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We prove strong nonlinear illposedness results for the generalized SQG equation $$\partial_t \theta + \nabla^\perp \Gamma[\theta] \cdot \nabla \theta = 0 $$ in any sufficiently regular Sobolev spaces, when $\Gamma$ is a singular in the sense that its symbol satisfies $|\Gamma(\xi)|\to\infty$ as $|\xi|\to\infty$ with some mild regularity assumptions. The key mechanism is degenerate dispersion, i.e., the rapid growth of frequencies of solutions around certain shear states, and the robustness of our method allows one to extend linear and nonlinear illposedness to fractionally dissipative systems, as long as the order of dissipation is lower than that of $\Gamma$. Our illposedness results are completely sharp in view of various existing wellposedness statements as well as those from our companion paper. Key to our proofs is a novel construction of degenerating wave packets for the class of linear equations $$\partial_t \phi + ip(t,X,D)\phi = 0$$ where $p(t,X,D)$ is a pseudo-differential operator which is self-adjoint in $L^2$, degenerate, and dispersive. Degenerating wave packets are approximate solutions to the above linear equation with spatial and frequency support localized at $(X(t),\Xi(t))$, which are solutions to the bicharacteristic ODE system associated with $p(t,x,\xi)$. These wave packets explicitly show degeneration as $X(t)$ approaches a point where $p$ vanishes, which in particular allows us to prove illposedness in topologies finer than $L^2$. While the equation for the wave packet can be formally obtained from a Taylor expansion of the symbol near $\xi=\Xi(t)$, the difficult part is to rigorously control the error in sufficiently long timescales, which is obtained by sharp estimates for not only degenerating wave packets but also for oscillatory integrals which naturally appear in the error estimate., Comment: 74 pages

We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani, $$\partial_t \theta - \nabla^\perp \log(10+(-\Delta)^{\frac12})\theta \cdot \nabla \theta = 0 ,$$ and establish local existence and uniqueness of smooth solutions in the scale of Sobolev spaces with exponent decreasing with time. Such a decrease of the Sobolev exponent is necessary, as we have shown in the companion paper that the problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence of the Sobolev exponent can be removed when there is a dissipation term strictly stronger than log. These results improve wellposedness statements by Chae, Constantin, C\'{o}rdoba, Gancedo, and Wu in \cite{CCCGW}. This well-posedness result can be applied to describe the long-time dynamics of the $\delta$-SQG equations, defined by $$\partial_t \theta + \nabla^\perp (10+(-\Delta)^{\frac12})^{-\delta}\theta \cdot \nabla \theta = 0,$$ for all sufficiently small $\delta>0$ depending on the size of the initial data. For the same range of $\delta$, we establish global well-posedness of smooth solutions to the logarithmically dissipative counterpart: $$\partial_t \theta + \nabla^\perp (10+(-\Delta)^{\frac12})^{-\delta}\theta \cdot \nabla \theta + \log(10+(-\Delta)^{\frac12})\theta = 0.$$, Comment: 21 pages

We study systems interpolating between the 3D incompressible Euler and electron--MHD equations, given by \begin{equation*} \partial_t B + V \cdot \nabla B = B\cdot \nabla V, \qquad V = -\nabla\times (-\Delta)^{-a} B, \qquad \nabla\cdot B = 0, \end{equation*} where $B$ is a time-dependent vector field in $\mathbb{R}^3$. 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

We introduce an active vector system, which generalizes both the 3D Euler equations and the electron--magnetohydrodynamic equations (E--MHD). We may as well view the system as singularized systems for the 3D Euler equations, in which case the equations of (E--MHD) correspond to the order two more singular one than the 3D Euler equations. The generalized surface quasi-geostrophic equation (gSQG) can be also embedded into a special case of our system when the unknown functions are constant in one coordinate direction. We investigate some basic properties of this system as well as the conservation laws. In the case when the system corresponds up to order one more singular than the 3D Euler equations, we prove local well-posedness in the standard Sobolev spaces. The proof crucially depends on a sharp commutator estimate similar to the one used for (gSQG) in the work of Chae, Constantin, C\'{o}rdoba, Gancedo, and Wu. Since the system covers many areas of both physically and mathematically interesting cases, one can expect that there are various related problems to be investigated, parts of which are discussed here., Comment: 18 pages, to appear in Nonlinearity

