Journal of Mathematical Analysis and Applications, May 1, 2016
In this work, a regularity criterion is proved for local strong solutions of the Navier-Stokes eq... more In this work, a regularity criterion is proved for local strong solutions of the Navier-Stokes equations in the presence of mass diffusion.
In this note, a logarithmically improved regularity criterion for the Navier-Stokes equations is ... more In this note, a logarithmically improved regularity criterion for the Navier-Stokes equations is established in terms of the pressure in the nonhomogeneous Besov space.
In this work, a regularity criterion is proved for local strong solutions of the Navier-Stokes eq... more In this work, a regularity criterion is proved for local strong solutions of the Navier-Stokes equations in the presence of mass diffusion.
We derive various anisotropic versions of the Troisi inequality for the study of the conditional ... more We derive various anisotropic versions of the Troisi inequality for the study of the conditional regularity of solutions to the Navier–Stokes equations. We impose additional conditions on one or two non-diagonal entries of the velocity gradient, namely or and . Formulating our results in the frame of the anisotropic Lebesgue spaces we improve two recent results from the literature. Further we present two new criteria.
Nonlinear Analysis-real World Applications, Jun 1, 2017
Abstract We study the regularity criteria for the incompressible Navier–Stokes equations in the w... more Abstract We study the regularity criteria for the incompressible Navier–Stokes equations in the whole space R 3 based on one velocity component, namely u 3 , ∇ u 3 and ∇ 2 u 3 . We use a generalization of the Troisi inequality and anisotropic Lebesgue spaces and prove, for example, that the condition ∇ u 3 ∈ L β ( 0 , T ; L p ) , where 2 / β + 3 / p = 7 / 4 + 1 / ( 2 p ) and p ∈ ( 2 , ∞ ] , yields the regularity of u on ( 0 , T ] .
Journal of Mathematical Fluid Mechanics, Jun 13, 2019
In this paper we establish regularity conditions for the three dimensional incompressible Navier-... more In this paper we establish regularity conditions for the three dimensional incompressible Navier-Stokes equations in terms of one entry of the velocity gradient tensor, say for example, ∂ 3 u 3. We show that if ∂ 3 u 3 satisfies certain integrable conditions with respect to time and space variables in anisotropic Lebesgue spaces, then a Leray-Hopf weak solution is actually regular. The anisotropic Lebesgue space helps us to almost reach the Prodi-Serrin level 2 in certain special case. Moreover, regularity conditions on non-diagonal element of gradient tensor ∂ 1 u 3 are also established, which covers some previous literature.
Journal of Mathematical Analysis and Applications, Feb 1, 2018
We prove, among others, the following regularity criterion for the solutions to the Navier-Stokes... more We prove, among others, the following regularity criterion for the solutions to the Navier-Stokes equations: If u is a global weak solution satisfying the energy inequality and ω = ∇ × u, then u is regular on (0, T), T > 0, if two components of ω belong to the space L q (0, T ; Ḃ −3/p ∞,∞) for p ∈ (3, ∞) and 2/q + 3/p = 2. This result is an improvement of the results presented by Chae and Choe (1999) [7] or Zhang and Chen (2005) [38]. Our method of the proof uses a suitable application of the Bony decomposition and can also be used for the proofs of some other kin criteria. One such example is presented in Appendix.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Some results on regularity of weak solutions to the Navier-Stokes equations published recently in... more Some results on regularity of weak solutions to the Navier-Stokes equations published recently in [3] follow easily from a classical theorem on compact operators. Further, weak solutions of the Navier-Stokes equations in the space L2(0, T, W 1,3(Ω)3) are regular.
We study the large time behavior of solutions to the Navier-Stokes equations. The emphasis is put... more We study the large time behavior of solutions to the Navier-Stokes equations. The emphasis is put on the rate of the decay of the solutions and their behavior in the frequency space (see [1] and [2]).
