Search

We are interested in the multi-dimentional compressible viscoelastic flows of Oldroyd type, which is one of non-Newtonian fluids exhibiting the elastic behavior. In order to capture the damping effect of the additional deformation tensor, to the best of our knowledge, the "div-curl" structural condition plays a key role in previous efforts. Our aim of this paper is to remove the structural condition and prove a global existence of strong solutions to compressible viscoelastic flows in critical spaces. The new ingredient lies in the introduction of effective flux $(\theta,\mathcal{G})$, which enables us to capture the dissipation arising from \textit{combination} of density and deformation tensor. In absence of compatible conditions, the partial dissipation is found in non-Newtonian compressible fluids, which is weaker than that of usual Navier-Stokes equations., Comment: title changed and some proofs in the paper are rearranged

In this paper, we consider the global existence and convergence of smooth solutions for the three dimensional spherically symmetric compressible Euler equations with time-dependent damping and physical vacuum. The damping coefficient decays with time and the sound speed is C 1/2-Hölder continuous across the physical vacuum boundary. Both the degeneration of the damping coefficient at time infinity and the non C 1 continuity of the sound speed across the vacuum boundary will cause difficulty in proving the global existence of smooth solutions. Under suitable assumptions on the decayed damping coefficients, the globally in-time smooth solutions and convergence to the modified Barenblatt solution will be given. Also obtained are the pointwise convergence rate of the density, velocity and the expanding rate of the physical vacuum boundary. Our result extends that in Zeng (2017 Arch. Ration. Mech. Anal. 226 33–82) by considering the degenerate damping coefficient instead of the constant damping coefficient and that in Pan (2021 Calc. Var. Partial Differ. Equ. 60 5) from the one dimensional case to the three dimensional case with spherically symmetric data.

In this paper, the one-side physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the physical vacuum boundary, the sound speed is C 1 / 2 -Holder continuous. The coefficient of the time-dependent damping is given by μ ( 1 + t ) λ , ( 0 λ , 0 μ ) which decays by order −λ in time. First we give an one-side physical vacuum background solution whose density and velocity have a growing order with respect to time. Then the main purpose of this paper is to prove the stability of this background solution under the assumption that 0 λ 1 , 0 μ or λ = 1 , 2 μ . The pointwise convergence rate of the density, velocity and the expanding rate of the physical vacuum boundary are also given. The proof is based on the space-time weighted energy estimates, elliptic estimates and the Hardy inequality in the Lagrangian coordinates.

In this paper, we are concerned with the compressible viscoelastic flows in whole space \begin{document}$ \mathbb{R}^n $\end{document} with \begin{document}$ n\geq2 $\end{document} . We aim at extending the global existence in energy spaces (see [ 18 ] by Hu & Wang and [ 30 ] by Qian & Zhang) such that it holds in more general \begin{document}$ L^p $\end{document} critical spaces, which allows to the case of large highly oscillating initial velocity. Precisely, We define "two effective velocities" which are used to eliminate the coupling between the density, velocity and deformation tensor. Consequently, the global existence in the \begin{document}$ L^p $\end{document} critical framework is constructed by elementary energy approaches. In addition, the optimal time-decay estimates of strong solutions are firstly shown in the \begin{document}$ L^p $\end{document} framework, which improve recent decay efforts for compressible viscoelastic flows.

In this paper, we derive the Liouville theorem of D-solutions to the stationary MHD system under the asymptotic assumption: one of the velocity field and the magnetic field approaches zero and the other approaches a non zero constant vector at infinity. Our result extends the corresponding one of D-solutions to the Navier–Stokes equations when the velocity approaches a non zero constant vector at infinity.

In this paper, the smooth solution of the physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the vacuum boundary, the sound speed is $C^{1/2}$-H\"{o}lder continuous. The coefficient of the damping depends on time, given by this form $\frac{\mu}{(1+t)^\lambda}$, $\lambda$, $\mu>0$, which decays by order $-\lambda$ in time. Under the assumption that $0, Comment: 44 pages. All comments are welcome. arXiv admin note: substantial text overlap with arXiv:1407.6111 by other authors

We prove the time analyticity for weak solutions of inhomogeneous parabolic equations with measurable coefficients in the half space with either the Dirichlet boundary condition or the conormal boundary condition under the assumption that the solution and the source term have the exponential growth of order $2$ with respect to the space variables. We also obtain the time analyticity for bounded mild solutions of the incompressible Navier-Stokes equations in the half space with the Dirichlet boundary condition. Our work is an extension of the recent work by Zhang [Proc. Amer. Math. Soc. 148 (2020)] and Dong-Zhang [J. Funct. Anal. (2020)], where the authors proved the time analyticity of solutions to the homogeneous heat equation and the Navier-Stokes equations in the whole space., 20pages

