gpt 6 astra ultra attempt on Math-391

Bourgain's unbounded Sobolev orbit problem for cubic defocusing NLS on the square two-torus

Definitions and mathematical statement

On the square torus $\mathbb T^2=({\mathbb{R}}/2\pi{\mathbb{Z}})^2$, consider the defocusing cubic equation \[ i\partial_tu+\Delta u=|u|^2u. \] For Fourier coefficients $\widehat u(k)$, take $\|u\|_{H^s}^2=\sum_{k\in{\mathbb{Z}}^2}(1+|k|^2)^s|\widehat u(k)|^2$ with a fixed harmless normalization. The question asks whether there exist one global smooth solution and one real $s>1$ such that \[ \limsup_{t\to+\infty}\|u(t)\|_{H^s}=+\infty. \] The same solution must exhibit arbitrarily large growth at arbitrarily late times. Choosing a different initial condition for each desired amplification does not establish this orbit statement. Irrational tori or equations with a convolution potential are related variants, not the fixed square-torus equation here.

Formulation and concept references: [S1].

Short English statement

Can one smooth solution of cubic defocusing NLS on the square two-torus develop unbounded higher Sobolev norms along arbitrarily late times?

Research attempt

A single orbit, rather than a sequence of experiments.

The inspected introduction of [S1] distinguishes the unbounded-orbit question from arbitrary but finite growth. Its quoted finite-growth theorem for the square torus gives a solution depending on the desired amplification. Its own convolution-potential and irrational-torus results are not substituted for the selected equation. We retain the same square torus, the same cubic defocusing nonlinearity and a single smooth solution throughout the target.

This attempt derives the conservation constraints, identifies the rectangle resonance that permits redistribution, and states an exact topological condition that would turn finite growth into a single unbounded orbit. The missing step is verifying that condition, or a uniform infinite cascade, for this equation.

Conserved quantities and what they actually control.

Use normalized Haar measure on the torus so that Fourier Parseval has no volume factor. For a smooth solution, multiplication by $\bar u$ and integration give conservation of mass \[ M(u)=\int|u|^2. \] Indeed $u_t=i\Delta u-i|u|^2u$, and the real part of $\int\bar u u_t$ vanishes after integration by parts. The energy is \[ E(u)=\frac12\int|\nabla u|^2+\frac14\int|u|^4. \tag{387.1} \] Its derivative equals $\operatorname{Re}\int(-\Delta\bar u+|u|^2\bar u)u_t=0$ by the equation. Thus \[ \|u(t)\|_{H^1}^2\leq M(u(0))+2E(u(0)) \tag{387.2} \] for all times on which the smooth solution exists. These identities are compatible with the established global smooth theory used in the formulation; they alone do not prove a uniform $H^s$ bound for $s>1$.

The gap cannot be filled by an inequality bounding $H^s$ using only the values on the right of $387.2$. The smooth functions $v_N(x)=N^{-1}e^{iNx_1}$ have uniformly bounded mass and energy, but \[ \|v_N\|_{H^s}=N^{-1}(1+N^2)^{s/2}\longrightarrow\infty \quad(s>1). \tag{387.3} \] This is a statement about possible states, not a construction of a trajectory visiting them. The equation may have dynamical constraints beyond the two conserved scalar quantities.

Conversely, every plane wave \[ u(t,x)=a\exp\left(i k\cdot x-i(|k|^2+|a|^2)t\right), \qquad k\in\mathbb Z^2, \tag{387.4} \] is an exact global solution and has constant Sobolev norms of every order. Thus a high-frequency initial state by itself produces no growth. The problem requires transfer among frequencies over time.

No fixed Fourier truncation can display the target behavior.