Nonlinear Analysis: Theory, Methods & Applications, 2014
We improve a regularity criterion for the solutions to the Navier-Stokes equations in the full th... more We improve a regularity criterion for the solutions to the Navier-Stokes equations in the full three-dimensional space involving the gradient of one velocity component. Revising the method used in Pokorný and Zhou (2009, 2010), we show that a weak solution u is regular on (0, T) provided that ∇u 3 ∈ L t
Nonlinear Analysis: Theory, Methods & Applications, 2014
Abstract We study the lower bounds of decay rates for turbulent solutions to the Navier–Stokes eq... more Abstract We study the lower bounds of decay rates for turbulent solutions to the Navier–Stokes equations in the norms of Besov spaces. We focus on solutions satisfying 0 lim inf t → ∞ t γ ‖ u ( t ) ‖ ≤ lim sup t → ∞ t γ ‖ u ( t ) ‖ ∞ for some γ ∈ ( 0 , 5 / 4 ] and prove among others that such solutions, measured in the norm of the Besov space B 2 , ∞ − 2 γ , are estimated for large times from below by some positive constant. These estimates stem from sufficiently fast rate of large time energy concentration in low frequencies occurring in the studied solutions. Our results have simple proofs and improve and extend substantially the results published by Miyakawa (2002) and by Schonbek and Wiegner (1996).
Journal of Mathematical Analysis and Applications, 2013
Abstract In the paper we study the large-time behavior of solutions to the Navier–Stokes equation... more Abstract In the paper we study the large-time behavior of solutions to the Navier–Stokes equations in the frequency space. We describe in detail the large-time energy concentration which occurs in every (turbulent) solution. If the energy of the solution decreases exponentially then it concentrates in frequencies localized in an annulus in the frequency space. The annulus can be taken arbitrarily narrow and its diameter determines the rate of the exponential decay. All the other solutions are characterized by the concentration of the energy in the frequencies localized in a ball with an arbitrarily small diameter centered at the origin of the coordinates. It will follow from the presented results that the frequencies outside the annulus or the ball and especially the higher frequencies die out very quickly. We will further observe the concentration occurring in any time derivative of the solution or in the vorticity and its time derivatives with the same annulus or the ball for the particular solution.
Journal of Mathematical Analysis and Applications, May 1, 2016
In this work, a regularity criterion is proved for local strong solutions of the Navier-Stokes eq... more In this work, a regularity criterion is proved for local strong solutions of the Navier-Stokes equations in the presence of mass diffusion.
In this note, a logarithmically improved regularity criterion for the Navier-Stokes equations is ... more In this note, a logarithmically improved regularity criterion for the Navier-Stokes equations is established in terms of the pressure in the nonhomogeneous Besov space.
In this work, a regularity criterion is proved for local strong solutions of the Navier-Stokes eq... more In this work, a regularity criterion is proved for local strong solutions of the Navier-Stokes equations in the presence of mass diffusion.
We derive various anisotropic versions of the Troisi inequality for the study of the conditional ... more We derive various anisotropic versions of the Troisi inequality for the study of the conditional regularity of solutions to the Navier–Stokes equations. We impose additional conditions on one or two non-diagonal entries of the velocity gradient, namely or and . Formulating our results in the frame of the anisotropic Lebesgue spaces we improve two recent results from the literature. Further we present two new criteria.
Nonlinear Analysis-real World Applications, Jun 1, 2017
Abstract We study the regularity criteria for the incompressible Navier–Stokes equations in the w... more Abstract We study the regularity criteria for the incompressible Navier–Stokes equations in the whole space R 3 based on one velocity component, namely u 3 , ∇ u 3 and ∇ 2 u 3 . We use a generalization of the Troisi inequality and anisotropic Lebesgue spaces and prove, for example, that the condition ∇ u 3 ∈ L β ( 0 , T ; L p ) , where 2 / β + 3 / p = 7 / 4 + 1 / ( 2 p ) and p ∈ ( 2 , ∞ ] , yields the regularity of u on ( 0 , T ] .