In this paper, we derive asymptotic properties of both the velocity and the vorticity fields to the 3-dimensional axially symmetric Navier-Stokes equations at infinity under the generalized D-solution assumption $\int_{\mathbb{R}^3}|\nabla u|^qdx, Comment: 24 pages. Final version, to appear in Commun. Contemp. Math

In this paper, the global well-posedness and the low Mach number limit for compressible viscous magnetohydrodynamic equations in the critical L p -type Besov space are considered. More precisely, we will show that, in the isentropic case, the solution of compressible magnetohydrodynamic equations will converge to that of incompressible magnetohydrodynamic equations when the Mach number tends to zero in the critical L p framework. Moreover, the convergence rate will be obtained.

In this paper, we study Liouville theorems of D-solutions to the stationary magnetohydrodynamic system in a slab. We will prove trivialness of the velocity and the magnetic field with various boundary conditions. In some boundary conditions, the additional assumption that the horizontal angular component(s) of the velocity or (and) the magnetic field is (are) axially symmetric is needed. More precisely, five types of boundary conditions will be considered: the vertical periodic boundary condition for the velocity and the magnetic field, the Navier-slip boundary condition for the velocity, the perfectly conducting or insulating boundary condition for the magnetic field, the non-slip boundary condition for the velocity, and the perfectly conducting or insulating boundary condition for the magnetic field. One of our innovations is that we do not impose finite Dirichlet integral assumption on the magnetic field compared with previous works.

In this paper, we study the regularity of 3d axisymmetric Navier-Stokes equations under a prior point assumption on \(v^{r}\) or \(v^{z}\). That is, the weak solution of the 3d axisymmetric Navier-Stokes equations \(v\) is smooth if $$ rv^{r}\geq-1; \quad\mbox{or}\quad r\bigl|v^{r}(t,x)\bigr|\leq Cr^{\alpha}, \ \alpha\in(0,1];\quad\mbox{or} \quad r\bigl|v^{z}(t,x)\bigr| \leq Cr^{ \beta},\ \beta\in[0,1]; $$ where \(r\) is the distance from the point \(x\) to the symmetric axis.

In this paper, we will show that solutions of the three-dimensional non-resistive and non-diffusive MHD-Boussinesq system are globally regular if the initial data is axisymmetric and the swirl components of the velocity and the magnetic vorticity are zero. Our main result extends previous ones on the three-dimensional non-resistive MHD system and non-diffusive Boussinesq system, and the method used here can also be applied to the magnetic Rayleigh-B\'{e}nard convection system., Comment: 17pages

In this paper, we study the stationary magnetohydrodynamics system in $\mathbb{R}^2\times\mathbb{T}$. We prove trivialness of D-solutions (the velocity field $u$ and the magnetic field $h$) when they are swirl-free. Meanwhile, this Liouville type theorem also holds provided $u$ is swirl-free and $h$ is axially symmetric, or both $u$ and $h$ are axially symmetric. Our method is also valid for certain related boundary value problems in the slab $\mathbb{R}^2\times[-\pi,\,\pi]$., Comment: 16 pages

In the paper \cite{KNSS:1}, the authors make the following conjecture: {\it any bounded ancient mild solution of the 3D axially symmetric Navier-Stokes equations is constant.} And it is proved in the case that the solution is swirl free. Our purpose of this paper is to improve their result by allowing that the solution can grow with a power smaller than 1 with respect to the distance to the origin. Also, we will show that such a power is optimal to prove the Liouville type theorem since we can find counterexamples for the Navier-Stokes equations such that the Liouville theorem fails if the solution can grow linearly., Comment: 10 pages

We show that weak solutions to parabolic equations in divergence form with conormal boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain integrability conditions.

We study the 1-d isentropic Euler equations with time-dependent damping { ∂ t ρ + ∂ x ( ρ u ) = 0 , ∂ t ( ρ u ) + ∂ x ( ρ u 2 ) + ∂ x p ( ρ ) = − μ ( 1 + t ) λ ρ u , ρ | t = 0 = 1 + e ρ 0 ( x ) , u | t = 0 = e u 0 ( x ) . In a previous paper [8] , we have proven that, when λ = 1 , μ > 2 , the 1-D Euler equations have global existence of small data solutions. However in this paper, we will show that, when the damping, with respect to time, decays faster or equal to 2 1 + t , the C 1 solution of the above system will blow up in finite time. More precisely, when λ = 1 , 0 ≤ μ ≤ 2 or λ > 1 , μ ≥ 0 , we will give a finite upper bound for the lifespan. Combining the results in this paper and [8] , we see that, when the damping decays with time like μ ( 1 + t ) λ , the critical exponents for λ , μ to separate the global existence and finite-time blow up of small data solutions are λ = 1 , μ = 2 .

