Strong solution to the compressible magnetohydrodynamic equations with vacuum

We consider the compressible magnetohydrodynamic (MHD) equations with nonnegative thermal conductivity or infinite electric conductivity. We prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set.     

粘性系数依赖于密度的Korteweg型不可压流体的强解问题

研究一类Korteweg型不可压流体模型的强解问题.针对粘性系数依赖于密度的情形,当初始值满足兼容性条件(9)对,证明了强解的局部存在性和唯一性.我们在这指出,本文允许初始真空存在.     

Classical solutions to Navier–Stokes equations for nonhomogeneous incompressible fluids with non-negative densities

We study the Navier–Stokes equations for nonhomogeneous incompressible fluids in a bounded domain Ω of R³. We first prove the existence and uniqueness of local classical solutions to the initial boundary value problem of linear Stokes equations and then we obtain the existence and uniqueness of local classical solutions to the Navier–Stokes equations with vacuum under the assumption that the data satisfies a natural compatibility condition.     

Existence results for viscous polytropic fluids with vacuum

We consider the full Navier-Stokes equations for viscous polytropic fluids with nonnegative thermal conductivity. We prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Moreover our results hold for both bounded and unbounded domains.     

ON THE EXISTENCE OF LOCAL CLASSICAL SOLUTION FOR A CLASS OF ONE-DIMENSIONAL COMPRESSIBLE NON-NEWTONIAN FLUIDS

In this paper, the aim is to establish the local existence of classical solutions for a class of compressible non-Newtonian fluids with vacuum in one-dimensional bounded intervals, under the assumption that the data satisfies a natural compatibility condition. For the results, the initial density does not need to be bounded below away from zero.     

Global existence of strong solutions of the Navier-Stokes equations for isentropic compressible fluids with density-dependent viscosity

This paper is concerned with global strong solutions of the isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient in one-dimensional bounded intervals. Precisely, the viscosity coefficient μ=μ(ρ) and the pressure P is proportional to ργ with γ>1. The important point in this paper is that the initial density may vanish in an open subset. We also show that the strong solution obtained above is unique provided that the initial data satisfies additional regularity and a compatible condition. Compared with former results obtained by Hyunseok Kim in [H. Kim, Global existence of strong solutions of the Navier-Stokes equations for one-dimensional isentropic compressible fluids, available at: http://com2mac.postech.ac.kr/papers/2001/01-38.pdf], we deal with density-dependent viscosity coefficient.     

Local strong solutions to a compressible non‐Newtonian fluid with density‐dependent viscosity

We consider the initial boundary problem for a compressible non‐Newtonian fluid with density‐dependent viscosity. The local existence of strong solution is established that is based on some compatibility condition. Moreover, it is also proved that the solutions are to blow up, and the maximum norm of velocity gradients controls the possible break down of the strong solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Global solutions to the shallow water system with a method of an additional argument

The classical system of shallow water (Saint–Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasi-linear hyperbolic system for a wide class of initial data, we find a sufficient condition on the initial data that guarantee existence of a global classical solution continued from a local solution. The sufficient conditions can be easily satisfied for the fluid flow propagating in one direction with two characteristic velocities of the same sign and two monotonically increasing Riemann invariants. We prove that these properties persist in the time evolution of the classical solutions to the shallow water equations and provide no shock wave singularities formed in a finite time over a half-line or an infinite line. On a technical side, we develop a novel method of an additional argument, which allows to obtain local and global solutions to the quasi-linear hyperbolic systems in physical rather than characteristic variables.     

带真空无磁扩散不可压磁流体方程柯西问题的局部适定性

该文讨论了在真空远场的密度条件下,二维不可压零磁耗散磁流体力学方程组柯西问题的局部适定性.在初始密度和磁场具有一定的衰减性时,证明了磁流体方程具有唯一的局部强解.当初值满足兼容性条件和适当的正则性条件时,该强解就是经典解.除此之外,文中还给出了一个仅与磁场有关的爆破准则.     

