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In this paper, we deal with the $2\frac{1}{2}$ dimensional Hall MHD by taking the velocity field $u$ and the magnetic field $B$ of the form $u(t,x,y)=\left(\nabla^{\perp}\phi(t,x,y), W(t,x,y)\right)$ and $B(t,x,y)=\left(\nabla^{\perp}\psi(t,x,y), Z(t,x,y)\right)$. We begin with the Hall equations (without the effect of the fluid part). We first show the long time behavior of weak solutions and weak-strong uniqueness. We then proceed to prove the existence of unique strong solutions locally in time and to derive a blow-up criterion. We also demonstrate that the strong solution exists globally in time and decay algebraically if some smallness conditions are imposed. We further improve the decay rates of $\psi$ using the structure of the equation of $\psi$. As a consequence of the decay rates of $(\psi,Z)$, we find the asymptotic profiles of $(\psi,Z)$. We finally show that a small perturbation of initial data near zero can be extended to a small perturbations near harmonic functions. In the presence of the fluid filed, the results, by comparison, fall short of the previous ones in the absence of the fluid part. We prove two results: the existence of unique strong solutions locally in time and a blow-up criterion, and the existence of unique strong solutions globally in time with some smallness condition on initial data.

We study the Stokes system with the localized boundary data in the half-space. We are concerned with the local regularity of its solution near the boundary away from the support of the given boundary data which are product forms of each spatial variable and the temporal variable. We first show that if the boundary data are smooth in time, the corresponding solutions are also smooth in space and time near the boundary, even if the boundary data are only spatially integrable. Secondly, if the normal component of the boundary data is absent, we are able to construct a solution such that its second normal derivatives of the tangential components become singular near the boundary. Perturbation argument enables us to construct solutions of the Navier-Stokes equations with similar singular behaviors near the boundary in the half-space as the case of Stokes system. Lastly, we provide specific types of the localized boundary data to obtain the pointwise bounds of the solutions up to second derivatives. It turns out that such solutions are globally strong, and the second normal derivatives are, however, unbounded near the boundary. These results can be compared to the previous works in which only the normal component is present. In fact, the temporally non-smooth tangential boundary data can also cause spatially singular behaviors near the boundary, although such behaviors are milder than those caused by the normal boundary data of the same type., Comment: 52 pages, 3 figures

We study porous medium equations with a divergence form of drift terms in a bounded domain with no-flux lateral boundary conditions. We establish $L^q$-weak solutions for $ 1\leq q < \infty$ in Wasserstein space under appropriate conditions on the drift, which is an extension of authors' previous works done in the whole space into the case of bounded domains. Applying existence results to a certain Keller-Segel equation of consumption type, construction of $L^q$-weak solutions is also made, in case that the equation of a biological organism is of porous medium type., 37 pages. arXiv admin note: text overlap with arXiv:2111.04278

We prove that there exists a weak solution of the Stokes system with a non-zero external force and no-slip boundary conditions in a half space of dimensions three and higher so that its normal derivatives are unbounded near boundary. A localized and divergence free singular force causes, via non-local effect, singular behaviors of normal derivatives for the solution near boundary, although such boundary is away from the support of the external force. The constructed one is a weak solution that has finite energy globally, and it can be comparable to the one in \cite{Seregin-Sverak10} as a form of a shear flow that is of only locally finite energy. Similar construction is performed for the Navier-Stokes equations as well.

The paper is devoted to investigating the well-posedness, stability and large-time behavior near the hydrostatic balance for the 2D Boussinesq equations with partial dissipation. More precisely, the global well-posedness is obtained in the case of partial viscosity and without thermal diffusion for the initial data belonging to $H^{\delta}(\mathbb{R}^2) \times H^{s}(\mathbb{R}^2)$ for $\delta \in [s-1,s+1]$ if $s \in \mathbb{R}, s > 2$, for $\delta \in (1,s+1]$ if $s \in (0,2]$ and for $\delta \in [0,1]$ if $s = 0$. In addition, if one has either horizontal or vertical thermal diffusion then the stability and large-time behavior are provided in $H^m(\mathbb{R}^2)$, $m \in \mathbb{N}$ and in $\dot{H}^{m-1}(\mathbb{R}^2)$ with $m \in \mathbb{N}$, $m \geq 2$, respectively.