Suppose a solution of any mass-conserving finite-mode model remains supported in $|k|\leq K$. Then \[ \|u(t)\|_{H^s}^2 \leq(1+K^2)^{s-1}\|u(t)\|_{H^1}^2 \quad(s>1). \tag{387.5} \] In particular the Hamiltonian Galerkin truncation, obtained by orthogonally projecting the nonlinearity to these modes, conserves the restrictions of mass and energy. To check energy conservation, its vector field is the Hamiltonian gradient of $387.1$ on the finite Fourier subspace, so the same calculation works with the orthogonal projection inserted. The mass conservation is equally direct. Its finite-dimensional trajectories therefore have a uniform $H^s$ bound depending on $K$ and the initial invariants.

Finite-mode computations can detect amplification and test a local transfer mechanism. They cannot prove an unbounded norm on a fixed orbit unless there is an argument passing to infinitely many active scales while maintaining a single admissible initial condition. Letting $K$ increase between experiments changes the system being tested and does not remove this requirement.

The exact resonant geometry on the square torus.

Write $u(t,x)=\sum_k u_k(t)e^{ik\cdot x}$ and remove the linear oscillation by $a_k(t)=e^{i|k|^2t}u_k(t)$. The Fourier equation is \[ i\dot a_k=\sum_{k_1-k_2+k_3=k} a_{k_1}\bar a_{k_2}a_{k_3} e^{-it\Omega},\qquad \Omega=|k_1|^2-|k_2|^2+|k_3|^2-|k|^2. \tag{387.6} \] Using the momentum relation to expand the last square gives \[ \Omega=-2(k_1-k_2)\cdot(k_3-k_2). \tag{387.7} \] Thus a nondegenerate resonant quadruple consists of the vertices of a lattice rectangle. For example \[ 0,\quad a=(N,0),\quad b=(0,N),\quad d=a+b \] obey $a\cdot b=0$ and $|0|^2+|d|^2=|a|^2+|b|^2$. This equality permits an exchange between the opposite pairs without changing the quadratic energy. For the Sobolev weight $w(k)=(1+|k|^2)^s$, strict convexity gives \[ w(0)+w(d)-w(a)-w(b) =1+(1+2N^2)^s-2(1+N^2)^s>0 \quad(s>1). \tag{387.8} \] Thus the higher norm is not algebraically forced to remain constant by this particular resonance.

To make that observation dynamical without overstating it, consider only the four-mode interaction Hamiltonian \[ H_{\rm int}=Z+\bar Z,\qquad Z=a_0a_d\bar a_a\bar a_b, \qquad i\dot a_j=\partial_{\bar a_j}H_{\rm int}. \] Writing $I_j=|a_j|^2$ gives the exact identities \[ \dot I_0=\dot I_d=-2\operatorname{Im}Z,\qquad \dot I_a=\dot I_b=2\operatorname{Im}Z. \tag{387.9} \] They follow from $\dot I_j=2\operatorname{Im}(\bar a_j\partial_{\bar a_j}H_{\rm int})$. Mass and the quadratic Fourier energy are preserved, while the weighted sum has derivative \[ \frac{d}{dt}\sum_jw(j)I_j =2\operatorname{Im}Z\,[w(a)+w(b)-w(0)-w(d)]. \tag{387.10} \] Suitable phases make this nonzero. This is a verified local mechanism in the stated interaction model. It is not asserted that the full NLS stays on this four-mode subsystem: other cubic interactions and nonresonant terms are present in $387.6$.

Why averaging has a finite-time qualification.

Terms with $\Omega\ne0$ in $387.6$ oscillate. Integrating one such term by parts produces a factor $1/\Omega$, a boundary term, and an integral involving derivatives of the amplitudes. Even though the square-torus frequencies make nonzero $\Omega$ integers, the number of interacting terms, amplitude sizes and duration all enter the remainder estimate. Dropping oscillatory terms is therefore a finite-time approximation requiring explicit control, not an exact identity for arbitrarily long trajectories.

