Non-unique solutions to the periodic gKdV equation

Abstract: In this paper we utilize a convex integration scheme to construct non-trivial weak solutions to the $k$-generalized KdV equation which lie in $$ \bigcap_{ε&gt; 0} C_t⁰ L_x^{k-ε}([0,1] \times \mathbb{T}) $$ and, when $k \ge 3$, it may also be chosen in [ \bigcap_{ε>0} C_t^0 H_x^{\frac{1}{2} - \frac{1}{k} - ε}([0,1] \times \mathbb{T}) ] attaining identically $0$ initial data. Since our solutions do not lie in $C_t⁰ L_x<sup>k$, this requires introducing a new notion of weak solution, which is in fact stronger than the classical notion of a weak solution when the nonlinearity is integrable. This result shows that a necessary condition for unconditional uniqueness for $k$-gKdV is that the nonlinearity lies in $C_t<sup>0L<sup>1_x$. In the case of KdV this is in fact also sufficient.

• The paper demonstrates the construction of nontrivial weak solutions for the periodic gKdV equation using convex integration, achieving non-uniqueness even with zero initial data.
• It employs a framework that manages oscillation, dispersion, and temporal errors in weighted Wiener spaces to secure convergence in low-regularity function classes.
• The results underline the critical threshold where unconditional uniqueness fails, offering new tools for analyzing dispersive PDEs at suboptimal regularity levels.

Non-Uniqueness in the Periodic Generalized KdV Equation: Convex Integration and Weak Singular Solutions

Introduction and Motivation

This paper establishes the existence of non-unique weak solutions for the $k$-generalized Korteweg–de Vries equation (gKdV) in the periodic setting. The authors approach this by constructing weak solutions in highly singular classes using convex integration. The analysis covers all $k \geq 2$, including the classical KdV ($k=2$) and its modified and higher-order variants. Notably, the constructed solutions attain identically zero initial data yet are nontrivial. This advances the understanding of the sharp thresholds for unconditional uniqueness and nonuniqueness in dispersive PDEs, connecting with lines of research on well-posedness theory at critical and subcritical regularities.

Technical Foundations

The $k$-gKdV equation is considered on the one-dimensional torus, with the following Cauchy problem:

$$\partial_t u + \partial_{xxx} u + \sigma\partial_x(u^k) = 0, \quad u(0, x) = u_0(x)$$

where $\sigma \in \{\pm 1\}$. For sufficiently regular data, the mass, momentum, and energy are formally conserved. Previous work provides very sharp results for well-posedness in various Sobolev spaces, with global uniqueness at $H^{-1}$ for KdV ($k=2$) and ill-posedness below that level. However, for larger $k$ and lower-regularity spaces, the landscape is more rigid, and the threshold of uniqueness versus nonuniqueness has been far from completely characterized.

The distinction between conditional and unconditional uniqueness is critical here. The constructed weak solutions circumvent uniqueness by existing below the regularity in which the standard contraction or energy methods operate.

Main Results

The core contribution is the convex integration-based construction of nontrivial weak solutions in the class:

$$u \in \bigcap_{\epsilon > 0} C_t^0 L_x^{k-\epsilon}([0,1]\times \mathbb{T})$$

for all $k \geq 2$0, with a further refinement for $k \geq 2$1 to

$k \geq 2$2

where the nonlinearity $k \geq 2$3 is defined in a "weighted Wiener space" sense, via absolute summability of the Fourier expansion in negative Sobolev spaces.

Key numerical claim: For $k \geq 2$4, nonuniqueness is established below $k \geq 2$5. At the $k \geq 2$6 endpoint, unconditional uniqueness is known to hold, confirming the sharpness of the result.

This demonstrates that unconditional uniqueness for $k \geq 2$7-gKdV requires the nonlinearity to lie in $k \geq 2$8u^k</sup>tobeintegrableinspace</em>;belowthis,convexintegrationcansucceedinproducingsingularsolutionsthatviolateuniqueness.</p><p>Acorresponding<strong>conjecture</strong>isformulated:for</sup> to be integrable in space</em>; below this, convex integration can succeed in producing singular solutions that violate uniqueness.</p> <p>A corresponding <strong>conjecture</strong> is formulated: for k \geq 2$9, unconditional local well-posedness should hold in $k=2$0 for all $k=2$1.

