Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right‐hand side in L^p (p⩾1)

Abstract

SUMMARYAccurate modelling of heat transfer in high-temperature situations requires accounting for the effect of heat radiation. In complex industrial applications involving dissipative heating, we hardly can expect from the mathematical theory that the heat sources will be in a better space than L 1 . In this paper, we focus on a stationary heat equation with nonlocal boundary conditions and L p right-hand side, with p 1 being arbitrary. Thanks to new coercivity results, we are able to produce energy estimates that involve only the L p norm of the heat sources and to prove the existence of weak solutions. Copyright q 2008 John Wiley & Sons, Ltd.KEY WORDS: existence of generalized solutions; nonlinear PDE of elliptic type; radiative heat transfer; nonlocal boundary condition; right-hand side L 1 INTRODUCTIONAccurate modelling of heat transfer in high-temperature situations requires accounting for the effect of heat radiation. In the field of industrial applications, crystal growth, for example, has motivated a lot of mathematical work on this topic [1][2][3][4][5][6]. For this type of applications, situations are relevant in which a transparent medium is enclosed by one or several opaque or diffuse * Correspondence to: Pierre-Étienne Druet, Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany. † E-mail: druet@wias-berlin.de The heat radiation incoming in a part of the boundary of the transparent cavity 0 depends on the radiation outgoing at the other parts of the surface that it can see. This effect has to be modelled by means of nonlocal radiation boundary conditions for the conductive heat flux (see, for example, [2,6]).From the point of view of mathematical analysis, an important result was attained in the paper [7], in which existence and uniqueness of generalized solutions were proved for the heat equation with radiation boundary conditions and heat sources in the class [W 1,2 ] * .In the present paper, we want to extend these results to the case where the heat source density might be less regular. In many applications, the heat sources have to be computed from Maxwell's equations (resistive/inductive heating) or from the Navier-Stokes equations (heat-conducting fluids). From the viewpoint of the presently available regularity theory, this leads in complex situations (temperature-dependent coefficients, nonsmooth surfaces) to heat source densities that belong only to L 1 , or at most to L 1+ . The latter observations have motivated research on elliptic problems with L 1 right-hand sides (see, for example, [8,9]). An L 1 -theory is also particularly attractive for the heat equation in that it leads to natural energy estimates, that is, estimates in terms of the total heating power, the quantity which is actually controlled in applications. The mathematical problemWe assume that 1 , . . . , m are disjoint bounded domains in R 3 , separated from each other by a transparent medium 0 . They represent opaque bodies with different material properties. The bounded domain...

Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary p-summable right-hand side

“…This proves (2). (3) For θ δ , according to Proposition 4.2 we are allowed to introduce the solution R δ of the radiosity equation (0.10),…”

“…This can be generalized to the case of a piecewise C 1 -boundary (see for example [3], [15], [2]). The second constitutive relation between R and J is then given by (0.8)…”

“…As in the stationary case, we can prove the existence of weak solutions only if the surface Σ is sufficiently smooth (cf. [2]). In addition, we obtain uniform estimates only in the case of a nowhere vanishing reflexivity, that is…”

Existence for the stationary MHD-equations coupled to heat transfer with nonlocal radiation effects

“…P r o o f. For the proof of the point (1), see [14]. The proof of point (2) in [5] is correct if ε does not take the value one.…”

“…Now we prove the convergence property for the boundary integral. Since the employed techniques are similar to the ones used in [5], we will only give the main ideas. First, we prove that the sequence of numbers M δ given by (68) is bounded.…”