This paper is devoted to presenting new interior regularity criteria in terms of one velocity component for weak solutions to the Navier-Stokes equations in three dimensions. It is shown that the velocity is regular near a point $z$ if its scaled $L^p_tL^q_x$-norm of some quantities related to the velocity field is finite and the scaled $L^p_tL^q_x$-norm of one velocity component is sufficiently small near $z$.

This paper deals with a parabolic-elliptic chemotaxis-consumption system with tensor-valued sensitivity $S(x,n,c)$ under no-flux boundary conditions for $n$ and Robin-type boundary conditions for $c$. The global existence of bounded classical solutions is established in dimension two under general assumptions on tensor-valued sensitivity $S$. One of main steps is to show that $\nabla c(\cdot,t)$ becomes tiny in $L^{2}(B_{r}(x)\cap \Omega)$ for every $x\in\overline{\Omega}$ and $t$ when $r$ is sufficiently small, which seems to be of independent interest. On the other hand, in the case of scalar-valued sensitivity $S=\chi(x,n,c)\mathbb{I}$, there exists a bounded classical solution globally in time for two and higher dimensions provided the domain is a ball with radius $R$ and all given data are radial. The result of the radial case covers scalar-valued sensitivity $\chi$ that can be singular at $c=0$.

We study the regular sets of local energy solutions to the Navier-Stokes equations in terms of conditions on the initial data. It is shown that if a weighted $L^2$ norm of the initial data is finite, then all local energy solutions are regular in a region confined by space-time hypersurfaces determined by the weight. This result refines and generalizes Theorems C and D of Caffarelli, Kohn and Nirenberg (Comm. Pure Appl. Math. 35; 1982) and our recent paper [15] as well., arXiv admin note: text overlap with arXiv:2006.13145

For the Stokes system in the half space, Kang [Math.~Ann.~2005] showed that a solution generated by a compactly supported, H\"older continuous boundary flux may have unbounded normal derivatives near the boundary. In this paper we first prove explicit global pointwise estimates of the above solution, showing in particular that it has finite global energy and its derivatives blow up everywhere on the boundary away from the flux. We then use the above solution as a profile to construct solutions of the Navier-Stokes equations which also have finite global energy and unbounded normal derivatives due to the flux. Our main tool is the pointwise estimates of the Green tensor of the Stokes system proved by us in \cite{Green} (arXiv:2011.00134). We also examine the Stokes flows generated by dipole bumps boundary flux, and identify the regions where the normal derivatives of the solutions tend to positive or negative infinity near the boundary., Comment: 47 pages. Any comment wecolme

We consider degenerate porous medium equations with a divergence type of drift terms. We establish the existence of $L^{q}$-weak solutions (satisfying energy estimates or even further with moment and speed estimates in Wasserstein spaces), in case the drift term belongs to a sub-scaling (including scaling invariant) class depending on $q$ and $m$ caused by the nonlinear structure of diffusion, which is a major difference compared to that of a linear case. It is noticeable that the classes of drift terms become wider if the drift term is divergence-free. Similar conditions of gradients of drift terms are also provided to ensure the existence of such weak solutions. Uniqueness results follow under an additional condition on the gradients of the drift terms with the aid of methods developed in Wasserstein spaces. One of our main tools is so called the splitting method to construct a sequence of approximated solutions, which implies, bypassing to the limit, the existence of weak solutions satisfying not only an energy inequality but also moment and speed estimates. One of the crucial points in the construction is uniform H\"{o}lder continuity up to initial time for homogeneous porous medium equations, which seems to be of independent interest. As an application, we improve a regularity result for solutions of a repulsive Keller-Segel system of porous medium type., Comment: 78 pages

We are concerned with local regularity of the solutions for the Stokes and Navier-Stokes equations near boundary. Firstly, we construct a bounded solution but its normal derivatives are singular in any $L^p$ with $1

We prove an $��$-regularity criterion for the 3D Navier-Stokes equations in terms of initial data. It shows that if a scaled local $L^2$ norm of initial data is sufficiently small around the origin, a suitable weak solution is regular in a set enclosed by a paraboloid started from the origin. The result is applied to the estimate of the regular set for local energy solutions with initial data in weighted $L^2$ spaces. We also apply this result to studying energy concentration near a possible blow-up time and regularity of forward discretely self-similar solutions.

