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For Darcy-Bénard convection, the dimensionless Darcy law is $\mathbf{u}=-\nabla p+R\theta {\mathbf{e}}_z$, with incompressibility and a heat equation about $T_0=1-z$. Here $R$ is a Darcy thermal-time Rayleigh number, with permeability replacing the squared-depth dependence in the fluid-layer Rayleigh number. The stated stream function convention gives $\mathbf{u}=(-{\psi }_z,0,{\psi }_x)$, so $w={\psi }_x$. The vertical-plane curl of Darcy law gives ${\nabla }^2\psi =R{\theta }_x$; omitting the quadratic perturbation advection from the heat equation gives ${\theta }_t={\psi }_x+{\nabla }^2\theta$.

On the impermeable connected square boundary, zero normal velocity makes $\psi$ constant along the boundary; choose that constant as zero. The prescribed conductive boundary temperature fixes $\theta =0$ on all four sides. Thus the perfect-conductor wording is interpreted through the supplied homogeneous perturbation data: the side temperatures retain the background vertical profile, rather than imposing a different isothermal side state.

At marginal stability put $q=\sqrt{R}$ and $P=\psi +iq\theta$. The two coupled equations implyThe gauge transformation for conducting-square Darcy onset $P=e^{-iq(x-1/2)/2}F$ removes the first derivative:The square's Dirichlet Laplacian eigenfunctions are $\sin (m\pi x)\sin (n\pi z)$, with eigenvalues of $-{\nabla }^2$ equal to ${\pi }^2(m^2+n^2)$, $m,n\ge 1$. The first marginal value is thereforeIt is genuinely the first instability threshold, not just an isolated neutral value. Eliminating $\psi$ gives the temperature linear operator $L_R={\Delta }_D+R{\partial }_x{\Delta }_D^{-1}{\partial }_x$, where ${\Delta }_D$ is the Dirichlet realization of an elliptic operator for the Laplacian. Integration by parts shows the second term is self-adjoint and nonnegative, since its quadratic form is $-⟨{\theta }_x,{\Delta }_D^{-1}{\theta }_x⟩\ge 0$. Thus the spectrum is real, starts negative at $R=0$, and its leading eigenvalue cannot cross into growth without a zero eigenvalue. The smallest zero value is the one found above.

Let $S=\sin \pi x\sin \pi z$, $\xi =x-1/2$ and $a=q_c/2=\sqrt{2}\pi$. Multiplying the lowest real $F=S$ by independent real and imaginary constants gives two real physical eigenfunctions:The subscripts indicate even and odd parity about $x=1/2$. Because $S$ is even in $\xi$, the first has even $\psi$ and odd $\theta$, while the second reverses those parities. They are linearly independent and satisfy all four Dirichlet boundary conditions. The lowest complex $F$ eigenfunction is simple over the complex numbers, but its arbitrary complex multiplier supplies two independent real pairs; this explains the physical degeneracy.

The assertion about all $H_0^1(U)\cap H^k(U)$ is false as printed in the original PDF. Take $U=(0,\pi )$, $L=-d^2/d{x}^2$, and $u=x(\pi -x)$. This is a nonzero smooth Dirichlet function, but $Lu=2$ and $L^{2}u=0$. In particularAlready at $k=3$, $B[Lu,Lu]$ is not defined on its stipulated domain, because $Lu=2\notin H_0^1$. Extending $B$ to $H^1$ for this example would give zero, not an inner product. Additional boundary conditions are essential.

The corrected spaces are the Sobolev domains of powers of an elliptic Dirichlet operator. Let $A_D$ be the Dirichlet realization of an elliptic operator, and setIn particular $X_1=H_0^1$, $X_2=H^2\cap H_0^1$, and $X_3$ also requires $T(Lu)=0$. The formulas in the question define inner products on these spaces.

Here is the norm-equivalence proof on the corrected domains. The base cases are the $L^2$ inner product at $k=0$ and part (i) at $k=1$. Standard Dirichlet elliptic regularity, together with invertibility from the Lax-Milgram theorem, gives for every integer $r\ge 0$The first estimate applies when $Tu=0$; invertibility absorbs the usual lower-order $L^2$ term. Moreover, $A_D:X_{r+2}\to X_r$ is an isomorphism. To see surjectivity, solve $Lu=f$ with zero Dirichlet trace for $f\in X_r$; regularity gives $u\in H^{r+2}$, and $T(L^{j}u)=T(L^{j-1}f)=0$ for each further required trace. The formulas satisfyInduction therefore givesInjectivity of each iterate of $A_D$ gives positive definiteness. These spaces are $D(A_D^{k/2})$; the spectral characterization of elliptic Dirichlet domains in the next part makes the half-powers precise. On the paper's uncorrected spaces the claim is valid at $k=0,1,2$, but fails in general beyond them.

For a strictly positive Dirichlet realization of an elliptic operator with smooth coefficients and smooth boundary, put $X_0=L^2(U)$. For each integer $k\ge 1$,These are the domains $D(A_D^{k/2})$, with norms equivalent to the indicated Sobolev space norms. The bilinear forms $((u,v){)}_{2l}=(L^{l}u,L^{l}v{)}_{L^2}$ and $((u,v){)}_{2l+1}=B[L^{l}u,L^{l}v]$ are inner products on $X_{2l}$ and $X_{2l+1}$. The higher boundary conditions are essential: for $L=-d^2/d{x}^2$ and $u=x(\pi -x)$, $u$ is smooth and zero at the boundary but $L^{2}u=0$, so $\Vert L^{2}u{\Vert }_2$ is not a norm on all of $H^4\cap H_0^1$. Repeated elliptic regularity proves $A_D:X_{k+2}\to X_k$ is an isomorphism, which gives the norm equivalence inductively from $X_0$ and $X_1=H_0^1$.

Let $(w_m)$ be the orthonormal basis of Dirichlet eigenfunctions for a strictly positive Dirichlet realization of an elliptic operator, with eigenvalues ${\lambda }_m$. ThenFor integers $k$, these domains are the Sobolev domains of powers of an elliptic Dirichlet operator. At $k=0$ the condition is just Parseval identity and imposes no boundary condition. At $k=1,2$ it describes $H_0^1$ and $H^2\cap H_0^1$, respectively. Higher $k$ impose traces of powers of $L$, not only the trace of $u$. For $L=-d^2/d{x}^2$ on $(0,\pi )$, the smooth Dirichlet function $u=x(\pi -x)$ has sine coefficients proportional to $m^{-3}$ for odd $m$, so the weighted sum diverges at $k=3$ despite its ordinary $H^3$ regularity.

For a time-independent strictly positive Dirichlet realization of an elliptic operator and $\psi \in L^2(U)$, the eigenfunction expansionsolves $u_t+Lu=0$ with homogeneous Dirichlet boundary conditions. For $t\ge \epsilon >0$, every power ${\lambda }_m^N$ times the exponential is bounded, so the series and all its time derivatives converge in all the Sobolev domains of powers of an elliptic Dirichlet operator. Elliptic regularity and the Sobolev embedding theorem give smoothness up to the spatial boundary for positive time. Parseval identity and the dominated convergence theorem give $u(t)\to \psi$ in $L^2$ as $t\downarrow 0$.