Flow (mathematics) - Examples - Solutions of Heat Equation

Solutions of Heat Equation

Let be a subdomain (bounded or not) of ( is an integer). We denote its boundary (supposed smooth). Let us consider the following Heat Equation on (for ):

$\begin{array}{rcll} u_t - \Delta u & = & 0 & \mbox{ in } \Omega \times (0,T), \\ u & = & 0 & \mbox{ on } \Gamma \times (0,T), \end{array}$

with the following initial boundary condition .

Equation corresponds to the Homogeneous Dirichlet boundary condition. The mathematical setting for this problem can be the semigroup approach. To use this tool, we introduce the unbounded operator defined on by its domain (see the classical Sobolev spaces with and is the closure of the infinitely differentiable functions with compact support in for the norm). For any, we have

$\Delta_D v = \Delta v = \sum_{i=1}^n \frac{\partial^2 }{\partial x_i^2} v.$

With this operator, the heat equation becomes and . Thus, the flow corresponding to this equation is (see notations above)

$\varphi(u^0,t) = \mbox{e}^{t\Delta_D}u^0$ where is the (analytic) semigroup generated by .

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