Density-dependent incompressible viscous fluids in critical spaces

By Raphaël Danchin. Proceedings of the Royal Society of Edinburgh. Section A Mathematics, 2003

251 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0f9f202a745fd86885eca07fccc15170a7ae9ab2

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Citations in the corpus

Global Unique Solvability of Inhomogeneous Navier-Stokes Equations with Bounded Density (2013, background): “In this framework, it has been stated in [1, 7] that for the initial data (ρ0, u0) satisfying (ρ0 − 1) ∈ Ḃ d p p,1(R ), u0 ∈ Ḃ d p −1 p,1 (R ) with div u0 = 0 and that for a small enough constant c ‖ρ0 − 1‖ Ḃ d p p,1 + ‖u0‖ Ḃ d p−1 p,1 ≤ c, we have for any p ∈ [1, 2d) • existence of global solution ”
Dissipative quasi-geostrophic equations in critical Sobolev spaces: smoothing effect and global well-posedness (2007, unknown): “We essentially follow the idea of Proposition 2 in [17] (see also earlier [3], [7] and [ 8 ] for similar estimates).”
A New Bernstein’s Inequality and the 2D Dissipative Quasi-Geostrophic Equation (2006, background): “Proof: The proof is a straightforward adaptation of Lemma A.1 in [ 16 ] which is a version of the commutator estimate in Besov space.”
On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity (2007, background): “We state now a classical result for the linearized momentum equation (for a more general study see Proposition 2.2 [ 8 ] and the references therein).”
Global smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with variable viscosity (2009, background): “if 1 9 ] for the constant viscosity case.”
A Lagrangian Approach for the Incompressible Navier‐Stokes Equations with Variable Density (2011, methodology): “[1, 2, 6, 7]) we want to solve the above system in critical homogeneous Besov spaces.”
Local existence and uniqueness for the non-resistive MHD equations in homogeneous Besov spaces (2016, background): “We state the following logarithmic interpolation inequality which will be useful in the sequel (see [5]).”
Les theoremes de Leray et de Fujita-Kato pour le systeme de Boussinesq partiellement visqueux. The Leray and Fujita-Kato theorems for the Boussinesq system with partial viscosity (2008, unknown): “14 (qui se généralisent de façon immédiate aux espaces L̃ρT (Ḃ s p,r) voir par exemple [11] ou [13]), et (19), on obtient les majorations ‖div (u2 ⊗ δu)‖e L(1)t (Ḃ N,∞) .”
Global Well-posedness of Incompressible Inhomogeneous Fluid Systems with Bounded Density or Non-Lipschitz Velocity (2012, background): “This result generalized the corresponding results in [10, 11] and was improved by Abidi and Paicu in [2] when μ̃(a) is a positive constant by using different Lebesgue indices for the density and for the velocity.”
Equation de Navier-Stokes avec densité et viscosité variables dans l'espace critique (2007, background): “La question à laquelle nous allons tenter de répondre dans ce travail est la suivante : peut-on généraliser le résultat cité de [8] lorsque la viscosité μ n’est plus constante mais dépend de la densité et vérifie (1.”
Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces (2008, unknown): “On the other hand, due to the weak regularity assumptions, we shall be in the limit case for product laws in Proposition 4 so that a logarithmic interpolation argument (similar to that which has been used in [8] and [10]) will be required.”
Global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces (2006, background): “Now let us we recall the following commutator lemma (see [3,10] and the references therein).”
Global solutions to the 3-D incompressible inhomogeneous Navier–Stokes system (2012, result): “In [10], Danchin studied in general space dimension N the unique solvability of the system (INS) with constant viscosity coefficient and in scaling invariant (or critical) homogeneous Besov spaces, which generalized the celebrated results by Fujita and Kato [14] devoted to the classical Navier–Stoke”
Global Well-Posedness for Compressible Viscoelastic Fluids near Equilibrium (2010, unknown): “[9] Let s ∈ R.”