Strichartz estimates for Schrödinger equations with nonlinear boundary interactions

Technical Summary: Strichartz Estimates for Schrödinger Equations with Nonlinear Boundary Interactions

Problem Statement
This paper investigates the dispersive properties and well-posedness of a Schrödinger equation defined in the upper half-space $\mathbb{R}^{d+1}_+ = \mathbb{R}^d_x \times \mathbb{R}^+_z$, featuring a nonlinear Neumann boundary interaction driven by a singular Bessel operator $B^{(a)}_z = \partial_{zz} + \frac{a}{z}\partial_z$ with $a > -1$. The primary Cauchy problem is given by:
$$\begin{cases} \partial_t U - i(\Delta_x U + B^{(a)}_z U) = F(X, t), & X \in \mathbb{R}^{d+1}_+, t \in \mathbb{R}, \\ \lim_{z \to 0^+} z^a \partial_z U(X, t) = -\mu |U(x, 0, t)|^{p-1}U(x, 0, t), & x \in \mathbb{R}^d, t \in \mathbb{R}, \\ U(X, 0) = u_0(X). \end{cases}$$
The authors identify two complementary motivations for this problem:

1. Interface Problems: When $a=0$, the equation describes a Schrödinger evolution in the whole space with a nonlinear source concentrated on the codimension-one hypersurface $\{z=0\}$. For general $a > -1$, it represents a singular interaction on the interface coupled with a singular Bessel operator.
2. Extension Problems: The system arises naturally as an extension formulation for a nonlocal nonlinear Schrödinger (NLS) equation with memory, analogous to the Caffarelli-Silvestre extension for fractional Laplacians. The authors note that while the extension operator is of Schrödinger type (involving oscillatory phenomena rather than positivity), the theory developed here is intended to analyze such nonlocal problems in future work.

Methodology and Key Contributions
The core of the analysis focuses on the associated linear inhomogeneous problem with non-homogeneous Neumann data. The authors develop a complete linear theory to serve as a foundation for the nonlinear analysis.

• New Duhamel Representation Formula: A central contribution is the derivation of a generalized Duhamel formula that separates the evolution into bulk and boundary components. The solution $U$ is represented as:
  $$U = T^*_a(u_0) + D_a(F) + \Theta^*_a(\Phi),$$
  where $T^*_a$ and $D_a$ describe the bulk evolution (initial data and forcing term), and $\Theta^*_a$ encodes the boundary interaction. This decomposition allows for the independent analysis of bulk and boundary dispersive behaviors.

• Identification of Dispersive Regimes: The analysis reveals a fundamental dichotomy based on the parameter $a$:

  • Regime $a \ge 0$: The equation behaves like a classical Schrödinger evolution. The propagator obeys the natural scaling $(X, t) \to (\lambda X, \lambda^2 t)$, and bulk and boundary exhibit unified dispersive behavior.
  • Regime $-1 < a < 0$ (Anomalous): The dispersive structure becomes anomalous. The bulk propagator no longer follows standard scaling laws; instead, it behaves as if $a=0$ only after introducing nontrivial weights in the transverse variable $z$. This regime necessitates weighted estimates and functional frameworks involving sums and intersections of Lebesgue spaces.

• Strichartz Estimates: The authors establish sharp Strichartz estimates adapted to the geometry of the half-space and the singular structure of the Bessel operator.

  • For $a \ge 0$, they derive estimates compatible with natural scaling, identifying admissible triples $(q, r, m)$ satisfying $\frac{2}{q} + \frac{d}{r} + \frac{a+1}{m} = \frac{d+a+1}{2}$.
  • For $-1 < a < 0$, they prove estimates involving weighted norms and mixed spaces ($L^{q+q_\infty}_t$, $L^{q' \cap q'_\infty}_t$), reflecting the discrepancy between bulk and boundary dynamics.
  • Notably, the boundary propagator $\Theta^*_a$ exhibits a uniform dispersive behavior for all $a > -1$, distinct from the bulk dichotomy.

• Trace Theory: A critical technical step involves establishing the existence and continuity of traces on the boundary $\{z=0\}$. While Strichartz estimates provide uniform control in $z$, they do not guarantee convergence to the boundary manifold. The authors prove a separate trace theorem ensuring that solutions belong to $C_b^z L^q_t L^r_x$, which is essential for defining the nonlinear boundary term.

Main Results
Using the linear theory, the authors prove local and global well-posedness results for the nonlinear problem (1.1) in the $L^2_a$-mass critical and subcritical regimes.

• Case $a \ge 0$:

  • Critical Case ($p = p_c$): Global well-posedness is established for sufficiently small critical data.
  • Subcritical Case ($1 < p < p_c$): Local well-posedness is proven for arbitrary data, with global existence for small data or when the imaginary part of the coupling constant $\mu$ is zero (conservation of mass).

• Case $-1 < a < 0$:

  • Critical Case: Existence and uniqueness are proven on arbitrary finite time intervals for sufficiently small critical data.
  • Subcritical Case: Local well-posedness is established.

The critical exponents $p_c$ differ between regimes:

• For $a \ge 0$: $p_c = 1 + \frac{2(1-a)}{d+a+1}$.
• For $-1 < a < 0$: $p_c = 1 + \frac{2}{d+1}$.

Significance and Scope
The paper claims to provide a unified dispersive framework for Schrödinger equations with nonlinear boundary interactions and singular extension structures associated with nonlocal-in-time dynamics. It extends and generalizes previous work by the authors on one-dimensional Bessel Schrödinger equations and recent studies on NLS with boundary nonlinearities (specifically the case $a=0$).

The authors emphasize that the singular extension structure considered here leads to a "considerably richer dispersive theory" requiring new analytical tools not present in the $a=0$ case. The results are presented as a necessary step toward analyzing the nonlocal NLS with memory mentioned in the motivation, though the specific analysis of that nonlocal problem is reserved for a forthcoming work. Additionally, the derived Strichartz estimates are used to prove a Tomas-Stein restriction theorem for the Fourier-Hankel transform restricted to a paraboloid.