A vanishing dynamic capillarity limit equation with discontinuous flux

M Graf,D Vujadinovic,M Kunzinger,D Mitrovic

doi:10.1007/s00033-020-01432-3

Abstract

We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the dynamics capillarity equation ∂tuε,δ+divfε,δ(x,uε,δ)=εΔuε,δ+δ(ε)∂tΔuε,δ,x∈M,t≥0u|t=0=u0(x).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\partial _t u_{\\varepsilon ,\\delta } +\\mathrm {div} {\\mathfrak f}_{\\varepsilon ,\\delta }(\\mathbf{x}, u_{\\varepsilon ,\\delta })=\\varepsilon \\Delta u_{\\varepsilon ,\\delta }+\\delta (\\varepsilon ) \\partial _t \\Delta u_{\\varepsilon ,\\delta }, \\ \\ \\mathbf{x} \\in M, \\ \\ t\\ge 0\\\\ u|_{t=0}=u_0(\\mathbf{x}). \\end{array}\\right. } \\end{aligned}$$\\end{document}Here, {{mathfrak {f}}}_{varepsilon ,delta } and u_0 are smooth functions while varepsilon and delta =delta (varepsilon ) are fixed constants. Assuming {{mathfrak {f}}}_{varepsilon ,delta } rightarrow {{mathfrak {f}}}in L^p( {mathbb {R}}^dtimes {mathbb {R}};{mathbb {R}}^d) for some 1<p<infty , strongly as varepsilon rightarrow 0, we prove that, under an appropriate relationship between varepsilon and delta (varepsilon ) depending on the regularity of the flux {{mathfrak {f}}}, the sequence of solutions (u_{varepsilon ,delta }) strongly converges in L^1_{loc}({mathbb {R}}^+times {mathbb {R}}^d) toward a solution to the conservation law ∂tu+divf(x,u)=0.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\partial _t u +\\mathrm {div} {{\\mathfrak {f}}}(\\mathbf{x}, u)=0. \\end{aligned}$$\\end{document}The main tools employed in the proof are the Leray–Schauder fixed point theorem for the first part and reduction to the kinetic formulation combined with recent results in the velocity averaging theory for the second. These results have the potential to generate a stable semigroup of solutions to the underlying scalar conservation laws different from the Kruzhkov entropy solutions concept.

Highlights

Introduction and notationFlow in a two-phase porous medium is governed by the Darcy law [7]q = −K(S) (∇p + ρged), (1.1)where ed = (0, . . . , 0, 1), is the direction of gravity
The Darcy law represents conservation of momentum and, in order to close the system, we need the conservation of mass
Usually one assumes a constitutive relation between the pressure p, the capillary pressure Pc, and the saturation S

Summary

Introduction

We conclude the introduction by noting that the subsequence of (uε) given in (i) does not converge toward an entropy admissible solution to the underlying conservation law since its flux is not smooth and for such fluxes the well-posedness theory is not developed yet in full generality (compare with item (iii) above). 3, we derive necessary estimates from (1.7), (1.10) and use it to prove the strong convergence toward a weak solution to the underlying conservation law.

Results

Conclusion