Global Well-Posedness for the Fourth-Order Nonlinear Schrödinger Equation with Potential in the Energy-Critical Case

1. Problem Statement

The paper investigates the fourth-order nonlinear Schrödinger equation (4NLS) with an external potential $V(x)$ in the energy-critical regime. The equation is given by:
$$i\partial_t u + \Delta^2 u + V u + \lambda |u|^{p-1}u = 0, \quad x \in \mathbb{R}^n, \, t \in \mathbb{R}$$
where:

• $\Delta^2 = \Delta \circ \Delta$ is the biharmonic operator.
• $V$ is a real-valued potential.
• $\lambda > 0$ corresponds to the defocusing case.
• The nonlinearity power is critical: $p = 2^\sharp - 1$, where $2^\sharp = \frac{2n}{n-4}$ is the critical Sobolev exponent for the $H^2$ space. This implies $p = \frac{n+4}{n-4}$.
• The dimension is restricted to $n \ge 5$.

The primary goal is to establish global well-posedness (existence, uniqueness, and continuous dependence on initial data) and scattering (asymptotic behavior approaching a linear solution) for radial initial data $u_0 \in H^2(\mathbb{R}^n)$.

2. Methodology

The author extends the groundbreaking work of Pausader [24] (who treated the case $V=0$) to the case where a potential $V$ is present. The methodology relies on several advanced analytical tools adapted to handle the potential:

A. Equivalence of Sobolev Norms

A major technical hurdle is that the propagator $e^{it(\Delta^2+V)}$ cannot be represented via the standard Fourier transform when $V \neq 0$. To overcome this, the paper establishes the equivalence of homogeneous Sobolev norms:
$$\| (\Delta^2 + V)^{s/4} u \|_{L^p} \sim \| |\nabla|^s u \|_{L^p}$$
This is achieved under Assumption 1.3 (smallness of $V$ in $L^{n/4, \infty}$ and non-negativity). This equivalence allows the transfer of estimates from the free operator $\Delta^2$ to the perturbed operator $H = \Delta^2 + V$.

B. Strichartz and Decay Estimates

The proof relies heavily on Strichartz estimates for the operator $H$.

• Strichartz Estimates: Derived using the equivalence of norms and existing results for potentials (from [23]).
• $L^p-L^q$ Decay Estimates: Since the Fourier representation is unavailable, the author utilizes the boundedness of wave operators ($W_\pm$) on $L^p$ spaces (established in [11] under Assumption 1.1). The decay is derived via the relation $e^{itH}P_{ac}(H) = W_\pm e^{it\Delta^2} W_\pm^*$.

C. Concentration Compactness and Morawetz Estimates

To prove global existence in the critical case, the paper employs the concentration-compactness/rigidity method (inspired by Kenig-Merle and Pausader):

1. Almost Conservation Laws: A localized mass estimate is derived to show that mass does not concentrate too rapidly.
2. Morawetz Estimate: A localized Morawetz-type estimate is established using the virial identity. Crucially, Assumption 1.2 ($x \cdot \nabla V \le 0$) is required to handle the potential term in the virial identity, ensuring the estimate remains positive.
3. Stability Theory: Short-time and long-time perturbation lemmas are proven to show that approximate solutions stay close to true solutions, allowing the extension of local solutions to global ones.

D. Scattering Analysis

Scattering is proven by showing that the solution $u(t)$ asymptotically approaches a free linear solution $e^{it\Delta^2}u_\pm$. This requires proving that the wave operators exist on $H^2$ and that the solution scatters to the free evolution, not just the perturbed evolution.

3. Key Assumptions

The results hold under specific conditions on the potential $V$:

• Assumption 1.1 (Decay and Regularity): $V$ decays sufficiently fast at infinity ($|V(x)| \lesssim \langle x \rangle^{-\beta}$) and satisfies spectral conditions (no positive eigenvalues, zero energy is regular). This ensures the boundedness of wave operators.
• Assumption 1.2 (Monotonicity): $x \cdot \nabla V \le 0$. This is essential for the Morawetz estimate to control the potential energy term.
• Assumption 1.3 (Smallness and Positivity): $V \ge 0$, $V \in L^\infty \cap L^{n/4}$, and the weak $L^{n/4}$ norm is sufficiently small. This ensures the equivalence of Sobolev norms.

4. Main Results

Theorem 1.4: Global Well-Posedness

For $n \ge 5$, $\lambda > 0$, and $p = \frac{n+4}{n-4}$, if $V$ is a radial potential satisfying Assumptions 1.1, 1.2, and 1.3, then for any radial initial data $u_0 \in H^2$, there exists a unique global solution $u \in C(\mathbb{R}, H^2)$.

• Significance: This extends the global existence result from the free case ($V=0$) to the case with a decaying, non-negative potential.

Theorem 1.5: Scattering

Under the same assumptions, any radial global solution $u$ scatters to a free linear solution. Specifically, there exist radial functions $u_\pm \in H^2$ such that:
$$\| u(t) - e^{it\Delta^2} u_\pm \|_{H^2} \to 0 \quad \text{as } t \to \pm \infty.$$

• Significance: This confirms that despite the presence of the potential, the solution asymptotically behaves like a solution to the free fourth-order Schrödinger equation, not just the perturbed one.

5. Significance and Contributions

1. Extension to Potentials: This is a significant step forward in the theory of higher-order dispersive equations. While the $V=0$ case was settled by Pausader, the inclusion of a potential introduces substantial spectral and decay complications.
2. Technical Innovation: The paper successfully navigates the lack of Fourier representation for the propagator by rigorously establishing the equivalence of Sobolev norms and utilizing wave operator theory to derive decay estimates.
3. Radial Symmetry: The results are restricted to radial data, which is a standard but necessary constraint in energy-critical problems to prevent energy concentration at a point (which could lead to blow-up) without the full power of the concentration-compactness machinery for general data.
4. Scattering to Free Evolution: The proof that solutions scatter to the free linear evolution ($e^{it\Delta^2}$) rather than the perturbed evolution ($e^{itH}$) is non-trivial and highlights the dispersive nature of the equation dominating the potential effects at large times.

In summary, Nakayama's work provides a robust framework for analyzing the energy-critical 4NLS with potentials, bridging the gap between the free case and perturbed systems using modern harmonic analysis and PDE techniques.