Global existence for a thin film equation with subcritical mass
Jian-Guo Liu Jinhuan Wang
In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin film equation in one-dimensional case
$h_t+\partial_x (h^n\,\partial_{xxx} h)+\partial_x (h^{n+2}\partial_{x} h)=0,$
$n≥q 1$
. There exists a critical mass
found by Witelski et al.(2004 Euro. J. of Appl. Math. 15,223-256) for
. We obtain global existence of a non-negative entropy weak solution if initial mass is less than
. For
$n≥q 4$
, entropy weak solutions are positive and unique. For
, a finite time blow-up occurs for solutions with initial mass larger than
. For the Cauchy problem with
and initial mass less than
, we show that at least one of the following long-time behavior holds:the second moment goes to infinity as the time goes to infinity or
$ h(·, t_k)\rightharpoonup 0$
for some subsequence
${t_k} \to \infty $
keywords: Long-wave instability free-surface evolution equilibrium the Sz. Nagy inequality long-time behavior
Uniform $L^{∞}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model
Wenting Cong Jian-Guo Liu

This paper investigates the existence of a uniform in time $L^{∞}$ bounded weak entropy solution for the quasilinear parabolic-parabolic Keller-Segel model with the supercritical diffusion exponent $0<m<2-\frac{2}{d}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(2-m)}{2}}$ norm of initial data is smaller than a universal constant. Moreover, the weak entropy solution $u(x,t)$ satisfies mass conservation when $m>1-\frac{2}{d}$. We also prove the local existence of weak entropy solutions and a blow-up criterion for general $L^1\cap L^{∞}$ initial data.

keywords: Chemotaxis fast diffusion critical space semi-group theory global existence
Error estimates of the aggregation-diffusion splitting algorithms for the Keller-Segel equations
Hui Huang Jian-Guo Liu
In this paper, we discuss error estimates associated with three different aggregation-diffusion splitting schemes for the Keller-Segel equations. We start with one algorithm based on the Trotter product formula, and we show that the convergence rate is $C\Delta t$, where $\Delta t$ is the time-step size. Secondly, we prove the convergence rate $C\Delta t^2$ for the Strang's splitting. Lastly, we study a splitting scheme with the linear transport approximation, and prove the convergence rate $C\Delta t$.
keywords: positivity preserving. chemotaxis random particle method Newtonian aggregation
Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos
Hui Huang Jian-Guo Liu
This paper investigates the generalized Keller-Segel (KS) system with a nonlocal diffusion term $-\nu(-\Delta)^{\frac{\alpha}{2}}\rho~(1<\alpha<2)$. Firstly, the global existence of weak solutions is proved for the initial density $\rho_0\in L^1\cap L^{\frac{d}{\alpha}}(\mathbb{R}^d)~(d\geq2)$ with $\|\rho_0\|_{\frac {d}{\alpha}} < K$, where $K$ is a universal constant only depending on $d,\alpha,\nu$. Moreover, the conservation of mass holds true and the weak solution satisfies some hyper-contractive and decay estimates in $L^r$ for any $1< r<\infty$. Secondly, for the more general initial data $\rho_0\in L^1\cap L^2(\mathbb{R}^d)$$~(d=2,3)$, the local existence is obtained. Thirdly, for $\rho_0\in L^1\big(\mathbb{R}^d,(1+|x|)dx\big)\cap L^\infty(\mathbb{R}^d)(~d\geq2)$ with $\|\rho_0\|_{\frac{d}{\alpha}} < K$, we prove the uniqueness and stability of weak solutions under Wasserstein metric through the method of associating the KS equation with a self-consistent stochastic process driven by the rotationally invariant $\alpha$-stable Lévy process $L_{\alpha}(t)$. Also, we prove the weak solution is $L^\infty$ bounded uniformly in time. Lastly, we consider the $N$-particle interacting system with the Lévy process $L_{\alpha}(t)$ and the Newtonian potential aggregation and prove that the expectation of collision time between particles is below a universal constant if the moment $\int_{\mathbb{R}^d}|x|^\gamma\rho_0dx$ for some $1<\gamma<\alpha$ is below a universal constant $K_\gamma$ and $\nu$ is also below a universal constant. Meanwhile, we prove the propagation of chaos as $N\rightarrow\infty$ for the interacting particle system with a cut-off parameter $\varepsilon\sim(\ln N)^{-\frac{1}{d}}$, and show that the mean field limit equation is exactly the generalized KS equation.
keywords: rotationally invariant $\alpha$-stable Lévy process log-Lipschitz continuity uniqueness of the weak solutions collision between particles. interacting particle system Newtonian potential aggregation stability in Wasserstein metric
A degenerate $p$-Laplacian Keller-Segel model
Wenting Cong Jian-Guo Liu
This paper investigates the existence of a uniform in time $L^{\infty}$ bounded weak solution for the $p$-Laplacian Keller-Segel system with the supercritical diffusion exponent $1 < p < \frac{3d}{d+1}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(3-p)}{p}}$ norm of initial data is smaller than a universal constant. We also prove the local existence of weak solutions and a blow-up criterion for general $L^1\cap L^{\infty}$ initial data.
keywords: fast diffusion monotone operator global existence Chemotaxis critical space non-Newtonian filtration.
Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system
Marion Acheritogaray Pierre Degond Amic Frouvelle Jian-Guo Liu
This paper deals with the derivation and analysis of the the Hall Magneto-Hydrodynamic equations. We first provide a derivation of this system from a two-fluids Euler-Maxwell system for electrons and ions, through a set of scaling limits. We also propose a kinetic formulation for the Hall-MHD equations which contains as fluid closure different variants of the Hall-MHD model. Then, we prove the existence of global weak solutions for the incompressible viscous resistive Hall-MHD model. We use the particular structure of the Hall term which has zero contribution to the energy identity. Finally, we discuss particular solutions in the form of axisymmetric purely swirling magnetic fields and propose some regularization of the Hall equation.
keywords: incompressible viscous flow Hall-MHD global weak solutions KMC waves. entropy dissipation kinetic formulation generalized Ohm's law resistivity
Positivity property of second-order flux-splitting schemes for the compressible Euler equations
Cheng Wang Jian-Guo Liu
A class of upwind flux splitting methods in the Euler equations of compressible flow is considered in this paper. Using the property that Euler flux $F(U)$ is a homogeneous function of degree one in $U$, we reformulate the splitting fluxes with $F^{+}=A^{+} U$, $F^{-}=A^{-} U$, and the corresponding matrices are either symmetric or symmetrizable and keep only non-negative and non-positive eigenvalues. That leads to the conclusion that the first order schemes are positive in the sense of Lax-Liu [18], which implies that it is $L^2$-stable in some suitable sense. Moreover, the second order scheme is a stable perturbation of the first order scheme, so that the positivity of the second order schemes is also established, under a CFL-like condition. In addition, these splitting methods preserve the positivity of density and energy.
keywords: limiter function positivity. flux splitting Conservation laws
Convergence analysis of the vortex blob method for the $b$-equation
Yong Duan Jian-Guo Liu
In this paper, we prove the convergence of the vortex blob method for a family of nonlinear evolutionary partial differential equations (PDEs), the so-called b-equation. This kind of PDEs, including the Camassa-Holm equation and the Degasperis-Procesi equation, has many applications in diverse scientific fields. Our convergence analysis also provides a proof for the existence of the global weak solution to the b-equation when the initial data is a nonnegative Radon measure with compact support.
keywords: Degasperis-Procesi equation peakon solution. Camassa-Holm equation space-time BV estimates b-equation global weak solution vortex blob method
Continuous and discrete one dimensional autonomous fractional ODEs
Yuanyuan Feng Lei Li Jian-Guo Liu Xiaoqian Xu

