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We provide a new ε -condition for the vorticity of a suitable weak solution to the Navier–Stokes equations that leads to partial regularity. This refines the well known limsup condition of the Caffarelli-Kohn-Nirenberg Theorem by a new condition on the vorticity, replacing limsup by a suitable range of the radius r of the parabolic cylinders. As a consequence, the partial regularity is obtained directly from this ε -condition of the vorticity without relying on the ε -condition of the velocity. Furthermore, by the local nature of the method this result holds for any local suitable weak solution of the Navier–Stokes equations in a general domain. [ABSTRACT FROM AUTHOR]

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]

In this paper we study the Liouville type problem for stationary magnetohydrodynamic equations in R n . We show that under certain anisotropic integrability conditions on the components of the velocity and the magnetic field the solution is trivial. [ABSTRACT FROM AUTHOR]

We remove (α , λ) -discretely self-similar blow up for solutions to the Euler equations for α ≥ 3 2 , for which we allow sublinear growth for the profile. More precisely, we show that there are only spatial constant (α , λ) -discretely self-similar solutions v = c (t) having the sublinear growth at the infinity. For the proof, we establish a new a priori L loc 2 (R 3) estimate for the 3D Euler equations. [ABSTRACT FROM AUTHOR]

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 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 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]

In this paper, we prove a Liouville-type theorem for the stationary MHD and the stationary Hall-MHD systems. Assuming suitable growth condition at infinity for the mean oscillations for the potential functions, we show that the solutions are trivial. These results generalize the previous results obtained by two of the current authors in Chae and Wolf (J Differ Equ 295:233–248, 2021). To prove our main theorems, we use a refined iteration argument. [ABSTRACT FROM AUTHOR]

In this paper we prove a Liouville type theorem for the stationary equations of a non-Newtonian fluid in R 3 with the viscous part of the stress tensor A p (u) = div (| D (u) | p − 2 D (u)) , where D (u) = 1 2 (∇ u + (∇ u) ⊤) and 9 5 < p < 3. We consider a weak solution u ∈ W l o c 1 , p (R 3) and its potential function V = (V i j) ∈ W l o c 2 , p (R 3) , i.e. ∇ ⋅ V = u. We show that there exists a constant s 0 = s 0 (p) such that if the L s mean oscillation of V for s > s 0 satisfies a certain growth condition at infinity, then the velocity field vanishes. Our result includes the previous results [5,6] as particular cases. [ABSTRACT FROM AUTHOR]

In this paper, we derive kinematic relations for quantities involving the rate of strain tensor and the Hessian of the pressure for solutions of the 3D Euler equations and the 2D Boussinesq equations. Using these kinematic relations, we prove new blow-up criteria and obtain conditions for the absence of type I singularity for these equations. We obtain both global and localized versions of the results. Some of the new blow-up criteria and type I conditions improve previous results of Chae and Constantin (Int Math Res Notices rnab014, 2021). [ABSTRACT FROM AUTHOR]

In this paper we prove a Liouville type theorem for the stationary magnetohydrodynamics (MHD) system in R 3. Let (v , B , p) be a smooth solution to the stationary MHD equations in R 3. We show that if there exist smooth matrix valued potential functions Φ , Ψ such that ∇ ⋅ Φ = v and ∇ ⋅ Ψ = B , whose L 6 mean oscillations have certain growth condition near infinity, namely ⨍ B (r) | Φ − Φ B (r) | 6 d x + ⨍ B (r) | Ψ − Ψ B (r) | 6 d x ≤ C r ∀ 1 < r < + ∞ , then v = B = 0 and p =constant. With additional assumption of r − 8 ∫ B (r) | B − B B (r) | 6 d x → 0 as r → + ∞ , similar result holds also for the Hall-MHD system. [ABSTRACT FROM AUTHOR]

