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New unilateral problems in stratigraphy

Stanislav N. Antontsev, Gérard Gagneux, Robert Luce, Guy Vallet (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

This work deals with the study of some stratigraphic models for the formation of geological basins under a maximal erosion rate constrain. It leads to introduce differential inclusions of degenerated hyperbolic-parabolic type 0 ∈ ∂ t u - d i v { H ( ∂ t u + E ) ∇ u } , where H is the maximal monotonous graph of the Heaviside function and E is a given non-negative function. Firstly, we present the new and realistic models and an original mathematical formulation, taking into account the weather-limited rate constraint in the conservation...

No production of entropy in the Euler fluid

R. F. Streater (2004)

Banach Center Publications

We derive the Euler equations as the hydrodynamic limit of a stochastic model of a hard-sphere gas. We show that the system does not produce entropy.

Non-autonomous 2D Navier–Stokes system with a simple global attractor and some averaging problems

V. V. Chepyzhov, M. I. Vishik (2002)

ESAIM: Control, Optimisation and Calculus of Variations

We study the global attractor of the non-autonomous 2D Navier–Stokes system with time-dependent external force g ( x , t ) . We assume that g ( x , t ) is a translation compact function and the corresponding Grashof number is small. Then the global attractor has a simple structure: it is the closure of all the values of the unique bounded complete trajectory of the Navier–Stokes system. In particular, if g ( x , t ) is a quasiperiodic function with respect to t , then the attractor is a continuous image of a torus. Moreover the...

Non-autonomous 2D Navier–Stokes system with a simple global attractor and some averaging problems

V. V. Chepyzhov, M. I. Vishik (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We study the global attractor of the non-autonomous 2D Navier–Stokes system with time-dependent external force g(x,t). We assume that g(x,t) is a translation compact function and the corresponding Grashof number is small. Then the global attractor has a simple structure: it is the closure of all the values of the unique bounded complete trajectory of the Navier–Stokes system. In particular, if g(x,t) is a quasiperiodic function with respect to t, then the attractor is a continuous image...

Non-Euclidean geometry and differential equations

A. Popov (1996)

Banach Center Publications

In this paper a geometrical link between partial differential equations (PDE) and special coordinate nets on two-dimensional smooth manifolds with the a priori given curvature is suggested. The notion of G-class (the Gauss class) of differential equations admitting such an interpretation is introduced. The perspective of this approach is the possibility of applying the instruments and methods of non-Euclidean geometry to the investigation of differential equations. The equations generated by the...

Non-existence of global classical solutions to 1D compressible heat-conducting micropolar fluid

Jianwei Dong, Junhui Zhu, Litao Zhang (2024)

Czechoslovak Mathematical Journal

We study the non-existence of global classical solutions to 1D compressible heat-conducting micropolar fluid without viscosity. We first show that the life span of the classical solutions with decay at far fields must be finite for the 1D Cauchy problem if the initial momentum weight is positive. Then, we present several sufficient conditions for the non-existence of global classical solutions to the 1D initial-boundary value problem on [ 0 , 1 ] . To prove these results, some new average quantities are...

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