosité variables dans l'espace critique (2007, background): “Alors pour pouvoir utiliser le théorème d’Ascoli, il suffit d’estimer la dérivée par rapport au temps de a et u (voir par exemple [11]).”