We prove the first ever pointwise estimates of the (unrestricted) Green tensor and the associated pressure tensor of the nonstationary Stokes system in the half-space, for every space dimension greater than one. The force field is not necessarily assumed to be solenoidal. The key is to find a suitable Green tensor formula which maximizes the tangential decay, showing in particular the integrability of Green tensor derivatives. With its pointwise estimates, we show the symmetry of the Green tensor, which in turn improves pointwise estimates. We also study how the solutions converge to the initial data, and the (infinitely many) restricted Green tensors acting on solenoidal vector fields. As applications, we give new proofs of existence of mild solutions of the Navier-Stokes equations in $L^q$, pointwise decay, and uniformly local $L^q$ spaces in the half-space., Comment: A missing error term V_{ij} added in Lemma 4.2; Subsection 5.2 added to estimate V_{ij}; Reference [21] updated; Grant information updated

We consider the Cauchy problem for coupled system of Vlasov and non-Newtonian fluid equations. We establish local well--posedness of the strong solutions, provided that the initial data are regular enough. Global existence of unique strong solutions for any given time interval is shown as well if the initial data are sufficiently small., 28 pages

We consider a coupled system of Keller-Segel type equations and the incompressible Navier-Stokes equations in spatial dimension two and three. In the previous work [19], we established the existence of a weak solution of a Fokker-Plank equation in the Wasserstein space using the optimal transportation technique. Exploiting this result, we constructed solutions of Keller-Segel-Navier-Stokes equations such that the density of biological organism belongs to the absolutely continuous curves in the Wasserstein space. In this work, we refine the result on the existence of a weak solution of a Fokker-Plank equation in the Wasserstein space. As a result, we construct solutions of Keller-Segel-Navier-Stokes equations under weaker assumptions on the initial data.

We prove short time regularity of suitable weak solutions of 3D incompressible Navier-Stokes equations near a point where the initial data is locally in $L^3$. The result is applied to the regularity problems of solutions with uniformly small local $L^3$ norms, and of forward discretely self-similar solutions.

We consider a coupled system consisting of a degenerate porous medium type of Keller-Segel system and Stokes system modeling the motion of swimming bacteria living in fluid and consuming oxygen. We establish the global existence of weak solutions and H\"older continuous solutions in dimension three, under the assumption that the power of degeneracy is above a certain number depending on given parameter values. To show H\"older continuity of weak solutions, we consider a single degenerate porous medium equation with lower order terms, and via a unified method of proof, we obtain H\"older regularity, which is of independent interest.

We are concerned with existence of regular solutions for non-Newtonian fluids in dimension three. For a certain type of non-Newtonian fluids we prove local existence of unique regular solutions, provided that the initial data are sufficiently smooth. Moreover, if the $H^3$-norm of initial data is sufficiently small, then the regular solution exists globally in time., Comment: 28 pages, updated version

A simplified model of the tumor angiogenesis can be described by a Keller-Segel equation \cite{FrTe,Le,Pe}. The stability of traveling waves for the one dimensional system has recently been known by \cite{JinLiWa,LiWa}. In this paper we consider the equation on the two dimensional domain $ (x, y) \in \mathbf R \times {\mathbf S^{\lambda}}$ for a small parameter $\lambda>0$ where $ \mathbf S^{\lambda}$ is the circle of perimeter $\lambda$. Then the equation allows a planar traveling wave solution of invading types. We establish the nonlinear stability of the traveling wave solution if the initial perturbation is sufficiently small in a weighted Sobolev space without a chemical diffusion. When the diffusion is present, we show a linear stability. Lastly, we prove that any solution with our front conditions eventually becomes planar under certain regularity conditions. The key ideas are to use the Cole-Hopf transformation and to apply the Poincar\'{e} inequality to handle with the two dimensional structure., Comment: 35 pages, 1 figures, submitted

