GLOBAL CLASSICAL SOLUTIONS AND THE CLASSICAL LIMIT OF THE NON-RELATIVISTIC VLASOV-DARWIN SYSTEM WITH SMALL INITIAL DATA

We investigate the global classical solutions of the non-relativistic Vlasov-D arwin system with generalized variables(VDG)in three dimensions.We first prove the global existence and uniqueness for small initial data and derive the decay estimates of the Darwin potentials.Then,we show in this framework that the solutions converge in a pointwise sense to solutions of the classical Vlasov-Poisson system(VP)at the asymptotic rate of 1/c2 as the speed of light c tends to infinity for all time.Moreover,we obtain rigorously an asymptotic estimate of the difference between the two systems.

POSITIVE CLASSICAL SOLUTIONS OF DIRICHLET PROBLEM FOR THE STEADY RELATIVISTIC HEAT EQUATION

In this paper,for a bounded C2 domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C2 boundary data.We have a non-existence result,which is the justification for taking into account the restricted boundary data.There is a smooth positive boundary datum that precludes the existence of the positive classical solution.

GLOBAL CLASSICAL SOLUTIONS TO THE 3-D ISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH GENERAL INITIAL ENERGY

We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the 3-D compressible Navier-Stokes equations under the assumption that the initial density ρ0 L∞ is appropriate small and 1 < γ < 65.Here the initial density could have vacuum and we do not require that the initial energy is small.

CLASSICAL SOLUTIONS OF THE 3D COMPRESSIBLE FLUID-PARTICLE SYSTEM WITH A MAGNETIC FIELD

This paper addresses the 3-D Cauchy problem of a fluid-particle system with a magnetic field.First,the local classical solutions of the linearized model on the sphere Br are obtained by some a priori estimates that do not depend on the radius r.Second,the classical solutions of the linearized model in R^(3) are obtained by combining the continuation and compactness methods.Finally,the classical solutions of the original system are proved by use of the picard iteration argument and the energy method.

LIFE-SPAN OF CLASSICAL SOLUTIONS TO HYPERBOLIC GEOMETRY FLOW EQUATION IN SEVERAL SPACE DIMENSIONS

In this article, we investigate the lower bound of life-span of classical solutions of the hyperbolic geometry flow equations in several space dimensions with "small" initial data. We first present some estimates on solutions of linear wave equations in several space variables. Then, we derive a lower bound of the life-span of the classical solutions to the equations with "small" initial data.

On local strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with vacuum

We consider the local well-posedness of strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with density containing vacuum initially.We first prove the local existence and uniqueness of the strong solutions,where the initial compatibility condition proposed by Cho et al.(2004),Cho and Kim(2006)and Choe and Kim(2003)is removed in a suitable sense.Then the continuous dependence of strong solutions on the initial data is derived under an additional compatibility condition.Moreover,for the initial data satisfying some additional regularity and the compatibility condition,the strong solution is proved to be a classical one.

Global axisymmetric classical solutions of full compressible magnetohydrodynamic equations with vacuum free boundary and large initial data

In this paper,the global existence of the classical solution to the vacuum free boundary problem of full compressible magnetohydrodynamic equations with large initial data and axial symmetry is studied.The solutions to the system(1.6)–(1.8) are in the class of radius-dependent solutions,i.e.,independent of the axial variable and the angular variable.In particular,the expanding rate of the moving boundary is obtained.The main difficulty of this problem lies in the strong coupling of the magnetic field,velocity,temperature and the degenerate density near the free boundary.We overcome the obstacle by establishing the lower bound of the temperature by using different Lagrangian coordinates,and deriving the uniform-in-time upper and lower bounds of the Lagrangian deformation variable r;by weighted estimates,and also the uniform-in-time weighted estimates of the higher-order derivatives of solutions by delicate analysis.

Blow-up of Classical Solutions to the Isentropic Compressible BarotropicNavier-Stokes-Langevin-Korteweg Equations

In this paper,we study the barotropic Navier-Stokes-Langevin-Korteweg system in R3.Assuming the derivatives of the square root of the density and the velocity field decay to zero at infinity,we can prove the classical solutions blow up in finite time when the initial energy has a certain upper bound.We obtain this blow up result by a contradiction argument based on the conservation of the total mass and the total quasi momentum.

Conical Sonic-Supersonic Solutions for the 3-D Steady Full Euler Equations

This paper concerns the sonic-supersonic structures of the transonic crossflow generated by the steady supersonic flow past an infinite cone of arbitrary cross section.Under the conical assumption,the three-dimensional(3-D)steady Euler equations can be projected onto the unit sphere and the state of fluid can be characterized by the polar and azimuthal angles.Given a segment smooth curve as a conical-sonic line in the polar-azimuthal angle plane,we construct a classical conical-supersonic solution near the curve under some reasonable assumptions.To overcome the difficulty caused by the parabolic degeneracy,we apply the characteristic decomposition technique to transform the Euler equations into a new degenerate hyperbolic system in a partial hodograph plane.The singular terms are isolated from the highly nonlinear complicated system and then can be handled successfully.We establish a smooth local solution to the new system in a suitable weighted metric space and then express the solution in terms of the original variables.