In this paper we prove local in time well-posedness for the incompressible Euler equations in R n for the initial data in L 1 (1) 1 (R n) , which corresponds to a critical case of the generalized Campanato spaces L q (N) s (R n). The space is studied extensively in our companion paper [9] , and in the critical case we have embeddings B ∞ , 1 1 (R n) ↪ L 1 (1) 1 (R n) ↪ C 0 , 1 (R n) , where B ∞ , 1 1 (R n) and C 0 , 1 (R n) are the Besov space and the Lipschitz space respectively. In particular L 1 (1) 1 (R n) contains non- C 1 (R n) functions as well as linearly growing functions at spatial infinity. We can also construct a class of simple initial velocity belonging to L 1 (1) 1 (R n) , for which the solution to the Euler equations blows up in finite time. [ABSTRACT FROM AUTHOR]

In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in R 3 , having the diffusion term A p (u) = ∇ · (| D (u) | p - 2 D (u)) with D (u) = 1 2 (∇ u + (∇ u) ⊤) , 3 / 2 < p < 3 . In the case 3 / 2 < p ≤ 9 / 5 , we show that a suitable weak solution u ∈ W 1 , p (R 3) satisfying lim inf R → ∞ | u B (R) | = 0 is trivial, i.e., u ≡ 0 . On the other hand, for 9 / 5 < p < 3 we prove the following Liouville type theorem: if there exists a matrix valued function V = { V ij } such that ∂ j V ij = u i (summation convention), whose L 3 p 2 p - 3 mean oscillation has the following growth condition at infinity, ∫ - B (r) | V - V B (r) | 3 p 2 p - 3 d x ≤ C r 9 - 4 p 2 p - 3 ∀ 1 < r < + ∞ , then u ≡ 0 . [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 prove continuation in time of the local smooth solutions satisfying various Type I conditions for the 2D inviscid Boussinesq equations. [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 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 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 study the regularity of weak solutions to the 3D valued stationary Hall magnetohydrodynamic equations on $${\mathbb{R}^2}$$ . We prove that every weak solution is smooth. Furthermore, we prove a Liouville type theorem for the Hall equations. [ABSTRACT FROM AUTHOR]

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]

In this paper we show that the incompressible Hall-MHD system without resistivity is not globally in time well-posed in any Sobolev space H m ( R 3 ) for any m > 7 2 . Namely, either the system is locally ill-posed in H m ( R 3 ) , or it is locally well-posed, but there exists an initial data in H m ( R 3 ) , for which the H m ( R 3 ) norm of solution blows-up in finite time if m > 7 2 . In the latter case we choose an axisymmetric initial data u 0 ( x ) = u 0 r ( r , z ) e r + b 0 z ( r , z ) e z and B 0 ( x ) = b 0 θ ( r , z ) e θ , and reduce the system to the axisymmetric setting. If the convection term survives sufficiently long time, then the Hall term generates the singularity on the axis of symmetry and we have lim ⁡ sup t → t ⁎ ⁡ sup z ∈ R ⁡ | ∂ z ∂ r b θ ( r = 0 , z ) | = ∞ for some t ⁎ > 0 , which will also induce a singularity in the velocity field. [ABSTRACT FROM AUTHOR]

We study partial regularity of suitable weak solutions of the steady Hall magnetohydrodynamics equations in a domain $${\Omega \subset \mathbb{R}^3}$$ . In particular, we prove that the set of possible singularities of the suitable weak solution has Hausdorff dimension at most one. Moreover, in the case $${\Omega=\mathbb{R}^3}$$ , we show that the set of possible singularities is compact. [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]

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 prove the global well posedness of the continuously stratified inviscid quasi-geostrophic equations in $$\mathbb {R}^3$$ . [ABSTRACT FROM AUTHOR]