Consider an axis-symmetric suitable weak solution of 3D incompressible Navier-Stokes equation with nontrivial swirl. If the solution satisfies a slightly supercritical assumption, we will prove that v is regular. This extends the results of Chen-Strain-Tsai-Yau, Koch-Nadirashvili-Seregin-Sverak and Lei-Zhang where regularities under critical assumptions were proven., 40pages

In this paper, we study the 1-d isentropic Euler equations with time-dependent damping. Our damping decays at a speed of order − 1 with respect to time which is a little weaker than the linear one. Under our assumption, we will prove the global existence of solutions to the Euler system and obtain L 2 and L ∞ estimates for the solutions. Our approach is based on some detailed energy estimates.

In this paper, we study the Liouville theorem of the stationary Navier-Stokes equations in R 3 . When the solution is periodic in two variables, we can prove that actually the solution is trivial (constant vector) under the assumption that one component of the velocity, vanishing at infinity, has finite Dirichlet integral and the other two components can have some growth with respect to the distance to the origin.

An old problem since Leray \cite{Le:1} asks whether homogeneous D solutions of the 3 dimensional Navier-Stokes equation in $\mathbb{R}^3$ or some noncompact domains are 0. In this paper, we give a positive solution to the problem in two special cases: (1) when the solution is axially symmetric and periodic in the vertical variable; (2) full 3 dimensional slab case $\mathbb{R}^2 \times [0, 1]$ with Dirichlet boundary condition. Other partial results are also presented. The paper is self contained comparing with the first part \cite{CPZ:1} although the general idea is related., Comment: revised after referee's suggestions

We study axially symmetric D-solutions of the 3 dimensional Navier-Stokes equations. The first result is an a priori decay estimate of the velocity for general domains. The second is an a priori decay estimate of the vorticity in $\bR^3$, which improves the corresponding results in the literature. In addition, we prove a similar decay of full 3d solutions except for a small set of angles. Next we turn to D-solutions which are periodic in the third variable and prove vanishing result under a reasonable condition. As a corollary we prove that axially symmetric D-solutions in the slab $\bR^2 \times I$ with suitable boundary condition is $0$. Here $I$ is any finite interval. To the best of our knowledge, this seems to be the first vanishing result on a 3 dimensional D-solution without extra integral or decay or smallness assumption on the solution. The tools used include Brezis-Gallouet inequality, dimension reduction, scaling, Green's function bound and Liouville theorems for Navier-Stokes equations., Comment: 40 pages incorporated (with thanks) referees' helpful suggestions

Two main results will be presented in our paper. First, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a $log$ supercritical assumption on the horizontally radial component $u^r$ and vertical component $u^z$, accompanied by a $log$ subcritical assumption on the horizontally angular component $u^\theta$ of the velocity. Second, the precise Green function for the operator $-(\Delta-\frac{1}{r^2})$ under the axially symmetric situation, where $r$ is the distance to the symmetric axis, and some weighted $L^p$ estimates of it will be given. This will serve as a tool for the study of axially symmetric Navier-Stokes equations. As an application, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a critical (or a subcritical) assumption on the angular component $w^\theta$ of the vorticity., Comment: Final version, to appear in Comm. Pure Appl. Anal

We study the isentropic Euler equations with time-dependent damping, given by $\frac{\mu}{(1+t)^\lambda}\rho u$. Here, $\lambda,\mu$ are two non-negative constants to describe the decay rate of damping with respect to time. We will investigate the global existence and asymptotic behavior of small data solutions to the Euler equations when $0, Comment: 35 pages

A new catalyst which exhibits superior activity, stability and selectivity in fluid catalytic cracking (FCC) reactions of hydrocarbons is defined as a new type of zeolite L, ultra stable zeolite L (USL), as an active additive component of catalyst. The zeolite L was synthesized by hydrothermal crystallization method and characterized by means of XRD, SEM, FTIR and N2 isotherm adsorption. The zeolite L with a SiO2/Al2O3 mole ratio of 5 ∼7 was modified by hydrothermal dealumination and metal element modifications in order to prepare zeolite USL with a higher framework SiO2 /Al2O3 ratio of 15 ∼ 30. The FCC catalysts introduced a novel zeolite RE-USL were evaluated by microactivity test (MAT) and fixed-fluidized bed (FFB) experiments using heavy oil as a feedstock. The results show that the FCC catalyst is excellent for improving the quality of gasoline, decreasing the olefin content, enhancing the content of isoparaffins, aromatic hydrocarbons and research octane number (RON).