On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier–Stokes equations with vacuum

This paper concerns the Cauchy problem of the barotropic compressible Navier–Stokes equations on the whole two-dimensional space with vacuum as far field density. In particular, the initial density can have compact support. When the shear and the bulk viscosities are a positive constant and a power function of the density respectively, it is proved that the two-dimensional Cauchy problem of the compressible Navier–Stokes equations admits a unique local strong solution provided the initial density decays not too slow at infinity. Moreover, if the initial data satisfy some additional regularity and compatibility conditions, the strong solution becomes a classical one.     

Strong solutions to the incompressible magnetohydrodynamic equations

In this paper, we are concerned with strong solutions to the Cauchy problem for the incompressible Magnetohydrodynamic equations. By the Galerkin method, energy method and the domain expansion technique, we prove the local existence of unique strong solutions for general initial data, develop a blow‐up criterion for local strong solutions and prove the global existence of strong solutions under the smallness assumption of initial data. The initial data are assumed to satisfy a natural compatibility condition and allow vacuum to exist. Copyright © 2010 John Wiley & Sons, Ltd.     

Some uniform estimates and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows

This paper concerns the global existence and the large time behavior of strong and classical solutions to the two-dimensional (2D) Stokes approximation equations for the compressible flows. We consider the unique global strong solution or classical solution to the 2D Stokes approximation equations for the compressible flows together with the space-periodicity boundary condition or the no-stick boundary condition or Cauchy problem for arbitrarily large initial data. First, we prove that the density is bounded from above independent of time in all these cases. Secondly, we show that for the space-periodicity boundary condition or the no-stick boundary condition, if the initial density contains vacuum at least at one point, then the global strong (or classical) solution must blow up as time goes to infinity.     

STRONG SOLUTIONS FOR THE INCOMPRESSIBLE FLUID MODELS OF KORTEWEG TYPE

In this article, we are concerned with the strong solutions for the incompressible fluid models of Korteweg type in a bounded domain Ω R3. We prove the existence and uniqueness of local strong solutions to the initial boundary value problem. We point out that in this article we allow the existence of initial vacuum provided initial data satisfy a compatibility condition.     

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.     

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.     

Strong solutions of the Navier-Stokes equations for isentropic compressible fluids

We study strong solutions of the isentropic compressible Navier-Stokes equations in a domain . We first prove the local existence of unique strong solutions provided that the initial data ρ₀ and u₀ satisfy a natural compatibility condition. The important point in this paper is that we allow the initial vacuum: the initial density may vanish in an open subset of Ω. We then prove a new uniqueness result and stability result. Our results are valid for unbounded domains as well as bounded ones.     

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.     

Global existence and exponential stability for a nonlinear thermoelastic Kirchhoff–Love plate

We study an initial–boundary-value problem for a quasilinear thermoelastic plate of Kirchhoff & Love-type with parabolic heat conduction due to Fourier, mechanically simply supported and held at the reference temperature on the boundary. For this problem, we show the short-time existence and uniqueness of classical solutions under appropriate regularity and compatibility assumptions on the data. Further, we use barrier techniques to prove the global existence and exponential stability of solutions under a smallness condition on the initial data. It is the first result of this kind established for a quasilinear non-parabolic thermoelastic Kirchhoff & Love plate in multiple dimensions.     

Mean Curvature Motion of Triple Junctions of Graphs in Two Dimensions

We consider a system of three surfaces, graphs over a bounded domain in ?², intersecting along a time-dependent curve and moving by mean curvature while preserving the pairwise angles at the curve of intersection (equal to 2π/3.) For the corresponding two-dimensional parabolic free boundary problem we prove short-time existence of classical solutions (in parabolic Hölder spaces), for sufficiently regular initial data satisfying a compatibility condition.     

Global existence of the radially symmetric strong solution to Navier-Stokes-Poisson equations for isentropic compressible fluids

We prove the two existence results of the radially symmetric strong solutions to the Navier- Stokes-Poisson equations for isentropic compressible fluids. The important point in this paper is that the initial density is vacuum. It is different from weak solutions. Now we need some compatibility condition.