Limiting Case of the Sobolev Inequality in BMO,¶with Application to the Euler Equations

By H. Kozono, Y. Taniuchi. 2000

247 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4e97ee9d2b75d728f6154f7d9d04c29ff53889fb

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Citations in the corpus

Geometric Properties and Nonblowup of 3D Incompressible Euler Flow (2004, background): “Very recently, Kozono and Taniuchi (2000) have shown that the above L norm estimate on vorticity in the Beale et al. blowup criterion can be replaced by a weaker BMO norm estimate.”
Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO (2008, background): “See also Kozono and Taniuchi [ 11 ]a nd Ozawa [16].”
The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations (2002, methodology): “Our result is also an extension of the inequality in term of BMO which appears in Kozono-Taniuchi [ 27 ]; observing the case p = ρ = ∞, σ = ∞ in (2.2) and noting � f � ˙”
Local existence and blow‐up criterion for the Euler equations in the Besov spaces (2004, unknown): “Theorem 1.1 (Main theorem) . if and only if ) blows , namely if and only if , the above theorem improves the original Beale–Kato–Majda criterion [3], and its reﬁned version by Kozono and Taniuchi [16] for the case of H s ( R n ).”
Nonlinear maximum principles for dissipative linear nonlocal operators and applications (2011, methodology): “[26]) ‖f‖L∞ ≤ c3 ( 1 + ‖f‖BMO(1 + log+ ‖f‖W s,p) ) (5.”
An Extension of the Beale-Kato-Majda Criterion for the Euler Equations (2003, unknown): “It was recently observed by Kozono-Taniuchi ([8]) that one could replace the L∞ norm in (3) by the BMO norm.”
Chapter 3 - Harmonic Analysis Tools for Solving the Incompressible Navier–Stokes Equations (2005, background): “The idea is that if some additional integrability property is satisfied by at least one weak solution, more exactly, if v2 ∈ L((0, T );Lr) and if 2/s + n/r = 1 with n < r ∞, then all weak solutions have to coincide with it (recently, Kozono and Taniuchi in [126] considered the marginal case s = 2, r”
Local Well-Posedness and Blowup Criterion of the Boussinesq Equations in Critical Besov Spaces (2010, background): “Since there on, different authors improved the similar result, see [10, 11, 12, 15, 8, 16] and so on.”
On the well‐posedness of the Euler equations in the Triebel‐Lizorkin spaces (2002, background): “Kozono and Taniuch [15] recently refined this criterion by replacing the L∞ norm of vorticity by the BMO norm for the vorticity, and H m(Rn) by W s,p(Rn) for the velocity.”
Extension criterion via two-components of vorticity on strong solutions to the 3 D Navier-Stokes equations (2004, background): “Remark. Beale-Kato-Majda [1] and Kozono-Taniuchi [15], [ 16 ] showed the similar extension criterion to Theorem 1 under the hypotheses that ω ∈ L 1 (0 ,T ; L ∞ ) and ω ∈ L 1 (0 ,T ; BMO), respectively.”
On the Euler equations of incompressible fluids (2007, background): “The Beale-Kato-Majda ([15]) criterion (see also ([94]) for an extension) says that the time integral of the maximum magnitude of the vorticity ∫ T”
The three-dimensional Euler equations : Where do we stand? (2008, background): “Kozono and Taniuchi [30] have more recently proved a version of this theorem in the BMOnorm (bounded mean oscillations) which is weaker than the L-norm.”
Remarks on the blowup criteria for Oldroyd models (2009, result): “There were many subsequent results improving the criterion, see for instance [13, 12, 23].”