A vanishing dynamic capillarity limit equation with discontinuous flux

Abstract

We 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.

A dynamic capillarity equation with stochastic forcing on manifolds: a singular limit problem

“…This is the capillary pressure, the pressure difference across the interface between capillaries' wetting and non-wetting phases. If one assumes that the capillary pressure changes dynamically depending on the fluid concentration [30,31], after a linearization procedure, one obtains a pseudo-parabolic equation, which we considered in [27] (see also [57]). Porous media flow phenomena often occur along a non-flat surface and are influenced by unpredictable (stochastic) sinks or sources.…”

A Godunov type scheme and error estimates for multidimensional scalar conservation laws with Panov-type discontinuous flux

“…Onedimensional conservation laws with discontinuous flux have been the subject of a large literature over the past several decades. The multidimensional case has received less attention, see e.g., [6,9,21,22,29,31,34,37,38]. Even for the case of one dimension (d = 1), mathematical analysis of these type of equations is complicated due to the presence of discontinuities in the spatial variable of the flux function A(•, •).…”