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We obtain a new regularity criterion in terms of the oscillation of time derivative of the pressure for the 3D Navier–Stokes equations in a domain D ⊂ R 3 . The key observation for the proof is that the head pressure defined by Q ¯ (x , t) = 1 2 | v (x , t) | 2 + p ¯ (x , t) , where p ¯ (x , t) = p (x , t) - ∫ t 0 t sup y ∈ Ω (∂ s p (y , s) - | ω (y , s) | 2) d s with p(x, t) the pressure and ω the vorticity satisfies parabolic maximum principle in Ω × (t 0 , T) with Ω ⋐ D . [ABSTRACT FROM AUTHOR]

We prove the existence of a forward discretely self-similar solutions to the Navier–Stokes equations in R 3 × ( 0 , + ∞ ) for a discretely self-similar initial velocity belonging to L l o c 2 ( R 3 ) . [ABSTRACT FROM AUTHOR]

We consider the Navier–Stokes equations in R 3 subject to the initial condition with initial velocity field in L loc 2 (R 3) such that lim sup R → + ∞ R − 1 ‖ u 0 ‖ L 2 (B (R)) < + ∞. Our aim is to show the local existence of a weak solution, global existence of weak solution if C = 0 and the partial regularity in the sense of Caffarelli–Kohn–Nirenberg. [ABSTRACT FROM AUTHOR]

In this brief note we study the Liouville type problem for the stationary Navier–Stokes equations in R 3. We show that under certain anisotropic integrability conditions on the components of the velocity implies that the solution is trivial. [ABSTRACT FROM AUTHOR]

We study the scenario of discretely self-similar blowup for Navier–Stokes equations. We prove that at the possible blow-up time such solutions have only one point singularity. In case of the scaling parameter λ near 1, we remove the singularity. [ABSTRACT FROM PUBLISHER]

We prove Liouville type theorems for the self-similar solutions to the Navier-Stokes equations. One of our results generalizes the previous ones by Nečas-Ru̇žička-Šverák and Tsai. Using a Liouville type theorem, we also remove a scenario of asymptotically self-similar blow-up for the Navier-Stokes equations with the profile belonging to $${L^{p, \infty} (\mathbb{R}^3)}$$ with $${p > \frac{3}{2}}$$ . [ABSTRACT FROM AUTHOR]

We prove geometrically improved version of Prodi–Serrin type blow-up criterion. Let v and ω be the velocity and the vorticity of solutions to the 3D Navier–Stokes equations and denote { f } + = max { f , 0 } , Q T = R 3 × ( 0 , T ) . If { ( v × ω ∣ ω ∣ ) ⋅ Λ β v ∣ Λ β v ∣ } + ∈ L x , t γ , α ( Q T ) with 3 / γ + 2 / α ≤ 1 for some γ > 3 and 1 ≤ β ≤ 2 , then the local smooth solution v of the Navier–Stokes equations on ( 0 , T ) can be continued to ( 0 , T + δ ) for some δ > 0 . We also prove localized version of a special case of this. Let v be a suitable weak solution to the Navier–Stokes equations in a space–time domain containing z 0 = ( x 0 , t 0 ) , let Q z 0 , r = B x 0 , r × ( t 0 − r 2 , t 0 ) be a parabolic cylinder in the domain. We show that if either { ( v × ω ∣ ω ∣ ) ⋅ ∇ × ω ∣ ∇ × ω ∣ } + ∈ L x , t γ , α ( Q z 0 , r ) with 3 γ + 2 α ≤ 1 , or { ( v ∣ v ∣ × ω ) ⋅ ∇ × ω ∣ ∇ × ω ∣ } + ∈ L x , t γ , α ( Q z 0 , r ) with 3 γ + 2 α ≤ 2 , ( γ ≥ 2 , α ≥ 2 ), then z 0 is a regular point for v . This improves previous local regularity criteria for the suitable weak solutions. [ABSTRACT FROM AUTHOR]

In this paper we prove three different Liouville type theorems for the steady Navier–Stokes equations in R 3 . In the first theorem we improve logarithmically the well-known L 9 2 ( R 3 ) result. In the second theorem we present a sufficient condition for the trivially of the solution ( v = 0 ) in terms of the head pressure, Q = 1 2 | v | 2 + p . The imposed integrability condition here has the same scaling property as the Dirichlet integral. In the last theorem we present Fubini type condition, which guarantee v = 0 . [ABSTRACT FROM AUTHOR]

We study scenarios of self-similar type blow-up for the incompressible Navier–Stokes and the Euler equations. The previous notions of the discretely (backward) self-similar solution and the asymptotically self-similar solution are generalized to the locally asymptotically discretely self-similar solution. We prove that there exists no such locally asymptotically discretely self-similar blow-up for the 3D Navier–Stokes equations if the blow-up profile is a time periodic function belonging to C 1 ( R ; L 3 ( R 3 ) ∩ C 2 ( R 3 ) ) . For the 3D Euler equations we show that the scenario of asymptotically discretely self-similar blow-up is excluded if the blow-up profile satisfies suitable integrability conditions. [ABSTRACT FROM AUTHOR]