We consider two dimensional Keller-Segel equations coupled with the Navier-Stokes equations modelled by Tuval et al.[32]. Assuming that the chemotactic sensitivity and oxygen consumption rate are nondecreasing and differentiable, we prove that there is no blow-up in a finite time for solutions with large initial data to chemotaxis-Navier-Stokes equations in two dimensions. In addition, temporal decays of solutions are shown, as time tends to infinity., Comment: This paper has been withdrawn by the author due to a crucial error in the proof of Lemma 5

We consider Stokes system in bounded domains and we present conditions of given data, in particular, boundary data, which ensure H��lder continuity of solutions. For H��lder continuous solutions for the Stokes system the normal component of boundary data requires a bit more regular than boundary data of H��lder continuous solutions for the heat equation. We also construct an example, which shows that H��lder continuity is no longer valid, unless the proposed condition of boundary data is fulfilled. As an application, we consider a certain general types of nonlinear systems coupled to fluid equations and local well-posedness is established in H��lder spaces.

We consider a coupled system consisting of the Navier-Stokes equations and a porous medium type of Keller-Segel system that model the motion of swimming bacteria living in fluid and consuming oxygen. We establish the global-in-time existence of weak solutions for the Cauchy problem of the system in dimension three. In addition, if the Stokes system, instead Navier-Stokes system, is considered for the fluid equation, we prove that bounded weak solutions exist globally in time., Comment: 24pages

We consider coupled system of Keller-Segel type equations and the incompressible Navier-Stokes equations in spatial dimension two. We show temporal decay estimates of solutions with small initial data and obtain their asymptotic profiles as time tends to infinity., 17 pages

This paper presents a mathematical framework for a flexible pressure-sensor model using electrical impedance tomography (EIT). When pressure is applied to a conductive membrane patch with clamped boundary, the pressure-induced surface deformation results in a change in the conductivity distribution. This change can be detected in the current-voltage data ({\it i.e.,} EIT data) measured on the boundary of the membrane patch. Hence, the corresponding inverse problem is to reconstruct the pressure distribution from the data. The 2D apparent conductivity (in terms of EIT data) corresponding to the surface deformation is anisotropic. Thus, we consider a constrained inverse problem by restricting the coefficient tensor to the range of the map from pressure to 2D-apparent conductivity. This paper provides theoretical grounds for the mathematical model of the inverse problem. We develop a reconstruction algorithm based on a careful sensitivity analysis. We demonstrate the performance of the reconstruction algorithm through numerical simulations to validate its feasibility for future experimental studies., Comment: electrical impedance tomography, pressure sensing, conductive membrane, inverse problem, prescribed mean curvature equation

We study the $3 \times 3$ elliptic systems $\nabla \times (a(x) \nabla\times u)-\nabla (b(x) \nabla \cdot u)=f$, where the coefficients $a(x)$ and $b(x)$ are positive scalar functions that are measurable and bounded away from zero and infinity. We prove that weak solutions of the above system are H��lder continuous under some minimal conditions on the inhomogeneous term $f$. We also present some applications and discuss several related topics including estimates of the Green's functions and the heat kernels of the above systems., Proof of Theorem 3.1 is corrected

For Schr��dinger maps from $\R^2\times\R^+$ to the 2-sphere $��^2$, it is not known if finite energy solutions can form singularities (``blowup'') in finite time. We consider equivariant solutions with energy near the energy of the two-parameter family of equivariant harmonic maps. We prove that if the topological degree of the map is at least four, blowup does {\it not} occur, and global solutions converge (in a dispersive sense -- i.e. scatter) to a fixed harmonic map as time tends to infinity. The proof uses, among other things, a time-dependent splitting of the solution, the ``generalized Hasimoto transform", and Strichartz (dispersive) estimates for a certain two space-dimensional linear Schr��dinger equation whose potential has critical power spatial singularity and decay. Along the way, we establish an energy-space local well-posedness result for which the existence time is determined by the length-scale of a nearby harmonic map., 41 pages

We present some new regularity criteria for ``suitable weak solutions'' of the Navier-Stokes equations near the boundary in dimension three. We prove that suitable weak solutions are H\"older continuous up to the boundary provided that the scaled mixed norm $L^{p,q}_{x,t}$ with $3/p+2/q\leq 2, 2