arXiv:1909.02377v1 [math.AP] 5 Sep 2019 PARABOLIC EQUATIONS WITH DYNAMIC BOUNDARY CONDITIONS AND DRIFT TERMS A. KHOUTAIBI, L. MANIAR, D. MUGNOLO, A. RHANDI Abstract. The aim of this paper is to study the wellposedness and L 2 -regularity, firstly for a linear heat equation with dynamic boundary conditions by using the approach of sesquilinear forms, and secondly for its backward adjoint equation using the Galerkin approximation and the extension semigroup to a negative Sobolev space. 1. Introduction In this paper, we consider the wellposedness of the heat equation with dynamic boundary conditions and drift terms in the interior and in the boundary                ∂ t y − dΔy + B(x).∇y + c(x)y = f in Ω T ∂ t y Γ − δΔ Γ y Γ + d∂ ν y + b(x).∇ Γ y Γ + ℓ(x)y Γ = g on Γ T , y |Γ (t,x)= y Γ (t,x) on Γ T , y(0, ·)= y 0 in Ω, y |Γ (0, ·)= y 0,Γ on Γ, (1.1) where Ω is a bounded domain of R N , with smooth boundary Γ = ∂ Ω of class C 2 , N ≥ 2. Further, y |Γ denotes the trace of a function y :Ω → R, ν (x) is the outer unit normal field to Ω in x ∈ Γ, ∂ ν y := (ν.∇y) |Γ , d, δ are positive real numbers, c ∈ L ∞ (Ω), ℓ ∈ L ∞ (Γ), B ∈ L ∞ (Ω) N and b ∈ L ∞ (Γ) N . Further, we denote by Ω T := (0,T ) × Ω, Γ T := (0,T ) × ∂ Ω, for T> 0, and by y Γ the trace on Γ of a function y :Ω → R. Finally, Δ is the Laplace operator, Δ Γ is the Laplace-Beltrami operator on the Riemannian sub manifold Γ, ∇ Γ is the tangential gradient on the Riemannian sub manifold Γ, y 0 ∈ L 2 (Ω), y 0,Γ ∈ L 2 (Γ) are the initial states, and the inhomogeneous terms f and g are respectively in L 2 ((0,T ) × Ω) and L 2 ((0,T ) × Γ). We emphasize that y 0,Γ is not necessarily the trace of y 0 . Recall that the boundary Γ of the open set Ω ⊂ R N can be viewed as a Riemannian manifold endowed with the natural metric inherited from R N , given in local coordinates by √ det Gdy 1 ...dy N−1 , where G =(g ij ) denotes the metric tensor. Putting (g ij )=(g ij ) −1 , the so called Laplace-Beltrami operator Δ Γ is given in local coordinates g by Δ Γ u = 1 √ det G N−1  i,j =1 ∂ ∂y i  √ det Gg ij ∂u ∂y j  . We recall also the surface divergence theorem  Γ Δ Γ uv dσ = −  Γ 〈∇ Γ u, ∇ Γ v〉 Γ dσ, (u,v) ∈ H 2 (Γ) × H 1 (Γ), (1.2) 2010 Mathematics Subject Classification. 35A15; 35K20; 47D06. Key words and phrases. 1