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Nontrivial weak solutions of the stationary KdV equation in sharp $L^p$ spaces

This paper establishes the sharpness of $L^2$ regularity for weak solutions to the stationary KdV equation by constructing nontrivial solutions in $L^p(\mathbb{T})$ for $p<2$ via convex integration while proving that any $L^2$ weak solution must be smooth.

Original authors: Mandon Pathak

Published 2026-03-16

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mandon Pathak

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Rough" vs. "Smooth" Water Wave

Imagine you are watching a river. Usually, when we model water waves (like the famous KdV equation), we expect the water to be smooth and flowing nicely. If you look at the water, you can draw a single, unbroken line along the surface. In math terms, we call this a smooth solution.

For a long time, mathematicians believed that if you have a "stationary" (not moving forward in time, just sitting there) wave that is somewhat well-behaved (specifically, if its energy is finite, or $L^2$ ), it must be smooth. It's like saying, "If a river has a finite amount of water, the surface must be calm and glassy."

This paper proves that intuition is wrong.

The author, Mandon Pathak, shows that if you look at the river with slightly less strict rules (specifically, in a space called $L^p$ where $p < 2$ ), you can find "waves" that are incredibly rough, jagged, and chaotic. They are so rough they aren't even functions in the traditional sense; they are "distributions" (mathematical ghosts).

The paper draws a sharp line in the sand:

If the wave is "smooth enough" ( $L^2$ ): It must be perfectly smooth ( $C^\infty$ ).
If the wave is "rough enough" ( $L^p$ with $p < 2$ ): You can build wild, non-smooth waves that still satisfy the laws of physics (the equation).

The Problem: The "Dispersion Error" Trap

To build these rough waves, the author uses a technique called Convex Integration. Think of this like building a sculpture out of clay. You start with a rough shape, then add tiny layers of clay to fix the imperfections, then add even tinier layers to fix those imperfections, and so on.

However, the KdV equation has a tricky feature called dispersion.

The Analogy: Imagine you are trying to smooth out a bumpy road. Every time you add a new layer of asphalt (a correction) to fix a bump, the physics of the road (dispersion) tries to create new bumps elsewhere.
The Trap: In standard math, when you try to fix the error, the "dispersion" creates a new error that is actually bigger than the one you just fixed. It's like trying to fill a hole in a wall, but the act of filling it makes the wall crumble even more. In one dimension (a 1D river), this usually makes it impossible to build these rough solutions.

The Solution: The "Intermittent Slab" and the "Negative Lens"

The author gets around this trap using two clever tricks:

1. The Intermittent Slab (The "Patchwork Quilt")

Instead of spreading the new clay (the correction) evenly across the whole river, the author uses intermittent building blocks.

The Analogy: Imagine instead of painting the whole wall blue, you only paint tiny, isolated patches of blue, leaving huge gaps of white space in between.
Why it works: By concentrating the "roughness" into tiny, isolated slabs, the author can control how the error behaves. It's like a quilt where the patches are so small and far apart that they don't interfere with each other in a bad way.

2. Measuring with a "Negative Lens" (The Homogeneous Sobolev Norm)

This is the most technical part, but here is the simple version:

The Problem: When you measure the "size" of the error using standard tools (like $L^p$ norms), the dispersion error looks huge because the corrections are very high-frequency (very fast wiggles).
The Trick: The author decides to measure the error through a special "negative lens" (a negative Sobolev norm, $\dot{H}^{-s}$ ).
The Analogy: Imagine you are trying to hear a high-pitched squeak (the error). If you use a standard microphone, it's deafeningly loud. But if you use a filter that only picks up low frequencies (the negative lens), that high-pitched squeak becomes almost silent.
The Result: By measuring the error this way, the "dispersion error" that usually blows up the math suddenly becomes tiny and manageable. This allows the author to keep adding layers of roughness without the whole structure collapsing.

The Main Results Explained

1. The Rigidity Result (The "Smooth" Rule)

Theorem: If a solution is in $L^2$ (finite energy), it is automatically smooth.
Analogy: If you have a river with a reasonable amount of water, nature forces it to be calm. You cannot have a "rough" river with finite energy. If you try to make it rough, the math forces it to smooth itself out instantly.