We consider the Navier-Stokes equations in $\mathbb{R}^3$ subject to the initial condition with initial velocity field in $L^{2}_{\rm loc} (\mathbb{R}^3)$ such that $\limsup_{R \to +\infty } R^{-1} \|u_{0} \|_{ L^{2}(B(R))} < +\infty$. 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., Comment: Similar result is already obtained by Z. Bradshaw, I. Kukavica and T-P. Tsai (Indiana Univ. Math. J., vol. 7(2022), pp. 191-212

In this paper we address the temperature patch problem of the 2D viscous Boussinesq system without heat diffusion term. The temperature satisfies the transport equation and the initial data of temperature is given in the form of non-constant patch, usually called the temperature front initial data. Introducing a good unknown and applying the method of striated estimates, we prove that our partially viscous Boussinesq system admits a unique global regular solution and the initial $C^{k,\gamma}$ and $W^{2,\infty}$ regularity of the temperature front boundary with $k\in \mathbb{Z}^+ = \{1,2,\cdots\}$ and $\gamma\in (0,1)$ will be preserved for all the time. In particular, this naturally extends the previous work by Danchin $\&$ Zhang (2017) and Gancedo $\&$ Garc\'ia-Ju\'arez (2017). In the proof of the persistence result of higher boundary regularity, we introduce the striated type Besov space $\mathcal{B}^{s,\ell}_{p,r,W}(\mathbb{R}^d)$ and establish a series of refined striated estimates in such a function space, which may have its own interest., Comment: 50 pages. The striated estimates are simplified

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 [6]. To prove our main theorems we use a refined iteration argument., Comment: 14 pages

In this paper we prove a Liouville type theorem for the stationary equations of a non-Newtonian fluid in $\mathbb{R}^3$ with the viscous part of the stress tensor $\mathbf{A}_p(u) = \mathrm{div} ( | \mathbf{D}(u) |^{p-2} \mathbf{D}(u) )$, where $\mathbf{D}(u) = \frac 12 ( \nabla u + ( \nabla u )^{\top})$ and $\frac 95 < p < 3$. We consider a weak solution $u \in W^{1,p}_{loc}(\mathbb{R}^3)$ and its potential function $\mathbf{V} = (V_{ij}) \in W^{2,p}_{loc}(\mathbb{R}^3)$, i.e. $\nabla \cdot \mathbf{V} = u$. We show that there exists a constant $s_0=s_0(p)$ such that if the $L^s$ mean oscillation of $\mathbf{V}$ for $s>s_0$ satisfies a certain growth condition at infinity, then the velocity field vanishes. Our result includes the previous results \cite{CW20, CW19} as particular cases., Comment: 17 pages

We show that the log-H\"{o}lder coefficients of a solution to the transport equation is preserved in time., Comment: 8 pages

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 [3]., Comment: 14 pages

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 $$\int_0 ^T \int_0 ^t \left\{\int_0 ^s \|D^2 p (\tau)\|_{L^\infty} d\tau \exp \left( \int_{s} ^t \int_0 ^{\s} \|D^2 p (\tau)\|_{L^\infty} d\tau d\s \right) \right\}dsdt \, = +\infty.$$ 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 $\int_0 ^T \|u(t)\|_{L^\infty (B(x_0, \rho))} dt <+\infty$, we obtain localized versions of these results., Comment: 10 pages

