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In this paper, we establish the existence of global self-similar solutions to the 3D Muskat equation when the two fluids have the same viscosity but different densities. These self-similar solutions are globally defined in both space and time, with exact cones as their initial data. Furthermore we estimate the difference between our self-similar solutions and solutions of the linearized equation around the flat interface in terms of critical spaces and some weighted $\dot{W}^{k,\infty}(\mathbb{R}^2)$ spaces for $k=1,2$. The main ingredients of the proof are new estimates in the sense of $\dot{H}^{s_1}(\mathbb{R}^2) \cap \dot{H}^{s_2}(\mathbb{R}^2)$ with $3/2

We consider the 3D incompressible Euler equations under the following three scale hierarchical situation: large-scale vortex stretching the middle-scale, and at the same time, the middle-scale stretching the small-scale. In this situation, we show that, the stretching effect of this middle-scale flow is weakened by the large-scale. In other words, the vortices being stretched could have the corresponding stress tensor being weakened., Comment: 17 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 consider the Cauchy problem for the Keller-Segel system of consumption type coupled with the incompressible Euler equations in $\mathbb{R}^2$. This coupled system describes a biological phenomenon in which aerobic bacteria living in slightly viscous fluids (such as water) move towards a higher oxygen concentration to survive. We firstly prove the local existence of smooth solutions for arbitrary smooth initial data. Then we show that these smooth solutions can be extended globally if the initial density of oxygen is sufficiently small. The main ingredient in the proof is the $W^{1,q}$-energy estimate $(q>2)$ motivated by the partially inviscid two-dimensional Boussinesq system in \cite{C06}. Our result improves the well-known global well-posedness of the two-dimensional Keller-Segel system of consumption type coupled with the incompressible Navier-Stokes equations., Comment: 17 pages

This paper deals with a hyperbolic Keller-Segel system of consumption type with the logarithmic sensitivity \begin{equation*} \partial_{t} \rho = - \chi\nabla \cdot \left (\rho \nabla \log c\right),\quad \partial_{t} c = - \mu c\rho\quad (\chi,\,\mu>0) \end{equation*} in $\mathbb{R}^d\; (d \ge1)$ for nonvanishing initial data. This system is closely related to tumor angiogenesis, an important example of chemotaxis. We firstly show the local existence of smooth solutions corresponding to nonvanishing smooth initial data. Next, through Riemann invariants, we present some sufficient conditions of this initial data for finite-time singularity formation when $d=1$. We then prove that for any $d\ge1$, some nonvanishing $C^\infty$-data can become singular in finite time. Moreover, we derive detailed information about the behaviors of solutions when the singularity occurs. In particular, this information tells that singularity formation from some initial data is not because $c$ touches zero (which makes $\log c$ diverge) but due to the blowup of $C^1\times C^2$-norm of $(\rho,c)$. As a corollary, we also construct initial data near any constant equilibrium state which blows up in finite time for any $d\ge1$. Our results are the extension of finite-time blow-up results in \cite{IJ21}, where initial data is required to satisfy some vanishing conditions. Furthermore, we interpret our results in a way that some kinds of damping or dissipation of $\rho$ are necessarily required to ensure the global existence of smooth solutions even though initial data are small perturbations around constant equilibrium states., Comment: 18 pages