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In this paper we exclude the scenario of rotated discretely self-similar singularities of the 3D incompressible Euler equations in the case when the singularity point is isolated and the vorticity satisfies appropriate decay condition at spatial infinity. For the proof we use the maximum principle for the transformed self-similar equations. [ABSTRACT FROM AUTHOR]

We show that the log-Hölder coefficients of a solution to the transport equation is preserved in time. [ABSTRACT FROM AUTHOR]

We prove a blow up criterion in terms of the Hessian of the pressure of smooth solutions |$u\in C([0, T); W^{2,q} (\mathbb R^3))$|⁠ , |$q>3$| of the incompressible Euler equations. We show that a blow up at |$t=T$| happens only if $$\begin{align*} &\int_0 ^T \int_0 ^t \left\{\int_0 ^s \|D^2 p (\tau)\|_{L^\infty} \textrm{d}\tau \exp \left(\int_{s} ^t \int_0 ^{\sigma} \|D^2 p (\tau)\|_{L^\infty} \textrm{d}\tau \textrm{d}\sigma \right) \right\} \textrm{d}s \textrm{d}t \, = +\infty.\end{align*}$$ As consequences of this criterion we show that there is no blow up at |$t=T$| if |$ \|D^2 p(t)\|_{L^\infty } \le \frac{c}{(T-t)^2}$| with |$c<1$| as |$t\nearrow T$|⁠. Under the additional assumption of |$\int _0 ^T \|u(t)\|_{L^\infty (B(x_0, \rho))} \textrm{d}t <+\infty $|⁠ , we obtain localized versions of these results. [ABSTRACT FROM AUTHOR]

We exclude Type I blow-up, which occurs in the form of atomic concentrations of the L 2 norm for the solution of the 3D incompressible Euler equations. As a corollary we prove nonexistence of discretely self-similar blow-up in the energy conserving scale. [ABSTRACT FROM AUTHOR]

In this paper we obtain new local blow-up criterion for smooth axisymmetric solutions to the three dimensional incompressible Euler equation. If the vorticity satisfies ∫ 0 t ∗ (t ∗ - t) ‖ ω (t) ‖ L ∞ (B (x ∗ , R 0)) d t < + ∞ for a ball B (x ∗ , R 0) away from the axis of symmetry, then there exists no singularity at t = t ∗ in the torus T (x ∗ , R) generated by rotation of the ball B (x ∗ , R 0) around the axis. This implies that possible singularity at t = t ∗ in the torus T (x ∗ , R) is excluded if the vorticity satisfies the blow-up rate ‖ ω (t) ‖ L ∞ (T (x ∗ , R)) = O 1 (t ∗ - t) γ as t → t ∗ , where γ < 2 , and the torus T (x ∗ , R) does not touch the axis. [ABSTRACT FROM AUTHOR]

We study systems interpolating between the 3D incompressible Euler and electron–MHD equations, given by ∂ t B + V ⋅ ∇ B = B ⋅ ∇ V , V = − ∇ × (− Δ) − a B , ∇ ⋅ B = 0 , where B is a time-dependent vector field in R 3. Under the assumption that the initial data is axi-symmetric without swirl, we prove local well-posedness of Lipschitz continuous solutions and existence of traveling waves in the range 1 / 2 < a < 1. These generalize the corresponding results for the 3D axisymmetric Euler equations and should be useful in the study of stability and instability for axisymmetric solutions. [ABSTRACT FROM AUTHOR]

We study the backward discretely self-similar solutions to the generalized Euler equations, where usual Biot-Savart kernel representing the velocity in terms of the vorticity is replaced by various power of the laplacian. Under milder sufficient conditions than the previous results on the Euler equations we show that Liouville type theorems hold for the time periodic solutions to the profile equations, which means that there exists no backward discretely self-similar solutions to the generalized Euler system having such profile. [ABSTRACT FROM AUTHOR]

