Global existence of strong solutions of Navier–Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids

We consider strong solutions to the initial boundary value problems for the isentropic compressible Navier–Stokes equations in one dimension: $$\rho\left\{\begin{array}{lll} t+(\rho u)_x=0\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\, {\rm in}\,(0,T)\times(0,1)\\ (\rho u )_t+(\rho u^2)_x+\rho \Phi_x-(\mu( \rho )u_x)_x+P_x=0\quad\quad {\rm in}\,(0,T)\times(0,1) \\\left(\left(\frac{\delta(\Phi_x)^2\,+\,1}{(\Phi_x)^2\,+\,\delta}\right)^{\frac{2-p}{2}}\Phi_x\right)_x=4\pi g(\rho-\frac{1}{|\Omega|}\int\nolimits_\Omega \rho dx\,\,\,\, )\quad\,\, {\rm in}\,(0,T)\times(0,1)\end{array}\right.$$ Here, the Φ is a non-Newtonian potential and strong solutions of the problem and obtains the uniqueness under the compatibility condition.     

Global existence of strong solutions of Navier-Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids

The aim of this paper is to discuss the global existence and uniqueness of strong solution for a class of the isentropic compressible Navier-Stokes equations with non-Newtonian in one-dimensional bounded intervals. We prove two global existence results on strong solutions of the isentropic compressible Navier-Stokes equations. The first result shows only the existence, and the second one shows the existence and uniqueness result based on the first result, but the uniqueness requires some compatibility condition.     

Finite Velocity of the Propagation of Perturbations for a Class of Non-Newtonian Fluids

In this paper, we consider the initial boundary value problem of a class of non-Newtonian fluids. We obtain that finite velocity of the propagation of perturbations.     

Global existence and uniqueness of solution of the initial boundary value problem for a class of non-Newtonian fluids with vacuum

The aims of this paper are to discuss global existence and uniqueness of solution for a class of non-Newtonian fluids with vacuum in one-dimensional bounded interval. The important point in this paper is that we allow the initial vacuum. In particular, these results are used to prove similar results for more general non-Newtonian fluids, and applied to numerical computation. This work is partially supported by the 985 program of Jilin University, China Postdoctoral Sciences Foundation, and NSF Grant ([10571072]; [10601009]).     

Existence and uniqueness of solutions for a class of non-Newtonian fluids with vacuum and damping

In this paper, we proved local existence and uniqueness of solutions for a class of non-Newtonian fluids with vacuum and damping in one-dimensional bounded intervals. The main difficulty is due to the strong nonlinearity of the system and initial vacuum.     

Global strong solutions for a class of compressible non‐Newtonian fluids with vacuum

We study a class of compressible non‐Newtonian fluids in one space dimension. We prove, by using iterative method, the global time existence and uniqueness of strong solutions provided that the initial data satisfy a compatibility condition and the initial density is small in its H¹‐norm. The main difficulty is due to the strong nonlinearity of the system and the initial vacuum. Copyright © 2010 John Wiley & Sons, Ltd.     

MOF‐Derived Double‐Layer Hollow Nanoparticles with Oxygen Generation Ability for Multimodal Imaging‐Guided Sonodynamic Therapy

The high reactive oxygen species (ROS) generation ability and simple construction of sonosensitizer systems remain challenging in sonodynamic therapy against the hypoxic tumor. In this work, we rationally prepared MOF‐derived double‐layer hollow manganese silicate nanoparticle (DHMS) with highly effective ROS yield under ultrasound irradiation for multimodal imaging‐guided sonodynamic therapy (SDT). The presence of Mn in DHMS increased ROS generation efficiency because it could be oxidized by holes to improve the electron–hole separation. Moreover, DHMS could produce oxygen in the tumor microenvironment, which helps overcome the hypoxia of the solid tumor and thus enhance the treatment efficiency. In vivo experiments demonstrated efficient tumor inhibition in DHMS‐mediated SDT guided by ultrasound and magnetic resonance imaging. This work presents a MOF‐derived nanoparticle with sonosensitive and oxygen generating ability, which provides a promising strategy for tumor hypoxia in SDT.     

Study of the solutions to a model porous medium equation with variable exponent of nonlinearity

The authors of this paper study the Dirichlet problem of the following equation

ut−div(|u|^ν(x,t)∇u)=f−|u|^p(x,t)−1u.     

Unique solvability for a class of full non-Newtonian fluids of one dimension with vacuum

We prove the local existence and uniqueness of the strong solutions for a class of full non-Newtonian fluids in one space dimension with the hypotheses that the initial data are small in some sense and satisfy some compatibility conditions. The initial density need not be positive, which means that we allow the initial vacuum.