In this note, we show that Luo-Hou's ansatz for the self-similar solution to the axisymmetric solution to the 3D Euler equations leads to triviality of the solution under suitable decay condition of the blow-up profile. The equations for the blow-up profile reduces to an over-determined system of partial differential equations, whose only solution with decay is the trivial solution. We also propose a generalization of Luo-Hou's ansatz. Using the vanishing of the normal velocity at the boundary, we show that this generalized self-similar ansatz also leads to a trivial solution. These results show that the self-similar ansatz may be valid either only in a time-dependent region which shrinks to the boundary circle at the self-similar rate, or under different boundary conditions at spatial infinity of the self-similar profile. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we establish an optimal blow-up criterion for classical solutions to the incompressible resistive Hall-magnetohydrodynamic equations. We also prove two global-in-time existence results of the classical solutions for small initial data, the smallness conditions of which are given by the suitable Sobolev and the Besov norms respectively. Although the Sobolev space version is already an improvement of the corresponding result in [4], the optimality in terms of the scaling property is achieved via the Besov space estimate. The special property of the energy estimate in terms of norm is essential for this result. Contrary to the usual MHD the global well-posedness in the dimensional Hall-MHD is wide open. [Copyright &y& Elsevier]

Abstract: We prove local existence of smooth solutions for large data and global smooth solutions for small data to the incompressible, resistive, viscous or inviscid Hall-MHD model. We also show a Liouville theorem for the stationary solutions. [Copyright &y& Elsevier]

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: We establish temporal decay estimates for weak solutions to the Hall-magnetohydrodynamic equations. With these estimates in hand we obtain algebraic time decay for higher order Sobolev norms of small initial data solutions. [Copyright &y& Elsevier]

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]

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 study the Euler-Poincaré equations in $${\mathbb {R}^N}$$ . We prove local existence of weak solutions in $${W^{2,p}(\mathbb {R}^N), p > N}$$ , and local existence of unique classical solutions in $${H^k (\mathbb {R}^N)}$$ , k > N/2 + 3, as well as a blow-up criterion. For the zero dispersion equation ( α = 0) we prove a finite time blow-up of the classical solution. We also prove that as the dispersion parameter vanishes, the weak solution converges to a solution of the zero dispersion equation with sharp rate as α → 0, provided that the limiting solution belongs to $${C([0, T);H^k(\mathbb {R}^N))}$$ with k > N/2 + 3. For the stationary weak solutions of the Euler-Poincaré equations we prove a Liouville type theorem. Namely, for α > 0 any weak solution $${\mathbf{u} \in H^1(\mathbb{R}^N)}$$ is u=0; for α= 0 any weak solution $${\mathbf{u} \in L^2(\mathbb{R}^N)}$$ is u=0. [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: This paper investigates the global (in time) regularity of solutions to a system of equations that generalize the vorticity formulation of the 2D Boussinesq–Navier–Stokes equations. The velocity in this system is related to the vorticity through the relations and , which reduces to the standard velocity–vorticity relation when . When either or , the velocity is more singular. The “quasi-velocity” determined by satisfies an equation of very special structure. This paper establishes the global regularity and uniqueness of solutions for the case when and . In addition, the vorticity is shown to be globally bounded in several functional settings such as for in a suitable range. [Copyright &y& Elsevier]

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]

Any classical solution of the two-dimensional incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component u of the velocity field u is determined by the scalar θ through $${u_j =\mathcal{R}\Lambda^{-1}P(\Lambda) \theta}$$ , where $${\mathcal{R}}$$ is a Riesz transform and Λ = (−Δ). The two-dimensional Euler vorticity equation corresponds to the special case P(Λ) = I while the SQG equation corresponds to the case P(Λ) = Λ. We develop tools to bound $${\|\nabla u||_{L^\infty}}$$ for a general class of operators P and establish the global regularity for the Loglog-Euler equation for which P(Λ) = (log( I + log( I − Δ))) with 0 ≦ γ ≦ 1. In addition, a regularity criterion for the model corresponding to P(Λ) = Λ with 0 ≦ β ≦ 1 is also obtained. [ABSTRACT FROM AUTHOR]