Solvability in the sense of sequences for certain non-Fredholm operators with a drift and Laplace and bi-Laplace operators

Detailed Technical Summary

Problem Statement
This work investigates the solvability of specific linear nonhomogeneous elliptic problems involving fourth-order differential operators that lack the Fredholm property. The primary focus is on equations defined on the whole real line ($\mathbb{R}$) and on a finite interval ($I = [0, 2\pi]$) with periodic boundary conditions. The general form of the equation considered is:
$$(-\Delta + V(x))u - au = f$$
In the specific one-dimensional cases analyzed, the operator takes the form:
$$L_{a,b}u = -\frac{d^2u}{dx^2} + \frac{d^4u}{dx^4} - b\frac{du}{dx} - au = f(x)$$
where $a \geq 0$ and $b \in \mathbb{R}, b \neq 0$ are constants. The term $-b\frac{du}{dx}$ represents a drift (or transport) term.

The central difficulty arises when $a=0$. In this case, the origin belongs to the essential spectrum of the operator $L_{a,b}$, causing the operator to fail the Fredholm property. Consequently, the image of the operator is not closed, and standard solvability theories do not apply directly. The paper addresses the "solvability in the sense of sequences": if a sequence of right-hand sides $f_n$ converges to $f$ in $L^2$, does the corresponding sequence of solutions $u_n$ converge to a solution $u$ of the limiting equation? The goal is to establish conditions under which this sequence convergence implies the existence of a strong solution in the appropriate Sobolev space ($H^4$).

Methodology
The analysis relies heavily on spectral theory and Fourier analysis tailored to the specific domains:

1. Fourier Transform on $\mathbb{R}$: The standard Fourier transform is applied to convert the differential equations into algebraic relations in the frequency domain. The essential spectrum is identified as $\lambda_{a,b}(p) = p^2 + p^4 - a - ibp$.
2. Fourier Series on $I$: For the periodic case, Fourier series are utilized, with the spectrum given by $\lambda_{a,b}(n) = n^2 + n^4 - a - ibn$ for $n \in \mathbb{Z}$.
3. Orthogonality Conditions: When $a=0$, the solvability is linked to orthogonality relations between the source term $f$ and the kernel of the adjoint operator (specifically, the constant function 1).
4. Regularity Arguments: The authors demonstrate that if a solution exists in $L^2$, the structure of the operator (specifically the presence of the bi-Laplacian and the drift term) ensures that the solution actually belongs to $H^4$.
5. Sequence Convergence: The proofs involve estimating the difference between approximate solutions $u_m$ and the limit solution $u$ in the $L^2$ norm, and subsequently showing that this implies convergence in the $H^4$ norm.

Key Contributions and Results

• Regularization by Drift: The paper establishes that the inclusion of the drift term ($b \neq 0$) provides a regularization effect. While the operator without the drift term (self-adjoint) is non-Fredholm and singular when $a=0$, the non-self-adjoint operator with the drift term allows for solvability under specific conditions. The drift term prevents the essential spectrum from containing the origin in a way that blocks solvability, provided orthogonality conditions are met.

• Results on the Whole Real Line ($\mathbb{R}$):

  • Case $a > 0$: The operator is Fredholm. The problem admits a unique solution $u \in H^4(\mathbb{R})$ for any $f \in L^2(\mathbb{R})$. Furthermore, if $f_m \to f$ in $L^2$, then $u_m \to u$ in $H^4$.
  • Case $a = 0$: The operator is non-Fredholm. A unique solution $u \in H^4(\mathbb{R})$ exists if and only if the orthogonality condition $(f, 1)_{L^2(\mathbb{R})} = 0$ holds, provided $xf(x) \in L^1(\mathbb{R})$. Under these conditions, if $f_m \to f$ in $L^2$ and $xf_m \to xf$ in $L^1$ (with $(f_m, 1) = 0$), then $u_m \to u$ in $H^4$.

• Results on the Finite Interval ($I$):

  • Case $a > 0$: Similar to the real line, unique solutions exist in $H^4(I)$, and sequence convergence holds.
  • Case $a = 0$: The problem is studied in the constrained subspace $H^4_0(I)$ (functions orthogonal to constants). A unique solution exists if and only if $(f, 1)_{L^2(I)} = 0$. Sequence convergence in $H^4_0(I)$ is established under the assumption that $f_m \to f$ in $C(I)$ and the orthogonality conditions are preserved for the sequence.

Significance and Claims
The paper claims to establish sufficient conditions for the equivalence of "solutions in the sense of sequences" and "usual solutions" for a class of non-Fredholm operators. Specifically, it determines the precise conditions on the sequences of right-hand sides ($f_n$) that guarantee the strong convergence of the corresponding solution sequences ($u_n$) in the $H^4$ norm.

The authors emphasize that the transport (drift) term is significant for the regularization of solutions. Without this term, the case $a=0$ is more singular. The work extends the understanding of solvability for fourth-order operators (bi-Laplacian) with drift, which are relevant to reaction-diffusion problems and pattern formation theories, though the paper itself focuses strictly on the mathematical solvability conditions rather than proposing new physical applications. The results unify the treatment of these operators on both unbounded and bounded periodic domains.