We prove local non blow-up theorems for the 3D incompressible Euler equations under local Type I conditions. More specifically, for a classical solution v∈L∞(-1,0;L2(B(x0,r)))∩Lloc∞(-1,0;W1,∞(B(x0,r))) of the 3D Euler equations, where B(x0,r) is the ball with radius r and the center at x0, if the limiting values of certain scale invariant quantities for a solution v(·, t) as t→0 are small enough, then ∇v(·,t) does not blow-up at t = 0 in B(x0, r). [ABSTRACT FROM AUTHOR]

We consider 1D equations with nonlocal velocity of the form w t + u w x + δ u x w = − ν Λ γ w where the nonlocal velocity u is given by (1) u = ( 1 − ∂ x x ) − β w , β > 0 or (2) u = H w ( H is the Hilbert transform). In this paper, we address several local well-posedness results with blow-up criteria for smooth initial data. We then establish the global well-posedness by using the blow-up criteria. [ABSTRACT FROM AUTHOR]

We present a simple, short, and elementary proof that if v is a Beltrami flow with a finite energyin R3,then v=0. In the case of the Beltrami flows satisfying v∈Lloc∞(R3)∩Lq(R3) with q ∈[2,3),or [ABSTRACT FROM AUTHOR]

We prove a unique continuation type theorem for the self-similar Euler equations in R 3 , assuming the time periodicity. Namely, if a time periodic solution V ( y , s ) of the time dependent self-similar Euler equations has the property that V ( 0 , s ) = 0 for all s ∈ [ 0 , S 0 ] , where S 0 is the time period, and y = 0 is a local extremal point of V ( y , s ) near the origin, then, V ( y , s ) = 0 for all ( y , s ) ∈ R 3 × [ 0 , S 0 ] . A similar result holds for more general system with arbitrary coefficients, and also for the inviscid incompressible magnetohydrodynamic (MHD) system. As a consequence we obtain new criteria for the absence of the discretely self-similar singularities for the 3D Euler equations and the inviscid 3D MHD. [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]

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]

The paper addresses the question of the existence of a locally self-similar blow-up for the incompressible Euler equations. Several exclusion results are proved based on the L-condition for velocity or vorticity and for a range of scaling exponents. In particular, in N dimensions if in self-similar variables $${u \in L^p}$$ and $${u \sim \frac{1}{t^{\alpha/(1+\alpha)}}}$$, then the blow-up does not occur, provided $${\alpha > N/2}$$ or $${-1 < \alpha \leq N\,/p}$$. This includes the L case natural for the Navier-Stokes equations. For $${\alpha = N\,/2}$$ we exclude profiles with asymptotic power bounds of the form $${ |y|^{-N-1+\delta} \lesssim |u(y)| \lesssim |y|^{1-\delta}}$$. Solutions homogeneous near infinity are eliminated, as well, except when homogeneity is scaling invariant. [ABSTRACT FROM AUTHOR]

The global regularity problem concerning the inviscid SQG and the 3D Euler equations remains an outstanding open question. This paper presents several geometric observations on solutions of these equations. One observation stems from a relation between what we call Eulerian and Lagrangian deformations and reflects the alignment of the stretching directions of these deformations and the tangent direction of the level curves for the SQG equation. Various spatial symmetries in solutions to the 3D Euler equations are exploited. In addition, two observations on the curvature of the level curves of the SQG equation are also included. [ABSTRACT FROM AUTHOR]

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]

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: We obtain new continuation principle of the local classical solutions of the 3D Euler equations, where the regularity condition of the direction field of the vorticiy and the integrability condition of the magnitude of the vorticity are incorporated simultaneously. The regularity of the vorticity direction field is most appropriately measured by the Triebel–Lizorkin type of norm. Similar result is also obtained for the inviscid 2D quasi-geostrophic equation. [Copyright &y& Elsevier]

Several inviscid models in hydrodynamics and geophysics such as the incompressible Euler vorticity equations, the surface quasi-geostrophic equation, and the Boussinesq equations are not known to have even local well-posedness in the corresponding borderline Sobolev spaces. Here Hs is referred to as a borderline Sobolev space if the L∞-norm of the gradient of the velocity is not bounded by the Hs-norm of the solution but by the Hs-norm for any s>s. This paper establishes the local well-posedness of the logarithmically regularized counterparts of these inviscid models in the borderline Sobolev spaces. [ABSTRACT FROM AUTHOR]