Journal of Mathematical Fluid Mechanics, Jun 13, 2019
In this paper we establish regularity conditions for the three dimensional incompressible Navier-... more In this paper we establish regularity conditions for the three dimensional incompressible Navier-Stokes equations in terms of one entry of the velocity gradient tensor, say for example, ∂ 3 u 3. We show that if ∂ 3 u 3 satisfies certain integrable conditions with respect to time and space variables in anisotropic Lebesgue spaces, then a Leray-Hopf weak solution is actually regular. The anisotropic Lebesgue space helps us to almost reach the Prodi-Serrin level 2 in certain special case. Moreover, regularity conditions on non-diagonal element of gradient tensor ∂ 1 u 3 are also established, which covers some previous literature.
Journal of Mathematical Analysis and Applications, Feb 1, 2018
We prove, among others, the following regularity criterion for the solutions to the Navier-Stokes... more We prove, among others, the following regularity criterion for the solutions to the Navier-Stokes equations: If u is a global weak solution satisfying the energy inequality and ω = ∇ × u, then u is regular on (0, T), T > 0, if two components of ω belong to the space L q (0, T ; Ḃ −3/p ∞,∞) for p ∈ (3, ∞) and 2/q + 3/p = 2. This result is an improvement of the results presented by Chae and Choe (1999) [7] or Zhang and Chen (2005) [38]. Our method of the proof uses a suitable application of the Bony decomposition and can also be used for the proofs of some other kin criteria. One such example is presented in Appendix.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Some results on regularity of weak solutions to the Navier-Stokes equations published recently in... more Some results on regularity of weak solutions to the Navier-Stokes equations published recently in [3] follow easily from a classical theorem on compact operators. Further, weak solutions of the Navier-Stokes equations in the space L2(0, T, W 1,3(Ω)3) are regular.
We study the large time behavior of solutions to the Navier-Stokes equations. The emphasis is put... more We study the large time behavior of solutions to the Navier-Stokes equations. The emphasis is put on the rate of the decay of the solutions and their behavior in the frequency space (see [1] and [2]).
Nonlinear Analysis: Theory, Methods & Applications, 2014
We improve a regularity criterion for the solutions to the Navier-Stokes equations in the full th... more We improve a regularity criterion for the solutions to the Navier-Stokes equations in the full three-dimensional space involving the gradient of one velocity component. Revising the method used in Pokorný and Zhou (2009, 2010), we show that a weak solution u is regular on (0, T) provided that ∇u 3 ∈ L t
Nonlinear Analysis: Theory, Methods & Applications, 2014
Abstract We study the lower bounds of decay rates for turbulent solutions to the Navier–Stokes eq... more Abstract We study the lower bounds of decay rates for turbulent solutions to the Navier–Stokes equations in the norms of Besov spaces. We focus on solutions satisfying 0 lim inf t → ∞ t γ ‖ u ( t ) ‖ ≤ lim sup t → ∞ t γ ‖ u ( t ) ‖ ∞ for some γ ∈ ( 0 , 5 / 4 ] and prove among others that such solutions, measured in the norm of the Besov space B 2 , ∞ − 2 γ , are estimated for large times from below by some positive constant. These estimates stem from sufficiently fast rate of large time energy concentration in low frequencies occurring in the studied solutions. Our results have simple proofs and improve and extend substantially the results published by Miyakawa (2002) and by Schonbek and Wiegner (1996).
Journal of Mathematical Analysis and Applications, 2013
Abstract In the paper we study the large-time behavior of solutions to the Navier–Stokes equation... more Abstract In the paper we study the large-time behavior of solutions to the Navier–Stokes equations in the frequency space. We describe in detail the large-time energy concentration which occurs in every (turbulent) solution. If the energy of the solution decreases exponentially then it concentrates in frequencies localized in an annulus in the frequency space. The annulus can be taken arbitrarily narrow and its diameter determines the rate of the exponential decay. All the other solutions are characterized by the concentration of the energy in the frequencies localized in a ball with an arbitrarily small diameter centered at the origin of the coordinates. It will follow from the presented results that the frequencies outside the annulus or the ball and especially the higher frequencies die out very quickly. We will further observe the concentration occurring in any time derivative of the solution or in the vorticity and its time derivatives with the same annulus or the ball for the particular solution.
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