The dynamical systems approach to the equations of a linearly viscous compressible barotropic fluid [PDF]
Proceedings of the ICM, Beijing 2002, vol. 3, 295--304, 2003 We develop a dynamical systems theory for the compressible Navier-Stokes equations based on global in time weak solutions. The following questions will be addressed: Global existence and critical values of the adiabatic constant; dissipativity in the sense of Levinson - bounded absorbing sets; asymptotic compactness; and the long-time behaviour and ...
arxiv
The existence, nonexistence and uniqueness of global positive coexistence of a nonlinear elliptic biological interacting model [PDF]
arXiv, 2003 In this paper, we give a sufficient condition for the existence, nonexistence and uniqueness of coexistence of positive solutions to a rather general type of elliptic competition system.
arxiv
Model Theory of Partial Differential Fields: From Commuting to Noncommuting Derivations [PDF]
arXiv, 2006 McGrail has shown the existence of a model completion for the universal theory of fields on which a finite number of commuting derivations act and, independently, Yaffe has shown the existence of a model completion for the univeral theory of fields on which a fixed Lie algebra acts as derivations. We show how to derive the second result from the first.
arxiv
Well-posedness of the IBVP for 2-D Euler Equations with Damping [PDF]
arXiv, 2008 In this paper we focus on the initial-boundary value problem of the 2-D isentropic Euler equations with damping. We prove the global-in-time existence of classical solution to the initial-boundary value problem by the method of energy estimates.
arxiv
Existence of axially symmetric solutions to the Vlasov-Poisson system depending on Jacobi's integral [PDF]
arXiv, 2008 We prove the existence of axially symmetric solutions to the Vlasov--Poisson system in a rotating setting for sufficiently small angular velocity. The constructed steady states depend on Jacobi's integral and the proof relies on an implicit function theorem for operators.
arxiv
Maximal parabolic regularity for divergence operators including mixed boundary conditions [PDF]
arXiv, 2009 We show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of $A$ are discontinuous and $A$ is complemented with mixed boundary conditions.
arxiv
Global Strong Solution to the Density-Dependent Incompressible Viscoelastic Fluids [PDF]
arXiv, 2009 The existence and uniqueness of the global strong solution with small initial data to the three-dimensional viscoelastic fluids is established.
arxiv
Strong Solutions to the Three-Dimensional Compressible Viscoelastic Fluids [PDF]
arXiv, 2010 The existence and uniqueness of the local strong solution to the three-dimensional compressible viscoelastic fluids near the equilibrium is established. In addition to the uniform estimates on the velocity, some essential uniform estimates on the density and the deformation gradient are also obtained.
arxiv
On Neumann boundary problem for strongly degenerate parabolic-hyperbolic equations on a bounded rectangle [PDF]
arXiv, 2012 We study a Neumann type initial-boundary value problem for strongly degenerate parabolic-hyperbolic equations under the nonlinearity-diffusivity condition. We suggest a notion of entropy solution for this problem and prove its uniqueness. The existence of entropy solutions is also discussed.
arxiv
Gevrey Smoothing Effect for Solutions of the Non-Cutoff Boltzmann Equation in Maxwellian Molecules Case [PDF]
arXiv, 2013 In this paper we study the Gevrey regularity for the weak solutions to the Cauchy problem of the non-cutoff spatially homogeneous Botlzmann equation for the Maxwellian molecules model with the singularity exponent $s\in (0,1)$. We establish that any weak solution belongs to the Gevrey spaces for any positive time.
arxiv