In this paper, we study 1D autonomous fractional ODEs $D_c^{γ}u=f(u), 0< γ <1$, where $u: [0,∞) \to \mathbb{R}$ is the unknown function and $D_c^{γ}$ is the generalized Caputo derivative introduced by Li and Liu (arXiv:1612.05103). Based on the existence and uniqueness theorem and regularity results in previous work, we show the monotonicity of solutions to the autonomous fractional ODEs and several versions of comparison principles. We also perform a detailed discussion of the asymptotic behavior for $f(u)=Au^p$. In particular, based on an Osgood type blow-up criteria, we find relatively sharp bounds of the blow-up time in the case $A>0, p>1$. These bounds indicate that as the memory effect becomes stronger ($γ \to 0$), if the initial value is big, the blow-up time tends to zero while if the initial value is small, the blow-up time tends to infinity. In the case $A<0, p>1$, we show that the solution decays to zero more slowly compared with the usual derivative. Lastly, we show several comparison principles and Grönwall inequalities for discretized equations, and perform some numerical simulations to confirm our analysis.

keywords: Fractional ODE Caputo derivative Volterra integral equation blow-up time discrete Grönwall inequality
The modified Camassa-Holm equation in Lagrangian coordinates
Yu Gao Jian-Guo Liu

In this paper, we study the modified Camassa-Holm (mCH) equation in Lagrangian coordinates. For some initial data $m_0$, we show that classical solutions to this equation blow up in finite time $T_{max}$. Before $T_{max}$, existence and uniqueness of classical solutions are established. Lifespan for classical solutions is obtained: $T_{max}≥ \frac{1}{||m_0||_{L^∞}||m_0||_{L^1}}.$ And there is a unique solution $X(ξ, t)$ to the Lagrange dynamics which is a strictly monotonic function of $ξ$ for any $t∈[0, T_{max})$: $X_ξ(·, t)>0$. As $t$ approaching $T_{max}$, we prove that the classical solution $m(·, t)$ in Eulerian coordinates has a unique limit $m(·, T_{max})$ in Radon measure space and there is a point $ξ_0$ such that $X_ξ(ξ_0, T_{max}) = 0$ which means $T_{max}$ is an onset time of collisions of characteristics. We also show that in some cases peakons are formed at $T_{max}$. After $T_{max}$, we regularize the Lagrange dynamics to prove global existence of weak solutions $m$ in Radon measure space.

keywords: Integrable system Lagrange dynamics finite time blow up local classical solutions global weak solutions

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