Integral equations method and the transmission problem for the stokes system

Abstract

Abstract. The transmission problem for the Stokes system is studied:is a bounded open set with Lipschitz boundary and G − is the corresponding complementary open set. Using the integral equation method we study the problem in homogeneous Sobolev spaces. Under assumption that ∂G + is of class C 1 we study this problem also in Besov spaces and L q -solutions of the problem. We show the unique solvability of the problem. Moreover, we solve the corresponding boundary integral equations by the successive approximation method.

Integral potential method for a transmission problem with Lipschitz interface in $$\mathbb{R}^{3}$$ R 3 for the Stokes and Darcy–Forchheimer–Brinkman PDE systems

“…In this paper we will mainly deal with the first three systems. The layer potential method has a well known role in the study of elliptic boundary value problems (see, e.g., [15,18,24,30,41,47,48,53,57,66,67]). Lang and Mendez [42] have used a layer potential technique to obtain optimal solvability results in weighted Sobolev spaces for the Poisson problem for the Laplace equation with Dirichlet or Neumann boundary conditions in an exterior Lipschitz domain.…”

“…Then the Brinkman double layer velocity potential can be also written asW α;∂Ω h = W ∂Ω h + αN α;R 3 W ∂Ω h in R 3 \ ∂Ω, ∀ h ∈ H (∂Ω) 3 . (A 48). In addition, the last term in (A.47) can be expressed as V △ (h · ν) = −Q R 3 W ∂Ω h. Indeed, by using again the property that [γu h ] = −h, where u h = W ∂Ω h, we obtain(Q R 3 u h ) (x) = − Ω−∪Ω+ x − y 4π|x − y| 3 · u h (y)dy = − ν y · [γu h (y)]dy + 1 4π Ω−∪Ω+ 1 |x − y| div u h (y)dy = V △ (ν · [γu h ])(x) = −V △ (h · ν)(x).Now, the continuity of the operators involved in (A.48) leads to the continuity of the first operators in (A.37) and (A.39).…”

A note on a transmission problem for the Brinkman system and the generalized Darcy-Forchheimer-Brinkman system in Lipschitz domains in R3

“…which show the continuity of the operators b 1 and b 2 . Note that, the operator J k,β,Ω+ can be written as: 17) and by employing the relations (2.16), we have that J k,β,Ω+ satisfies (2.11) with c 0 = c * + c * , as asserted. Also, by using similar arguments to those in the proof of [8, Lemma 5.1] and again relations (2.16), one shows that the operator J k,β,Ω+ satisfies the Lipschitz-like condition (2.12).…”

On transmission-type problems for the generalized Darcy-Forchheimer-Brinkman and stokes systems in complementary lipschitz domains in R3

“…Then, by using inequalities (18) and (21), we deduce that our operator J k,β,Γ + satisfies relation (12) with the constant c 0 := c * + c * and hence it is a bounded operator and positively homogeneous of order 2. By using again relations (18) and (21) one may show that the Lipschitz-like inequality (13) for the operator J k,β,Γ + holds (using similar arguments as in the proof of [14, Lemma 5.1]).…”

“…Mitrea and Wright [20] have studied transmission-type problems for the Stokes system in Sobolev and Besov spaces. Medkova [18] has used the integral equations method in order to treat a transmission problem for the Stokes system in a Lipschitz domain in R 3 . Medkova [17] has studied a transmission problem for the Brinkman system in Lipschitz domains in R n , n > 2.…”