Convex Integration Architecture

The convex integration framework adapts methods previously successful for Navier–Stokes and dispersive models but tailored here to the challenges of dispersive nonlinear terms. The critical innovations are:

• Fully intermittent building blocks for the perturbations, designed so that the $k=2$2 norm remains of order $k=2$3 while lower $k=2$4 and Sobolev norms of the increments are summable. This is achieved via the construction of "intermittent slabs" whose Sobolev and Wiener norms are sharply quantified.
• Error decomposition: Each step introduces a perturbation that cancels the previous error in the nonlinearity to leading order (oscillation error), with Nash, dispersion, and temporal errors handled perturbatively in negative Wiener norms.
• Weighted Wiener spaces: The errors are controlled not in standard Lebesgue or Sobolev spaces but in weighted Fourier $k=2$5 spaces with explicit polynomial weights, which permit summing the high-frequency tails from intermittent profiles.
• Notion of Solution: The essential technical novelty in the definition of solutions is the requirement that the Fourier expansion of the nonlinearity's nonzero modes is absolutely convergent in the negative Sobolev sense (see Definitions in the paper), which is strictly stronger than previously used notions based on paraproducts or Christ's cutoff regularizations.
• Inductive scheme: Amplitudes and frequencies of perturbations are escalated at each step to ensure rapid decay of error and absolute convergence in the target spaces.

Analytical and Structural Implications

The analysis contains several strong or previously unproven claims:

• The constructed solutions have zero spatial mean for all time—a consequence of the structure of the weak formulation and the mass conservation property isolated in the class considered.
• The convex integration approach cannot succeed (cannot close the estimates) above the integrability threshold for the nonlinearity, but is highly flexible below it. This matches sharp results for $k=2$6 and points to the paradigm that unconditional well-posedness "breaks" at precisely the point where the nonlinearity ceases to be a classical distribution.
• The methodology adapts directly to stationary versions, the generalized Benjamin-Ono (gBO) equation, and other translation-invariant dispersive systems, emphasizing the robustness of the scheme under variations in the linear dispersion.
• The notion of solution proposed implies, and is strictly stronger than, both the paraproduct-based and cutoff-based (Christ) notions, as rigorously proved in the appendix.

Connections to Open Problems and Future Directions

The work situates itself as part of a methodological program extending convex integration to dispersive PDEs, where the dispersive linear term introduces substantial technical obstacles absent in, for example, the Euler or Navier–Stokes cases. The sharp threshold found here for uniqueness versus nonuniqueness is conjectured to be the same as the (unconditional) well-posedness threshold for the periodic $k=2$7-gKdV class.

For applications to other dispersive PDEs, the main technical requirements are that the linear operator's symbol can be divided by the spatial derivative without incurring more than polynomial loss in frequency, which is widely satisfied in models of mathematical physics involving higher-order KdV or generalized dispersive flows.

Further developments may include extensions to non-periodic settings, variable coefficient or stochastic gKdV equations, and connections with turbulence and rough solution theory inspired by the Onsager program.

Conclusion

The paper rigorously demonstrates, using a precisely engineered convex integration scheme, the existence of nonunique, nontrivial weak solutions to the periodic $k=2$8-gKdV equation in sub-integer Lebesgue and Sobolev spaces, with zero initial data. The solutions live in classes below the Lebesgue threshold where the nonlinearity is a distribution, confirming the strictness of the uniqueness thresholds known from contraction principle theory. The analysis provides new technical tools and conceptual insights for the study of low-regularity behavior in dispersive evolutionary PDEs, with implications for both theoretical analysis and potential numerical exploration of solution bifurcation and turbulence models.

Reference: "Non-unique solutions to the periodic gKdV equation" (2606.06916)