A vanishing dynamic capillarity limit equation with discontinuous flux

Abstract

AbstractWe prove existence and uniqueness of a solution to the Cauchy problem corresponding to the dynamics capillarity equation $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u_{\varepsilon ,\delta } +\mathrm {div} {\mathfrak f}_{\varepsilon ,\delta }(\mathbf{x}, u_{\varepsilon ,\delta })=\varepsilon \Delta u_{\varepsilon ,\delta }+\delta (\varepsilon ) \partial _t \Delta u_{\varepsilon ,\delta }, \ \ \mathbf{x} \in M, \ \ t\ge 0\\ u|_{t=0}=u_0(\mathbf{x}). \end{array}\right. } \end{aligned}$$ ∂ t u ε , δ + div f ε , δ ( x , u ε , δ ) = ε Δ u ε , δ + δ ( ε ) ∂ t Δ u ε , δ , x ∈ M , t ≥ 0 u | t = 0 = u 0 ( x ) . Here, $${{\mathfrak {f}}}_{\varepsilon ,\delta }$$ f ε , δ and $$u_0$$ u 0 are smooth functions while $$\varepsilon $$ ε and $$\delta =\delta (\varepsilon )$$ δ = δ ( ε ) are fixed constants. Assuming $${{\mathfrak {f}}}_{\varepsilon ,\delta } \rightarrow {{\mathfrak {f}}}\in L^p( {\mathbb {R}}^d\times {\mathbb {R}};{\mathbb {R}}^d)$$ f ε , δ → f ∈ L p ( R d × R ; R d ) for some $$1<p<\infty $$ 1 < p < ∞ , strongly as $$\varepsilon \rightarrow 0$$ ε → 0 , we prove that, under an appropriate relationship between $$\varepsilon $$ ε and $$\delta (\varepsilon )$$ δ ( ε ) depending on the regularity of the flux $${{\mathfrak {f}}}$$ f , the sequence of solutions $$(u_{\varepsilon ,\delta })$$ ( u ε , δ ) strongly converges in $$L^1_{loc}({\mathbb {R}}^+\times {\mathbb {R}}^d)$$ L loc 1 ( R + × R d ) toward a solution to the conservation law $$\begin{aligned} \partial _t u +\mathrm {div} {{\mathfrak {f}}}(\mathbf{x}, u)=0. \end{aligned}$$ ∂ t u + div f ( x , u ) = 0 . 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.