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.