In this paper we prove Liouville type theorem for the stationary Navier-Stokes equations in $\Bbb R^3$ under the assumptions on the relative decays of velocity, pressure and the head pressure. More precisely, we show that any smooth solution $(u,p)$ of the stationary Navier-Stokes equations satisfying $u(x) \to 0$ as $|x|\to +\infty$ and the condition of finite Dirichlet integral $\int_{\Bbb R^3} | \nabla u|^2 dx <+\infty $ is trivial, if either $|u|/|Q|=O(1)$ or $|p|/|Q| =O(1) $ as $|x|\to \infty$, where $|Q|=\frac12 |u|^2 +p$ is the head pressure., Comment: 9 pages

In this paper we consider the regularity problem of the Navier-Stokes equations in $ \R^{3} $. We show that the Serrin-type condition imposed on one component of the velocity $ u_3\in L^p(0,T; L^q(\R^{3} ))$ satisfying $ \frac{2}{p}+ \frac{3}{q} <1$, $ 3

In this paper we prove local in time well-posedness for the incompressible Euler equations in $\Bbb R^n$ for the initial data in $\mathscr {L}^{ 1}_{ 1(1)}(\mathbb {R}^{n}) $, which corresponds to a critical case of the generalized Campanato spaces $ \mathscr {L}^{ s}_{ q(N)}(\mathbb {R}^{n})$. The space is studied extensively in our companion paper\cite{trans}, and in the critical case we have embeddings $ B^{1}_{\infty, 1} (\Bbb R^n) \hookrightarrow \mathscr {L}^{ 1}_{ 1(1)}(\mathbb {R}^{n}) \hookrightarrow C^{0, 1} (\Bbb R^n)$, where $B^{1}_{\infty, 1} (\Bbb R^n)$ and $ C^{0, 1} (\Bbb R^n)$ are the Besov space and the Lipschitz space respectively. In particular $\mathscr {L}^{ 1}_{ 1(1)}(\mathbb {R}^{n}) $ contains non-$C^1(\Bbb R^n)$ functions as well as linearly growing functions at spatial infinity. We can also construct a class of simple initial velocity belonging to $ \mathscr {L}^{ 1}_{ 1(1)}(\mathbb {R}^{n})$, for which the solution to the Euler equations blows up in finite time., Comment: 50 pages

In this paper we study the transport equation in $\mathbb{R}^n \times (0,T)$, $T >0$, \[ \partial _t f + v\cdot \nabla f = g, \quad f(\cdot ,0)= f_0 \quad \text{in}\quad \mathbb{R}^n \] in generalized Campanato spaces $\mathscr{L}^s_{ q(p, N)}(\mathbb{R}^n)$. The critical case is particularly interesting, and is applied to the local well-posedness problem in a space close to the Lipschitz space in our companion paper\cite{cw}. More specifically, in the critical case $s=q=N=1$ we have the embedding relations, $B^1_{\infty, 1}(\Bbb R^n) \hookrightarrow \mathscr{L}^{ 1}_{ 1(p, 1)}(\mathbb{R}^n) \hookrightarrow C^{0, 1} (\Bbb R^n)$, where $B^1_{\infty, 1} (\Bbb R^n)$ and $C^{0, 1} (\Bbb R^n)$ are the Besov space and the Lipschitz space respectively. For $f_0\in \mathscr {L}^{ 1}_{ 1(p, 1)}(\mathbb {R}^{n})$, $v\in L^1(0,T; \mathscr {L}^{ 1}_{ 1(p, 1)}(\mathbb {R}^{n}))),$ and $ g\in L^1(0,T; \mathscr {L}^{ 1}_{ 1(p, 1)}(\mathbb {R}^{n})))$, we prove the existence and uniqueness of solutions to the transport equation in $ L^\infty(0,T; \mathscr {L}^{ 1}_{ 1(p, 1)}(\mathbb {R}^{n}))$ such that \[ \|f\|_{L^\infty(0,T; \mathscr{L}^1_{ 1(p, 1)} (\mathbb{R}^n)))} \le C \Big( \|v\|_{L^1(0,T; \mathscr{L}^1_{1(p, 1)} (\mathbb{R}^n)))}, \|g\|_{ L^1(0,T; \mathscr{L}^1_{ 1(p, 1)}(\mathbb{R}^n)))}\Big). \] Similar results in the other cases are also proved., Comment: 52 pages