Suppose a resonant construction executes $J$ transfers and is shadowed by the full equation with an error bound depending on $J$. To take $J\to\infty$, one needs compatible initial data, summable errors and existence on every corresponding time interval. A theorem giving one independently chosen shadowing solution for each finite $J$ does not furnish these properties. Conservation laws may help control the low norms, but they do not by themselves guarantee high-frequency errors remain small in the norm in which the transfer is measured.

The precise logical gap in finite amplification.

Finite growth has the quantifier pattern \[ \text{for every }R\text{ there exist }u_R\text{ and }T_R \text{ with }\|u_R(T_R)\|_{H^s}>R. \] The question instead asks for one $u$ for which the inequality holds for every $R$ at some late time. Interchanging these quantifiers is not valid. The elementary family of functions $F_j(t)=j\sin^2t$ has arbitrarily large values across the family, while each individual function is bounded. It is a logical countercheck, not an NLS model.

Compactness of a sequence of initial data is insufficient as well. For example, a finite-growth theorem may choose $\|u_j(0)\|_{H^s}\to0$ while $\|u_j(T_j)\|_{H^s}\to\infty$. Continuity of the solution map on each fixed bounded time interval then makes the limiting solution the zero solution on every such interval. Necessarily the growth times escape those intervals. The large values at $T_j\to\infty$ do not pass through convergence that is only uniform on compact time intervals.

An exact Baire-category route, with the missing premise exposed.

Let $\mathcal X=C^\infty(\mathbb T^2;\mathbb C)$ with its usual complete metrizable Frechet topology. Use the established global smooth flow and its continuity on fixed time intervals. Fix $s>1$ and define \[ \mathcal U_m=\bigcup_{t\geq0} \{u_0\in\mathcal X:\|u(t;u_0)\|_{H^s}>m\}. \tag{387.11} \] Each $\mathcal U_m$ is open, since it is a union of inverse images of open norm intervals under continuous fixed-time solution maps. If every $\mathcal U_m$ were dense, Baire's theorem would give a dense intersection $\bigcap_{m\geq1}\mathcal U_m$. Every point in that intersection would generate a smooth orbit unbounded in $H^s$. Continuity of one smooth orbit bounds its norm on each compact time interval, so its arbitrarily large values must occur at arbitrarily late times; the required limsup would follow.

This reduction cleanly specifies the needed strengthening: large growth must be available in every nonempty smooth neighborhood, not merely from some small data near the origin. Such a neighborhood can prescribe closeness in an arbitrarily high but finite list of Sobolev norms. A high-frequency perturbation small in one chosen $H^s$ need not be small in that neighborhood. The finite-growth theorem cited in [S1] does not assert this density premise, and the present calculations do not establish it.

An alternative constructive route would select nested nonempty closed neighborhoods of initial data, each guaranteeing a new transfer at a specified finite time, with diameters tending to zero in a complete smooth metric. Their intersection would provide one orbit. Again the hard requirement is the ability to perform the next transfer while preserving all previous finite-time requirements and remaining in the selected neighborhood. Stating nestedness without constructing it would simply restate the desired infinite cascade.

Diagnostics and outcome.

The finite checks verify the rectangle identity $387.7$, the strict higher-weight imbalance $387.8$ for sample exponents, and the mass and quadratic-energy cancellations in $387.9$. The fixed-cutoff inequality $387.5$ is proved above rather than tested by the script. These are finite arithmetic diagnostics; no finite Fourier simulation is reported as an unbounded-orbit construction.

The conservation constraints permit higher-norm growth, and a resonant interaction can change the required weighted energy. The first unresolved step is turning such local or finite transfer information into the uniform density or compatible infinite shadowing required for one smooth square-torus solution. The selected problem remains unresolved by this attempt.

Sources

• S1 Filippo Giuliani, Sobolev instability in the cubic NLS equation with convolution potentials on irrational tori (2023 preprint; JDE 2025). https://arxiv.org/html/2308.13468. Formulation source.