Solutions in Lr of the Navier-Stokes initial value problem

Giga, Yoshikazu; Miyakawa, Tetsuro

doi:10.1007/bf00276875

Cited by 347 publications

(266 citation statements)

References 17 publications

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“…[12], [14]) that ∇h ∈ H ⊥ σ , from which (2.1) immediately follows. One can also deduce this directly by noting the identity v · ∇h = div(hv) − (div v)h = div(hv) and then using Green's Theorem.…”

Section: Preliminary Observations and Resultsmentioning

confidence: 78%

“…When n = 3 it is well known that the situation is quite different; the existence and uniqueness of global strong solutions remains an open question for arbitrary initial data. Indeed, for many years global existence was known only for small initial data in the domain of (−∆) 1/4 [12]; this result was generalized to allow small initial data in L 3 (Ω) in [14]. Recently, it was discovered that large initial data could be allowed if the domain Ω is "thin" enough.…”

Section: Introductionmentioning

confidence: 99%

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Incompressible reacting flows

Avrin¹

1997

Trans. Amer. Math. Soc.

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Abstract. We establish steady-state convergence results for a system of reaction-convection-diffusion equations that model in particular combustion phenomena in the presence of nontrivial incompressible fluid motion. Despite the presence of the convection terms, we find that the asymptotic behavior of the system is identical to the case we have previously considered in which the velocity field was set equal to zero. In particular we are again able to establish the convergence of solutions to steady-states and to explicitly calculate the steady-states from the initial and boundary data. Key to our analysis is the establishment of high-order uniform bounds on the temperature and mass fraction components, a process significantly complicated by the presence of the convection terms.

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Section: Preliminary Observations and Resultsmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Incompressible reacting flows

Avrin¹

1997

Trans. Amer. Math. Soc.

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“…In the case of a smooth bounded domain in R n , it was proved by Y. Giga and T. Miyakawa in [22] that the Dirichlet-Navier-Stokes system admits a local mild solution for initial values in L n (critical space for the system in dimension n). Their method relies on the fact that the Dirichlet-Stokes operator, as defined in Section 1, extends to all L p spaces and is the negative generator of an analytic semigroup there, which was proved in [21].…”

Section: Resultsmentioning

confidence: 99%

Stokes Problems in Irregular Domains with Various Boundary Conditions

Monniaux¹,

Shen²

2016

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

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Different boundary conditions for the Navier-Stokes equations in bounded Lipschitz domains in R 3 , such as Dirichlet, Neumann, Hodge or Robin boundary conditions are presented here. The situation is a little different from the case of smooth domains. The analysis of the problem involves a good comprehension of the behaviour near the boundary. The linear Stokes operator associated to the various boundary conditions is first studied. Then a classical fixed point theorem is used to show how the properties of the operator lead to local solutions or global solutions for small initial data.

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“…The uniqueness proof is done in a standard way, so it is omitted. To see that the solution ( u, w ) of (30) [7], Giga & Miyakawa [9], Kozono & Ogawa [13], we can show the Hölder continuity with respect to time for the nonlinear term in L 3 σ (Ω) × L α (Ω) and this allows us to show that ( u, w ) is, in fact, a strong solution for the system (29) in the sense given in Definition 2.3 (see Tanabe [29]), Theorem 3.3.4).…”

Section: Remark 53mentioning

confidence: 99%

Global existence and exponential stability for the micropolar fluid system

Villamizar-Roa

Rodríguez-Bellido

2007

Z. angew. Math. Phys.

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We consider the micropolar fluid system in a bounded domain of R 3 and prove the existence and the uniqueness of a global strong solution with initial data being a perturbation of the stationary solution, whose existence is also obtained. We prove that these solutions converge uniformly to the stationary solutions with exponential decay rate. The technique of our analysis is the semigroups approach in L p −spaces.

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Solutions in Lr of the Navier-Stokes initial value problem

Cited by 347 publications

References 17 publications

Incompressible reacting flows

Incompressible reacting flows

Stokes Problems in Irregular Domains with Various Boundary Conditions

Global existence and exponential stability for the micropolar fluid system

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