In this paper we prove a Liouville type theorem for the stationary magnetohydrodynamics(MHD) system in $\Bbb R^3$. Let $(v, B, p)$ be a smooth solution to the stationary MHD equations in $\Bbb R^3$. We show that if there exist smooth matrix valued potential functions ${\bf \Phi}$, ${\bf \Psi}$ such that $ \nabla \cdot {\bf \Phi} =v$ and $\nabla \cdot {\bf \Psi}= B$, whose $L^6$ mean oscillations have certain growth condition near infinity, namely $$-\!\!\!\!\!\int_{B(r)} |\mathbf{\Phi} - \mathbf{\Phi}_{ B(r)} |^6 dx + -\!\!\!\!\!\int_{B(r)} |\mathbf{\Psi}- \mathbf{\Psi}_{ B(r)} |^6 dx\le C r\quad \forall 1< r< +\infty,$$ then $v=B= 0$ and $p=$constant. With additional assumption of $$r^{-8}\int_{B(r)}|B-B_{B(r)}|^6dx\to 0\quad \mathrm{as}\quad r\to+\infty,$$ similar result holds also for the Hall-MHD system., Comment: 16 pages. arXiv admin note: text overlap with arXiv:1811.09051

In this paper we prove a Liouville type theorem for generalized stationary Navier-Stokes systems in $\Bbb R^3$, which model non-Newtonian fluids, where the Laplacian term $\Delta u$ is replaced by the corresponding non linear operator $\bA_p( u)=\nabla \cdot ( |\bD(u)|^{p-2} \bD(u))$ with $ \bD(u) = \frac{1}{2} (\nabla u + (\nabla u)^{ \top})$, $3/2

In this paper we study the Liouville type problem for the stationary Navier-Stokes equations in $\Bbb R^3$. We deduce an asymptotic formula for an integral involving the head pressure, $Q=\frac12 |v|^2 +p$, and its derivative over domains enclosed by level surfaces of $Q$. This formula provides us with new sufficient condition for the triviality of solution to the Navier-Stokes equations., Comment: 7 pages

We prove local blow-up criterion for smooth axisymmetric solutions to the 3D incompressible Euler equation. If the vorticity satisfies $ \intl_{0}^{t_*} (t_*-t) \| \omega (t)\|_{ L^\infty(B(x_{ \ast}, R_0))} dt <+\infty$ for a ball $B(x_{ \ast}, R_0)$ 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 $B(x_{ \ast}, R_0)$ 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 $ \|\o (t)\|_{L^\infty (T(x_*, R))}= O\left(\frac{1}{(t_*-t)^\gamma}\right)$ as $t\to t_*$, where $\gamma <2$ and the torus $T(x_*, R)$ does not touch the axis., Comment: 47 pages

We prove a localized non blow-up theorem of the Beale-Kato-Majda type for the solution of the 3D incompressible Euler equations., Comment: 26 pages

We prove continuation in time of the local smooth solutions satisfying various Type I conditions for the 2D inviscid Boussinesq equations., Comment: 11 pages

We prove local non blow-up theorems for the 3D incompressible Euler equations under local Type I conditions. More specifically, for a classical solution $v\in L^\infty (-1,0; L^2 ( B(x_0,r)))\cap L^\infty_{\rm loc} (-1,0; W^{1, \infty} (B(x_0, r)))$ of the 3D Euler equations, where $B(x_0,r)$ is the ball with radius $r$ and the center at $x_0$, if the limiting values of certain scale invariant quantities for a solution $v(\cdot, t)$ as $t\to 0$ are small enough, then $ \nabla v(\cdot,t) $ does not blow-up at $t=0$ in $B(x_0, r)$., Comment: 22 pages

