Navier-stokes flow in r<sup>3</sup>with measures as initial vorticity and morrey spaces

Giga, Yoshikazu; Miyakawa, Tetsuro

doi:10.1080/03605308908820621

Cited by 276 publications

(240 citation statements)

References 18 publications

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“…In order to prove the lemma, let us first remark that we can limit ourselves, without loss of generality, to its scalar version, say, the estimate ||A5(t)(/5)||<^)ll/IIIMI, (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) where A = (-A) -2 is the scalar Calderon operator and / and g are two arbitrary scalar functions in X. In fact, this is true if we recall that the vector operator PVis a pseudo-differential operator of degree 1 whose symbol is given by the following coefficients:…”

Section: Lemma 21 Ifx Is a Well-suited Banach Space Then There Eximentioning

confidence: 99%

“…(ii) there exists a sequence of reals r]j > 0, j G Z, such that 5^2-1^ <oo (2.9) and that \\^(f9)\\<Vj\\f\\\\g\\ (2)(3)(4)(5)(6)(7)(8)(9)(10) for any scalar distributions / and g in X.…”

Section: The Navier-stokes Equationsmentioning

confidence: 99%

“…His clever proof is essentially based upon a bilinear operator estimate which is expressed in a simple quadratic form using a wavelet frame. The paper of Taylor [11] deals with the more general case M|(R 3 ), including the limiting case p = 3, and yields extensions of the results obtained in both [3] and [4]. His proof analyzes some general semigroup estimates in the Morrey-Campanato spaces, but does not deal with the infrared problem, which is, on the other hand, treated in [3].…”

Section: Introductionmentioning

confidence: 99%

“…Later, Kato and Ponce [8] generalized the same arguments in the case of Lf (E 3 ) spaces with 1 < p < oo and s > 1 + 3/p. More recently, considerable effort has been devoted to the analysis of MorreyCampanato spaces M£(R 3 ) [3,4,7,11]. In particular, Federbush [2,3] has obtained such an existence and uniqueness theorem in the M^R 3 ) setting when p > 3.…”

Section: Introductionmentioning

confidence: 99%

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Littlewood–Paley decomposition and Navier–Stokes equations

Cannone¹,

Meyer²

1995

Methods and Applications of Analysis

125

212

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ABSTRACT. Using the dyadic decomposition of Littlewood-Paley, we find a simple condition that, when tested on an abstract Banach space X, guarantees the existence and uniqueness of a local strong solution v(t) G C([0,T); X) of the Cauchy problem for the Navier-Stokes equations in E 3 . Many examples of such Banach spaces are offered. We also prove some regularity results on the solution v(t) and we illustrate, by means of a counterexample, that the above-mentioned sufficient condition is, in general, not necessary.

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Section: Lemma 21 Ifx Is a Well-suited Banach Space Then There Eximentioning

confidence: 99%

“…(ii) there exists a sequence of reals r]j > 0, j G Z, such that 5^2-1^ <oo (2.9) and that \\^(f9)\\<Vj\\f\\\\g\\ (2)(3)(4)(5)(6)(7)(8)(9)(10) for any scalar distributions / and g in X.…”

Section: The Navier-stokes Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Littlewood–Paley decomposition and Navier–Stokes equations

Cannone¹,

Meyer²

1995

Methods and Applications of Analysis

125

212

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“…The same argument also shows that the Cauchy problem for (9) is locally well-posed in L 3/2 (R 3 ), without smallness assumption on the data. More generally, one can prove that (9) has global solutions for small data in the Morrey space M 3/2 (R 3 ), see [14].…”

Section: The Cauchy Problem For the Vorticity Equationmentioning

confidence: 99%

Long-time asymptotics of the Navier-Stokes and vorticity equations on ℝ³

Gallay

Wayne

2002

Philosophical Transactions of the Royal Society of London. Seri

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We use the vorticity formulation to study the long-time behavior of solutions to the Navier-Stokes equation on R 3 . We assume that the initial vorticity is small and decays algebraically at infinity. After introducing self-similar variables, we compute the long-time asymptotics of the rescaled vorticity equation up to second order. Each term in the asymptotics is a self-similar divergence-free vector field with Gaussian decay at infinity, and the coefficients in the expansion can be determined by solving a finite system of ordinary differential equations. As a consequence of our results, we are able to characterize the set of solutions for which the velocity field satisfies u(·, t) L 2 = o(t −5/4 ) as t → +∞. In particular, we show that these solutions lie on a smooth invariant submanifold of codimension 11 in our function space.

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