Optimal well-posedness for the inhomogeneous incompressible Navier–Stokes system with general viscosity

Burtea, Cosmin

doi:10.2140/apde.2017.10.439

Cited by 12 publications

(19 citation statements)

References 28 publications

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“…Firstly, Danchin [12] constructed a unique strong solution to (1.1) in the critical space (L ∞ (R 3 ) ∩ Ḃ3/2 2,∞ (R 3 )) × Ḃ1/2 2,1 (R 3 ) in the case when the initial density is close to a constant. Later, many authors tried to improve Danchin's result to allow different Lebesgue indices of the critical spaces, or to remove the smallness assumption on the initial density (see [1][2][3][4]9,15,34]). Secondly, it is interesting to lower the regularity of the density to allow discontinuity.…”

Section: Introductionmentioning

confidence: 99%

Maximal $L^1$ regularity for solutions to inhomogeneous incompressible Navier-Stokes equations

Xu¹

2021

Preprint

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This paper is devoted to the maximal L 1 regularity and asymptotic behavior for solutions to the inhomogeneous incompressible Navier-Stokes equations under a scalinginvariant smallness assumption on the initial velocity. We obtain a new global L 1 -in-time estimate for the Lipschitz seminorm of the velocity field. In the derivation of this estimate, we study the maximal regularity for a linear Stokes system with variable coefficients. The analysis tools are a use of the semigroup generated by a generalized Stokes operator to characterize some Besov norms and a new gradient estimate for a class of second-order elliptic equations of divergence form. Our method can be used to study maximal L 1 regularity for other incompressible or compressible viscous fluids.

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Section: Introductionmentioning

confidence: 99%

Maximal $L^1$ regularity for solutions to inhomogeneous incompressible Navier-Stokes equations

Xu¹

2021

Preprint

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“…For this we will use the new class of estimates obtained by the first author in [6] instead of the classical maximal regularity estimates obtained by Danchin and Mucha in [15,16]. As in [6] our result will be given for n = 2 with p ∈ (1, 4) and n = 3 with p ∈ (6/5, 4).…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…The third term is dealt with the classical product laws in Besov spaces (see section 4.1), using the fact that k(ρ 0 ) = 1 + k(ρ 0 ) − k(1) (recall that k(1) = 1) and (4.43): and thanks to Proposition 8 we obtain as in [6]:…”

Section: First Step: Well-posedness For (27)mentioning

confidence: 99%

“…As in [15,6], to overcome this problem, the idea is to introduce the free solution, let us define (u L , ∇P L ) the unique global solution of the following system:…”

Section: Second Step: Well-posedness For (25)mentioning

confidence: 99%

“…As we wish to take advantage of the smallness of the L 1 T norm, we cannot do as before for the last term (as the u L L ∞ TḂ n p −1 p,1 norm may not be small and would prevent any absorption by the left-hand side) and the idea is (as in [15,6]) to use the L 2 tḂ n p p,1 -norm of u L instead: ). (2.22) Thanks to the fact that ( v, ∇ Q) ∈ G ε T and condition (2.16) we end up for all t ≤ T with (we do not give details as the computations are the same as before),…”

Section: Second Step: Well-posedness For (25)mentioning

confidence: 99%

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Lagrangian Methods for a General Inhomogeneous Incompressible Navier--Stokes--Korteweg System with Variable Capillarity and Viscosity Coefficients

Burtea¹,

Charve²

2017

SIAM J. Math. Anal.

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We study the inhomogeneous incompressible Navier-Stokes system endowed with a general capillary term. Thanks to recent methods based on Lagrangian change of variables, we obtain local well-posedness in critical Besov spaces (even if the integration index p = 2) and for variable viscosity and capillary terms. In the case of constant coefficients and for initial data that are perturbations of a constant state, we are able to prove that the lifespan goes to infinity as the capillary coefficient goes to zero, connecting our result to the global existence result obtained by Danchin and Mucha for the incompressible Navier-Stokes system with constant coefficients.

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Global well‐posedness of the 3D incompressible nematic liquid crystal flows with density‐dependent viscosity coefficient

Liu

2020

Math Methods in App Sciences

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This paper deals with the 3D incompressible nematic liquid crystal flows with density-dependent viscosity in bounded domain. The global well-posedness of strong solutions are established in the vacuum cases, provided the assumption that̄+ ||∇d 0 || L 3 is suitably small with large velocity. Furthermore, the exponential decay of the solution is also obtained. KEYWORDS exponential decay, incompressible nematic liquid crystal flow, strong solutions, vacuum MSC CLASSIFICATION 76A15; 35B65; 35Q35Math Meth Appl Sci. 2020;43:5985-6010. wileyonlinelibrary.com/journal/mma © 2020 John Wiley & Sons, Ltd. 5985 LIUfor some positive constant ,̄. Without loss of generality, both and are normalized to 1. The symbol ∇d ⊙ ∇d, which exhibits the property of the anisotropy of the material, denotes the n × n matrix whose (i, )-th entry is given by i d · d, for i, = 1, 2, 3. We consider an initial boundary value problem for (1) with the following initial and boundary conditions:with d 0 ∈  2 being given with compatibility, div u 0 = 0 in Ω, and d 0 ∈ C 1 (Ω) satisfying ∇d 0 = 0 on the boundary Ω (see e.g., Hu and Wang 1 and Li 2 ). The nematic liquid crystal flows are regarded as slow-moving particles where the fluid velocity and the alignment of the particles affect each other. The hydrodynamics of nematic liquid crystals is developed by Ericksen 3 and Leslie 4 in the 1960s, but it still retains most important mathematical structures as well as most of the essential difficulties of the original Ericksen-Leslie model. Mathematically, System (1) is a strongly coupled system between the nonhomogeneous incompressible Navier-Stokes equations and the transported heat flows of harmonic map, and thus, its mathematical analysis is full of challenges.When d is a constant vector and |d| = 1, System (1) reduces to the well-known nonhomogeneous incompressible Navier-Stokes equations. Lions 5 , Chapter 2 established the global existence of weak solutions to nonhomogeneous Navier-Stokes equations in any space dimensions for the initial density allowing vacuum. Later, Desjardins 6 proved the global weak solution with more regularity for the two-dimensional case provided that ( 0 ) is a small perturbation of a positive constant in L ∞ norm. Cho-Kim 7 established the local strong solution by using the compatibility condition −div (2 ( 0 )∇u 0 ) + ∇p 0 = √ 0 g and div u 0 = 0 in Ω,

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Optimal well-posedness for the inhomogeneous incompressible Navier–Stokes system with general viscosity

Cited by 12 publications

References 28 publications

Maximal $L^1$ regularity for solutions to inhomogeneous incompressible Navier-Stokes equations

Maximal $L^1$ regularity for solutions to inhomogeneous incompressible Navier-Stokes equations

Lagrangian Methods for a General Inhomogeneous Incompressible Navier--Stokes--Korteweg System with Variable Capillarity and Viscosity Coefficients

Global well‐posedness of the 3D incompressible nematic liquid crystal flows with density‐dependent viscosity coefficient

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