We study a nonlinear parabolic system for a time dependent solenoidal vector field on ℝ3. The nonlinear term of these new model equations is obtained slightly modifying that of the Navier-Stokes equations. The system has the same scaling property and Galilean invariance as the Navier-Stokes equations. For such a system we prove the global regularity for smooth initial data. [ABSTRACT FROM AUTHOR]

We study a nonlinear parabolic system for a time dependent solenoidal vector field on ℝ3. The nonlinear term of these new model equations is obtained slightly modifying that of the Navier-Stokes equations. The system has the same scaling property and Galilean invariance as the Navier-Stokes equations. For such a system we prove the global regularity for smooth initial data. [ABSTRACT FROM AUTHOR]

Abstract: Let be a smooth solution pair of the velocity and the pressure for the Navier–Stokes (Euler) equations on , . We set the Bernoulli function . Under suitable decay conditions at infinity for we prove that for almost all and defined on there holds where is the vorticity. This shows that, in each region squeezed between two levels of the Bernoulli function, besides the energy dissipation due to the enstrophy, the energy flows into the region through the level hypersurface having the higher level, and the energy flows out of the region through the level hypersurface with the lower level. Passing and , we recover the well-known energy equality, . A weaker version of the above equality under the weaker decay assumption of the solution at spatial infinity is also derived. The stationary version of the equality implies the previous Liouville type results on the Navier–Stokes equations. [Copyright &y& Elsevier]

In this paper we study the Liouville-type properties for solutions to the steady incompressible Euler equations with forces in $${\mathbb {R}^N}$$ . If we assume 'single signedness condition' on the force, then we can show that a $${C^1 (\mathbb {R}^N)}$$ solution ( v, p) with $${|v|^2+ |p| \in L^{\frac{q}{2}}(\mathbb {R}^N),\,q \in (\frac{3N}{N-1}, \infty)}$$ is trivial, v = 0. For the solution of the steady Navier-Stokes equations, satisfying $${v(x) \to 0}$$ as $${|x| \to \infty}$$ , the condition $${\int_{\mathbb {R}^3} |\Delta v|^{\frac{6}{5}} dx < \infty}$$ , which is stronger than the important D-condition, $${\int_{\mathbb {R}^3} |\nabla v|^2 dx < \infty}$$ , but both having the same scaling property, implies that v = 0. In the appendix we reprove Theorem 1.1 (Chae, Commun Math Phys 273:203-215, ), using the self-similar Euler equations directly. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we prove the Liouville type theorem for the stationary Navier–Stokes equations on . More specifically, if a solution to the stationary Navier–Stokes system satisfies additional conditions characterized by the decays near infinity and by the oscillation, then we show that . [Copyright &y& Elsevier]

In this paper we derive various sufficient conditions on the pressure for vanishing velocity in the incompressible Navier-Stokes and the Euler equations in ℝ N . [ABSTRACT FROM PUBLISHER]

Abstract: We prove Liouville type theorems for weak solutions of the Navier–Stokes and the Euler equations. In particular, if the pressure satisfies with , then the corresponding velocity should be trivial, namely on . In particular, this is the case when , where , , the Hardy space. On the other hand, we have equipartition of energy over each component, if with . Similar results hold also for the magnetohydrodynamic equations. [Copyright &y& Elsevier]

We generalize and localize the previous results by the author on the study of self-similar singularities for the 3D Euler equations. More specifically we extend the restriction theorem for the representation for the vorticity of the Euler equations in a bounded domain, and localize the results on asymptotically self-similar singularities. We also present progress towards relaxation of the decay condition near infinity for the vorticity of the blow-up profile to exclude self-similar blow-ups. The case of the generalized Navier-Stokes equations having the laplacian with fractional powers is also studied. We apply the similar arguments to the other incompressible flows, e.g. the surface quasi-geostrophic equations and the 2D Boussinesq system both in the inviscid and supercritical viscous cases. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the Euler and the Navier–Stokes equations, where the sense of convergence and self-similarity are considered in various generalized senses. We improve substantially, in particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities. Generalization of the self-similar transforms is also considered, and by appropriate choice of the parameterized transform we obtain new a priori estimates for the Euler and the Navier–Stokes equations depending on a free parameter. [Copyright &y& Elsevier]

Abstract: We prove new a priori estimates for the 3D Euler, the 3D Navier–Stokes and the 2D quasi-geostrophic equations by the method of similarity transforms. [Copyright &y& Elsevier]

Abstract: In this paper we study 1D equations with nonlocal flux. These models have resemblance of the 2D quasi-geostrophic equation. We show the existence of singularities in finite time and construct explicit solutions to the equations where the singularities formed are shocks. For the critical viscosity case we show formation of singularities and global existence of solutions for small initial data. [Copyright &y& Elsevier]

In this paper we prove a Liouville type theorem for the stationary Navier–Stokes equations in R 3 . Let V = (V ij) be a potential function of a smooth solution u, which means u = ∇ · V . We show that if there exists 3 < s < + ∞ such that the L s mean oscillation of the potential function has certain growth condition near infinity, namely 1 | B (r) | ∫ B (r) | V - V B (r) | s d x ≤ C r min { s - 3 3 , s 6 } ∀ 1 < r < + ∞ , then u ≡ 0 . [ABSTRACT FROM AUTHOR]