2. The Flexibility Result (The "Rough" Discovery)

Theorem: If you allow the solution to be slightly less regular ( $L^p$ where $p < 2$ ), you can construct solutions that are infinitely rough and chaotic.
Analogy: If you relax the rules just a tiny bit (allowing the water to be a bit more "ghostly" or undefined), you can build a river that is a chaotic mess of jagged spikes. These aren't just small ripples; they are wild, non-smooth structures that technically obey the KdV equation but look nothing like a normal wave.

Why Does This Matter?

It's a Sharp Line: The paper finds the exact "tipping point" ( $p=2$ ) where the behavior of the equation changes from "everything must be smooth" to "chaos is allowed."
New Tools for Old Problems: For years, mathematicians thought the KdV equation was too "rigid" to use the Convex Integration method (which is usually used for fluid turbulence). This paper proves that by changing how you measure the error (the negative lens trick), you can apply these powerful construction tools to dispersive waves for the first time.
Understanding Reality: It suggests that in the world of nonlinear waves, there might be hidden, chaotic states that we haven't been able to see because we were looking with the wrong "lens."

Summary

The author built a mathematical machine that constructs "ghost waves" for the stationary KdV equation. By using patchwork corrections (intermittency) and measuring errors with a special filter (negative norms), they bypassed the usual mathematical roadblocks. They proved that while "normal" waves must be smooth, if you look closely enough at the rougher, stranger side of the equation, you can find an infinite variety of chaotic, non-smooth solutions.

1. Problem Statement

The paper addresses the regularity of weak solutions to the stationary Korteweg–de Vries (KdV) equation on the torus $\mathbb{T}$ :
$6uu' - u^{(3)} = 0$
where $u: \mathbb{T} \to \mathbb{R}$ .

While the well-posedness and regularity of the time-dependent KdV equation have been extensively studied (with sharp thresholds established in Sobolev spaces $H^{-s}$ ), the regularity of stationary solutions in low regularity spaces ( $L^p$ for $p < 2$ ) remained unexplored.

The central problem is to determine the threshold of regularity that separates smooth solutions from potentially singular (non-smooth) weak solutions. Specifically, the author investigates whether non-trivial weak solutions exist in $L^p(\mathbb{T})$ for $p < 2$ , and whether solutions in $L^2(\mathbb{T})$ must necessarily be smooth.

2. Methodology

The paper employs a Convex Integration scheme, a technique originally developed for fluid dynamics (Euler and Navier-Stokes equations) by De Lellis and Székelyhidi, adapted here for a dispersive PDE.

Key Technical Challenges and Solutions:

Definition of Weak Solutions:
- Standard weak formulations require $u^2$ to be well-defined, which typically necessitates $u \in L^2$ . However, the goal is to construct solutions in $L^p$ for $p < 2$ .
- Solution: The author adopts the notion of weak paraproduct solutions (inspired by works on 2D stationary Navier-Stokes). The nonlinearity $u^2$ is interpreted via a paraproduct decomposition ( $P_{\neq 0}(u^2)$ ) in homogeneous Sobolev spaces $\dot{H}^s$ with negative regularity, allowing the product to be defined for distributions.
The Dispersion Error:
- In standard convex integration for dispersive equations, the "dispersion error" term ( $E_D = -w_{q+1}''$ ) creates a derivative loss. Estimating this error in standard $L^p$ or $C^\alpha$ norms fails in 1D because the required intermittency (concentration of the solution) is insufficient to counteract the $\lambda^2$ growth from the second derivative.
- Solution: The author measures the error in a homogeneous Sobolev norm with negative regularity ( $\dot{H}^{-s}$ ). By choosing $s$ sufficiently large, the derivative loss is compensated by the negative regularity of the norm, allowing the error to decay as the frequency $\lambda_{q+1} \to \infty$ .
Intermittent Building Blocks:
- The construction utilizes intermittent slabs (functions $\rho_{\lambda, \epsilon}$ ) which are highly oscillatory and spatially concentrated.
- These blocks are modulated by a smooth amplitude function and a high-frequency cosine carrier.
- The specific choice of intermittency allows the solution to remain in $L^p$ for $p < 2$ while ensuring the nonlinear interactions (Nash and oscillation errors) vanish in the limit.

