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Friends in Partial Differential Equations
May 25, 2024 15:20, St. Petersburg, St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, online


Kružkov-type uniqueness theorem for the chemical flood conservation law system

N. V. Rastegaev

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences



Abstract: We consider the system
$$ \begin{cases} s_t + f(s, c)_x = 0, \\ (cs + a(c))_t + (cf(s,c))_x = 0, \end{cases} $$
most commonly used to describe the flood of the oil reservoir with a chemical solution. The flow function $f$ is commonly assumed to be S-shaped in $s$. The adsorption function $a$ is often concave and usually represented by the Langmuir adsorption isotherm. In our work, we limit ourselves to functions $f$ monotone in $c$.
This system is neither strictly hyperbolic nor genuinely non-linear, therefore known results for strictly hyperbolic genuinely non-linear systems of conservation laws are not directly applicable.
The solutions for some boundary-initial problems for this system were explored, for example, in [1] (Riemann problem), [2] and [3] (slug injection). The last two papers use the Lagrange coordinate transformation to split the equations and the characteristics method to construct solutions. However, the question of the uniqueness of the constructed solutions is not covered.
We use the proposed coordinate change to prove a Kružkov-type uniqueness theorem for the Cauchy problem with several limitations on the initial data and the class of weak solutions under consideration. The vanishing viscosity method is utilized to determine admissible shocks.
This talk is based on the joint work with S. G. Matveenko.

Language: English

References
  1. T. Johansen, R. Winther, “The solution of the Riemann problem for a hyperbolic system of conservation laws modeling polymer flooding”, SIAM journal on mathematical analysis, 19:3 (1988), 541–566  crossref  mathscinet  zmath
  2. A.P. Pires, P.G. Bedrikovetsky, and A.A. Shapiro, “A splitting technique for analytical modelling of two-phase multicomponent flow in porous media”, Journal of Petroleum Science and Engineering, 51:1–2 (2006), 54–67  crossref  mathscinet
  3. P.M. Ribeiro, A.P. Pires, “The displacement of oil by polymer slugs considering adsorption effects”, SPE Annual Technical Conference and Exhibition, 2008, September, SPE-115272


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