Very Weak Solutions to the Boundary-Value Problem of the Homogeneous Heat Equation

Abstract

We consider the homogeneous heat equation in a domain Ω in R n with vanishing initial data and the Dirichlet boundary condition. We are looking for solutions in W r,s p,q (Ω × (0, T )), where r < 2, s < 1, 1 ≤ p < ∞, 1 ≤ q ≤ ∞. Since we work in the Lp,q framework any extension of the boundary data and integration by parts are not possible. Therefore, the solution is represented in integral form and is referred as very weak solution. The key estimates are performed in the halfspace and are restricted to Lq(0, T ; W α p (Ω)), 0 ≤ α < 1 p and Lq(0, T ; W α p (Ω)), α ≤ 1. Existence and estimates in the bounded domain Ω follow from a perturbation and a fixed point arguments.2000 Mathematics Subject Classification. 35K05, 35K20.

Long time estimate of solutions to 3d Navier–Stokes equations coupled with heat convection

“…Lemma 2.8. (see [4]) Let ϕ ∈ L q (0, T ; L p (S)), p, q ∈ [1, ∞] then u ∈ L q (0, T ; L p (Ω)) and u L q (0,T ;L p (Ω)) ≤ c ϕ L q (0,T ;L p (S)) .…”

Well posedness and regularity for heat equation with the initial condition in weighted Orlicz–Slobodetskii space subordinated to Orlicz space like $$\lambda (\mathrm{log} \lambda )^\alpha $$ λ ( log λ ) α and the logarithmic weight

“…For regularity results dealing with the initial data in the classical Besov spaces B α,q p ( ) we refer to papers: [20,43,51,[58][59][60] and to their references. Our motivation to ask about regularity in the Orlicz setting comes from the fact that many mathematical models in the nonlinear elliptic and parabolic PDEs arising from the mathematical physics seem to have a good interpretation only when stated in Orlicz framework, see e.g.…”