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., Comment: 47 pages

We study the scenario of discretely self-similar blow-up for Navier-Stokes equations. We prove that at the possible blow-up time such solutions only one point singularity. In case of the scaling parameter $ \lambda $ near $ 1$ we remove the singularity., Comment: 14 pages

We prove the existence of a forward discretely self-similar solutions to the Navier-Stokes equations in $ \Bbb R^{3}\times (0,+\infty)$ for a discretely self-similar initial velocity belonging to $ L^2_{ loc}(\Bbb R^{3})$., Comment: 25 pages

We prove Liouville type theorems for the self-similar solutions to the Navier-Stokes equations. One of our results generalizes the previous ones by Ne\v{c}as-R\.{u}\v{z}i\v{c}ka-\v{S}verak and Tsai. Using the Liouville type theorem we also remove a scenario of asymtotically self-similar blow-up for the Navier-Stokes equations with the profile belonging to $L^{p, \infty} (\Bbb R^3)$ with $p> \frac{3}{2}$., Comment: 22 pages

We study the regularity of weak solutions to the 3D valued stationary Hall magnetohydrodynamic equations on $ \Bbb R^2$. We prove that every weak solution is smooth. Furthermore, we prove a Liouville type theorem for the Hall equations., Comment: 18 pages

We prove geometrically improved version of Prodi-Serrin type blow-up criterion. Let $v$ and $\omega$ be the velocity and the vorticity of solutions to the 3D Navier-Stokes equations and denote $\{f\}_+=\max\{f, 0\}$ , $Q_T=\Bbb R^3\times (0, T)$. If $\left\{\left( v \times \frac{\omega}{|\omega|} \right)\cdot \frac{\Lambda^{\beta}v}{|\Lambda^{\beta}v|}\right\}_+ \in L^{\gamma, \alpha}_{x,t} (Q_T)$ with $3/\gamma +2/\alpha \leq 1$ for some $\gamma >3$ and $1 \leq \beta \leq 2$, then the local smooth solution $v$ of the Navier-Stokes equations on $(0,T)$ can be continued to $(0, T+\delta)$ for some $\delta >0$. We also prove localized version of a special case of this. Let $v$ be a suitable weak solution to the Navier-tokes equations in a space-time domain containing $z_0= (x_0, t_0)$, let $Q_{z_0, r}=B_{x_0, r} \times (t_0-r^2, t_0)$ be a parabolic cylinder in the domain. We show that if either $\left\{\left( v \times \frac{\omega}{|\omega|}\right) \cdot \frac{\nabla \times \omega}{|\nabla \times \omega|}\right\}_{+} \in L^{\gamma, \alpha}_{x,t}(Q_{z_0, r})$ with $\frac{3}{\gamma}+\frac{2}{\alpha} \leq 1$, or $\left\{\left(\frac{v}{|v|} \times \omega\right) \cdot \frac{\nabla \times \omega}{|\nabla \times \omega|}\right\}_{+} \in L^{\gamma, \alpha}_{x,t}(Q_{z_0, r})$ with $\frac{3}{\gamma}+\frac{2}{\alpha} \leq 2$, ($\gamma \geq 2$, $\alpha \geq 2$), then $z_0$ is a regular point for $v$. This improves previous local regularity criteria for the suitable weak solutions., Comment: 12 pages

In this paper we prove three different Liouville type theorems for the steady Navier-Stokes equations in $\Bbb R^3$. In the first theorem we improve logarithmically the well-known $L^{\frac92} (\Bbb R^3)$ result. In the second theorem we present a sufficient condition for the trivially of the solution($v=0$) in terms of the head pressure, $Q=\frac12 |v|^2 +p$. 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 $v=0$., Comment: 17 pages

