We consider a chemotaxis system consisting of one parabolic equation and one ordinary differential equation in one space dimension with a logarithmic chemotactic sensitivity function and the exponential growth of dynamics for the chemical attractant. The structure of positive solutions is studied from the viewpoint of infinite-dimensional dynamical systems.

In a recent paper in collaboration with S. Benzoni, F. Rousset and K. Zumbrun, we uncovered the amazing fact that besides the classes of Hadamard unstable (no estimate at all) and of strongly stable hyperbolic IBVPs (estimates without loss of derivatives), there is a third “open” class, in the sense that it is stable under small disturbances of the coefficients (we apologize for the use of the word “stable” with two different meanings) in the PDEs and in the boundary conditions. We called the latter class (WR), because it has a Weak stability property (estimates with loss of one derivative) and because it is characterized by a “Real” characteristic set for the Lopatinsliĭ determinant.

We show here that with an appropriate filtering, systems in the class WR can be recast as strongly stable ones. It is remarkable that the same filtering is applied to both the solution and the data. Thus there is a hope to apply this linear theory to fixed point methods in non-linear problems, like the stability of two-dimensional vortex sheets when the jump in the velocity is larger than 2√2 times the sound speed.

A precise finite speed result is proved for general hyperbolic systems using neither Hamilton-Jacobi Theory as in J.L. Joly, G. Métivier, J. Rauch, Hyperbolic domains of determinacy and Hamilton-Jacobi equatiions, nor the theorem of Marchaud as in Leray. The proof is shorter than both J. Leray and J.L. Joly, G. Métivier, J. Rauch, and more general than the first. It does not give the additional information about spacelike deformations proved in J.L. Joly, G. Métivier, J. Rauch.

We study the time-asymptotic behavior of solutions to the general Navier-Stokes equations in odd multi-dimensions. Through the pointwise estimates of the Green’s function of the linearized system, we obtain explicit expressions of the time-asymptotic behavior of the solutions and exhibit the weak Huygen’s principle.

This paper investigates some properties of entropy solutions of hyperbolic conservation laws on a Riemannian manifold. First, we generalize the Total Variation Diminishing (TVD) property to manifolds, by deriving conditions on the flux of the conservation law and a given vector field ensuring that the total variation of the solution along the integral curves of the vector field is non-increasing in time. Our results are next specialized to the important case of a flow on the 2-sphere, and examples of flux are discussed. Second, we establish the convergence of the finite volume methods based on numerical flux-functions satisfying monotonicity properties. Our proof requires detailed estimates on the entropy dissipation, and extends to general manifolds an earlier proof by Cockburn, Coquel, and LeFloch in the Euclidian case.

Conservation laws have been used to model a variety of physical phenomena and therefore the theory for this class of equations is well developed. However, in many problems, such as transport of hot fluids and gases undergoing mass transfer, balance laws are required to describe the flow.

As an example, in this work we obtain the solutions for the basic one-dimensional profiles that appear in the clean up problem or in recovery of geothermal energy. We consider the injection of a mixture of steam and water in several proportions in a porous rock filled with a different mixture of water and steam. We neglect compressibility, heat conductivity and capillarity and present a physical model for steam injection based on the mass balance and energy conservation equations.

We describe completely all possible solutions of the Riemann problem. We find several types of shock between regions and develop a scheme to find the solution from these shocks. A new type of shock, the evaporation shock, is identified in the Riemann solution. This work generalizes the work of Bruining et. al., where the condensation shock appears. It is a step towards obtaining a general method for solving Riemann problems for a wide class of balance equations with phase changes.