3. Key Contributions and Results

A. Rigidity Result (The Sharp Threshold)

Theorem 1.3: If $u \in L^2(\mathbb{T})$ is a weak solution to the stationary KdV equation, then $u \in C^\infty(\mathbb{T})$ .

Significance: This establishes $L^2$ as the sharp threshold for regularity. Any solution with $L^2$ regularity is automatically smooth. The proof relies on bootstrapping: $u \in L^2 \implies u'' \in L^1 \implies u \in C^1 \implies u \in C^3 \implies u \in C^\infty$ .

B. Flexibility Result (Existence of Singular Solutions)

Theorem 1.4: There exist non-trivial weak solutions $u$ such that:
$u \in \bigcap_{0 < \epsilon < 1} \left( \dot{B}^{-\epsilon}_{\infty, \infty}(\mathbb{T}) \cap L^{2-\epsilon}(\mathbb{T}) \right) \setminus L^2(\mathbb{T})$

Significance: This proves that for any $p < 2$ , there exist non-smooth, non-trivial weak solutions. These solutions are not in $L^2$ and possess very low regularity (negative Besov regularity).

C. Comparison with Navier-Stokes

The paper draws a parallel with the 2D stationary Navier-Stokes equation.

Navier-Stokes: If $u \in L^p$ ( $p > 2$ ) with zero mean, then $u \equiv 0$ .
Stationary KdV: If $u \in L^p$ ( $p < 2$ ), non-trivial solutions exist.
Difference: Unlike Navier-Stokes, the stationary KdV equation does not force mean-zero smooth solutions to be identically zero; it only forces them to be smooth.

4. Technical Outline of the Construction

The proof of Theorem 1.4 proceeds via an inductive convex integration scheme:

Relaxed Equation: The equation is relaxed to $3(u^2)' - u^{(3)} = E'$ , where $E$ is an error term.
Iterative Step: Given a smooth approximate solution $(u_q, E_q)$ , a perturbation $w_{q+1}$ is added: $u_{q+1} = u_q + w_{q+1}$ .
Error Decomposition: The new error $E_{q+1}$ $E_{q + 1}$ is decomposed into:
- Oscillation Error ( $E_O$ ): Arising from the quadratic nonlinearity of the perturbation.
- Nash Error ( $E_N$ ): Arising from the interaction between the perturbation and the previous solution.
- Dispersion Error ( $E_D$ ): Arising from the third derivative of the perturbation.
Estimates:
- The perturbation $w_{q+1}$ is constructed using intermittent slabs $\rho_{q+1}$ and a high-frequency carrier $\cos(2\pi \sigma_{q+1} x)$ .
- By choosing the frequency $\sigma_{q+1}$ $σ_{q + 1}$ and the intermittency parameter $\epsilon_{q+1}$ $ϵ_{q + 1}$ carefully, the author ensures that:
  - The $L^p$ norm of the increment decays for $p < 2$ .
  - The $L^2$ norm of the increment remains bounded away from zero (ensuring the limit is not in $L^2$ ).
  - All error terms converge to zero in the $\dot{H}^{-s}$ norm.
Convergence: The sequence $u_q$ converges to a limit $u$ in $L^p$ ( $p<2$ ) and $\dot{B}^{-\epsilon}_{\infty, \infty}$ , but the limit is not in $L^2$ .

5. Significance

Sharp Regularity Threshold: The paper precisely identifies $L^2$ as the dividing line between smooth and singular solutions for the stationary KdV equation.
Extension of Convex Integration: It demonstrates that convex integration, previously limited to fluid equations due to derivative loss issues in dispersive PDEs, can be successfully applied to the KdV equation by utilizing negative Sobolev norms for error estimation.
Low Regularity Phenomena: It reveals that the stationary KdV equation admits a rich set of low-regularity solutions, challenging the intuition that integrable systems (like KdV) might only possess smooth solutions in low dimensions.
Paraproduct Framework: It solidifies the utility of paraproduct definitions for weak solutions in contexts where classical multiplication is ill-defined, bridging the gap between distribution theory and nonlinear PDE analysis.

In conclusion, Pathak's work provides a definitive answer to the regularity of stationary KdV solutions, proving that while $L^2$ implies smoothness, any lower $L^p$ regularity ( $p<2$ ) allows for the existence of highly irregular, non-trivial weak solutions constructed via a novel adaptation of convex integration.