In this paper we study Liouville properties of smooth steady axially symmetric solutions of the Navier-Stokes equations. First, we provide another version of the Liouville theorem of \cite{kpr15} in the case of zero swirl, where we replaced the Dirichlet integrability condition by mild decay conditions. Then we prove some Liouville theorems under the assumption $\|\f{u_r}{r}{\bf 1}_{\{u_r< -\f 1r\}}\|_{L^{3/2}(\mbR^3)}< C_{\sharp}$ where $C_{\sharp}$ is a universal constant to be specified. In particular, if $u_r(r,z)\geq -\f1r$ for $\forall (r,z)\in[0,\oo)\times\mbR$, then ${\bf u}\equiv 0$. Liouville theorems also hold if $\displaystyle\lim_{|x|\to \oo}\Ga =0$ or $\Ga\in L^q(\mbR^3)$ for some $q\in [2,\oo)$ where $\Ga= r u_{\th}$. We also established some interesting inequalities for $\Om\co \f{\p_z u_r-\p_r u_z}{r}$, showing that $\na\Om$ can be bounded by $\Om$ itself. All these results are extended to the axially symmetric MHD and Hall-MHD equations with ${\bf u}=u_r(r,z){\bf e}_r +u_{\th}(r,z) {\bf e}_{\th} + u_z(r,z){\bf e}_z, {\bf h}=h_{\th}(r,z){\bf e}_{\th}$, indicating that the swirl component of the magnetic field does not affect the triviality. Especially, we establish the maximum principle for the total head pressure $\Phi=\f {1}{2} (|{\bf u}|^2+|{\bf h}|^2)+p$ for this special solution class., Comment: 20 pages in Discrete and Continuous Dynamical Systems-Series A, 2016

We study Beltrami flows in the setting of weak solution to the stationary Euler equations in $\Bbb R^3$. For this weak Beltrami flow we prove the regularity and the Liouville property. In particular, we show that if tangential part of the velocity has certain decay property at infinity, then the solution becomes trivial. This decay condition of of the velocity is weaker than the previously known sufficient conditions for the Liouville property of the Betrami flows. For the proof we establish a mean value formula and other various formula for the tangential and the normal components of the weak solutions to the stationary Euler equations., Comment: 10 pages

We study a nonlinear parabolic system for a time dependent solenoidal vector field on $\Bbb R^3$. The nonlinear term of this new model equations is obtained slightly modifying that of the Navier-Stokes equations. The system has the same scaling property and the Galileian invariance as the Navier-Stokes equations. For such system we prove the global regularity for a smooth initial data., Comment: 9 pages(to appear in Analysis)

We study partial regularity of suitable weak solutions of the steady Hall magnetohydrodynamics equations in a domain $\Omega \subset \Bbb R^3$. 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 $\Omega=\Bbb R^3$, we show that the set of possible singularities is compact., Comment: 23 pages(to appear in Communications in Mathematical Physics)

We study the Liouville type problem for the stationary 3D Navier-Stokes equations on $\Bbb R^3$. Specifically, we prove that if $v$ is a smooth solution to (NS) satisfying $\omega={\rm curl}\,v \in L^q (\Bbb R^3) $ for some $\frac32 \leq q< 3$, and $|v(x)|\to 0$ as $|x|\to +\infty$, then either $v=0$ on $\Bbb R^3$, or $\int_{\Bbb R^6} \Phi_+ dxdy=\int_{\Bbb R^6} \Phi_- dxdy=+\infty$, where $\Phi(x,y) :=\frac{1}{4\pi}\frac{\omega (x)\cdot(x-y)\times (v(y)\times \omega(y) )}{|x-y|^3} $, and $\Phi_\pm:=\max\{ 0, \pm \Phi\}$. The proof uses crucially the structure of nonlinear term of the equations., Comment: 12 pages

We study partial regularity of weak solutions of the 3D valued non-stationary Hall magnetohydrodynamics equations on $ \Bbb R^2$. In particular we prove the existence of a weak solution whose set of possible singularities has the space-time Hausdorff dimension at most two., Comment: 31 pages