Global Existence of Smooth Solutions for the One-Dimensional Full Euler System for a Dusty Gas

We study the existence of global-in-time classical solutions for the one-dimensional nonisentropic compressible Euler system for a dusty gas with large initial data.Using the characteristic decomposition method proposed by Li et al.(Commun Math Phys 267:1–12,2006),we derive a group of characteristic decompositions for the system.Using these characteristic decompositions,we find a sufficient condition on the initial data to ensure the existence of global-in-time classical solutions.

GLOBAL CLASSICAL SOLUTION TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

In this paper, we consider the global existence of classical solution to the 3-D compressible Navier-Stokes equations with a density-dependent viscosity coefficient λ(ρ)provided that the initial energy is small in some sense. In our result, we give a relation between the initial energy and the viscosity coefficient μ, and it shows that the initial energy can be large if the coefficient of the viscosity μ is taken to be large, which implies that large viscosity μ means large solution.

GLOBAL SOLUTIONS TO A 3D AXISYMMETRIC COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY

In this paper,we consider the 3D compressible isentropic Navier-Stokes equations when the shear viscosityμis a positive constant and the bulk viscosity is λ(ρ)=ρ^(β) with β>2,which is a situation that was first introduced by Vaigant and Kazhikhov in[1].The global axisymmetric classical solution with arbitrarily large initial data in a periodic domain Ω={(r,z)|r=√x^(2)+y^(2),(x,y,z)∈R^(3),r∈I⊂(0,+∞),-∞<z<+∞} is obtained.Here the initial density keeps a non-vacuum state ρ>0 when z→±∞.Our results also show that the solution will not develop the vacuum state in any finite time,provided that the initial density is uniformly away from the vacuum.

Time-Periodic Solutions to Quasilinear Hyperbolic Systems on General Networks

For quasilinear hyperbolic systems on general networks with time-periodic boundary-interface conditions with a dissipative structure,the existence and stability of the time-periodic classical solutions are discussed.

BOUNDEDNESS AND EXPONENTIAL STABILIZATION IN A PARABOLIC-ELLIPTIC KELLER–SEGEL MODEL WITH SIGNAL-DEPENDENT MOTILITIES FOR LOCAL SENSING CHEMOTAXIS

In this paper we consider the initial Neumann boundary value problem for a degenerate Keller-Segel model which features a signal-dependent non-increasing motility function.The main obstacle of analysis comes from the possible degeneracy when the signal concentration becomes unbounded.In the current work,we are interested in the boundedness and exponential stability of the classical solution in higher dimensions.With the aid of a Lyapunov functional and a delicate Alikakos-Moser type iteration,we are able to establish a time-independent upper bound of the concentration provided that the motility function decreases algebraically.Then we further prove the uniform-in-time boundedness of the solution by constructing an estimation involving a weighted energy.Finally,thanks to the Lyapunov functional again,we prove the exponential stabilization toward the spatially homogeneous steady states.Our boundedness result improves those in[1]and the exponential stabilization is obtained for the first time.

Global Stability to Steady Supersonic Rayleigh Flows in One-Dimensional Duct

Heat exchange plays an important role in hydrodynamical systems,which is an interesting topic in theory and application.In this paper,the authors consider the global stability of steady supersonic Rayleigh flows for the one-dimensional compressible Euler equations with heat exchange,under the small perturbations of initial and boundary conditions in a finite rectilinear duct.

The global classical solution to a 1D two-fluid model with density-dependent viscosity and vacuum

In this paper, we consider the initial-boundary problem for a 1D two-fluid model with densitydependent viscosity and vacuum. The pressure depends on two variables but the viscosity only depends on one of the densities. We prove the global existence and uniqueness of the classical solution in the one-dimensional space with large initial data and vacuum. We use a new Helmholtz free energy function and the material derivative of the velocity field to deal with the general pressure with two variables, without the equivalence condition. We also develop a new argument to handle the general viscosity.

Survey on Path-Dependent PDEs

In this paper,the authors provide a brief introduction of the path-dependent partial differential equations(PDEs for short)in the space of continuous paths,where the path derivatives are in the Dupire(rather than Fréchet)sense.They present the connections between Wiener expectation,backward stochastic differential equations(BSDEs for short)and path-dependent PDEs.They also consider the well-posedness of path-dependent PDEs,including classical solutions,Sobolev solutions and viscosity solutions.

A FREE-BOUNDARY PROBLEM TO DYNAMIC SYSTEM FOR PURE FOREST

A free-boundary model of nonlinear dynamic system for pure forest is presented, in which the felling rate is unbounded nearby the free boundary. The effiect of unbounded function on a priori estimate and analysis of regularity is overcome, and the existence and uniqueness of the global classical solution to this system are proved.

On the Cahn-Hilliard-Brinkman Equations in R^(4):Global Well-Posedness

We study the global well-posedness of large-data solutions to the Cauchy problem of the energy critical Cahn-Hilliard-Brinkman equations in R^(4).By developing delicate energy estimates,we show that for any given initial datum in H^(5)(R^(4)),there exists a unique global-in-time classical solution to the Cauchy problem.As a special consequence of the result,the global well-posedness of large-data solutions to the energy critical Cahn-Hilliard equation in R^(4) follows,which has not been established since the model was first developed over 60 years ago.The proof is constructed based on extensive applications of Gagliardo-Nirenberg type interpolation inequalities,which provides a unified approach for establishing the global well-posedness of large-data solutions to the energy critical Cahn-Hilliard and Cahn-Hilliard-Brinkman equations for spatial dimension up to four.