Weak Solutions for Inviscid SQG with Lorentz Data

Technical Summary: Weak Solutions for Inviscid SQG with Lorentz Data

Problem Statement
The paper addresses the global existence of weak solutions to the inviscid Surface Quasi-Geostrophic (SQG) equation in $\mathbb{R}^2$ and smooth bounded domains. The equation describes the transport of a scalar $\theta$ by a velocity field $u$ derived from $\theta$ via a singular integral operator of order zero:
$$\partial_t \theta + u \cdot \nabla \theta = 0, \quad u = \nabla^\perp \Lambda^{-1} \theta,$$
where $\Lambda = (-\Delta)^{1/2}$.

The central challenge lies in the regularity of the initial data. While global weak solutions are known for $L^2$ data, the problem remains open for data with lower regularity. Previous results established existence for $L^p$ data with $p > 4/3$ or for the critical Lebesgue space $L^{4/3}$. However, the critical Lebesgue space $L^{4/3}$ is not the largest space for which the Hamiltonian (negative Sobolev energy) is controlled. The authors aim to construct global weak solutions for initial data in the critical Lorentz space $L^{4/3, 2}$, which strictly contains $L^{4/3}$ (since $L^{4/3} = L^{4/3, 4/3} \subsetneq L^{4/3, 2}$).

A critical difficulty at this regularity is that the product $u\theta$ is not well-defined in the classical sense, and weak convergence in natural negative Sobolev spaces does not automatically preserve the quadratic Hamiltonian $H(\theta) = \frac{1}{2}\|\Lambda^{-1/2}\theta\|_{L^2}^2$. Furthermore, sequences in $L^{4/3, 2}$ may concentrate, potentially leading to energy loss or non-uniqueness.

Methodology
The authors employ a regularization scheme tailored specifically to the properties of Lorentz spaces, avoiding the addition of viscosity.

1. Regularization Scheme:

   • The velocity operator $\Lambda^{-1}$ is regularized by truncating its heat semigroup representation: $\Lambda^{-1}_{\epsilon_1} f = c_0 \int_{\epsilon_1}^\infty e^{s\Delta} f s^{-1/2} ds$.
   • The initial data $\theta_0$ is smoothed via the heat semigroup: $\theta_0^{\epsilon_2} = e^{\epsilon_2 \Delta} \theta_0$.
   • This yields a sequence of approximate equations with smooth, divergence-free velocities, which admit global smooth solutions.

2. Symmetrization of the Nonlinearity:

   • To handle the singularity of the kernel $K(x,y) \sim |x-y|^{-3}$, the authors symmetrize the transport term. For a test function $\phi$, the term $\int u \cdot \nabla \theta \phi$ is rewritten using the kernel's antisymmetry (in $\mathbb{R}^2$) or a symmetrized form (in bounded domains).
   • This reduces the effective singularity of the kernel from order 2 to order 1 ($|x-y|^{-1}$), making the quadratic form continuous on $L^{4/3, 2} \times L^{4/3, 2}$.

3. Lorentz Space Estimates and Compactness:

   • Endpoint Control: The authors prove the sharp estimate $\|\Lambda^{-1/2} f\|_{L^2} \lesssim \|f\|_{L^{4/3, 2}}$. They demonstrate that the secondary Lorentz exponent $q=2$ is optimal; the estimate fails for $L^{4/3, q}$ with $q > 2$.
   • High Amplitude Cutoffs: A key technical tool is the modulus $\tau_M(f) = \|(|f|-M)_+\|_{L^{4/3, 2}}$. The authors show that $\tau_M$ is preserved by the heat semigroup and measure-preserving transport. Crucially, $\tau_M$ is weakly lower semicontinuous.
   • Uniform Bounds: The approximation scheme preserves the distribution functions of the initial data (up to the initial smoothing), ensuring that $\tau_M(\theta(t)) \leq \tau_M(\theta_0)$ uniformly in time and regularization parameters.
   • Weighted Estimates: Using the uniform bounds on $\tau_M$, the authors construct a non-decreasing weight function $\Phi$ such that a weighted Lorentz norm remains bounded uniformly in time. This prevents concentration of mass and ensures compactness.

4. Passage to the Limit:

   • The authors establish spatial tightness (in $\mathbb{R}^2$) and boundary decay (in bounded domains) to handle the non-compactness of the domain.
   • By splitting the quadratic kernels into near-diagonal and far-diagonal parts, they use the uniform bounds on $\tau_M$ to control the near-diagonal contribution, while the far-diagonal part converges via standard compactness arguments.

Key Results
The main theorem (Theorem 1.1) establishes the following for any initial datum $\theta_0 \in L^{4/3, 2}(\Omega)$:

1. Global Existence: There exists a global weak solution $\theta \in L^\infty_{loc}([0, \infty); L^{4/3, 2}(\Omega)) \cap C([0, \infty); \mathcal{D}'(\Omega))$.
2. Hamiltonian Conservation: The negative Sobolev energy is exactly conserved for all time:
   $$\|\theta(t)\|_{\dot{H}^{-1/2}(\Omega)} = \|\theta_0\|_{\dot{H}^{-1/2}(\Omega)}.$$
3. Distributional Control: The solution satisfies a uniform bound on high-amplitude cutoffs: $\tau_M(\theta(t)) \leq \tau_M(\theta_0)$ for all $M \geq 0$.
4. Weighted Norm Bound: There exists a time-independent weight $\Phi$ such that a weighted Lorentz norm of the solution remains bounded by the initial data.

Significance and Claims
The paper claims to resolve the existence of Hamiltonian-conserving weak solutions at the Lorentz endpoint $L^{4/3, 2}$. This is significant because:

• It extends the known existence range beyond the critical Lebesgue space $L^{4/3}$ to the larger Lorentz space $L^{4/3, 2}$, allowing for "rougher" initial data.
• It demonstrates that the secondary Lorentz exponent $q=2$ is the sharp threshold for controlling the Hamiltonian via the operator $\Lambda^{-1/2}$.
• The method avoids viscosity, relying instead on a tailored approximation that preserves the distribution function structure of the data, which is essential for maintaining the Hamiltonian identity in the limit.
• The result is complementary to recent non-uniqueness results for weak solutions with slightly higher regularity ($L^{\bar{p}}$ with $\bar{p} > 4/3$); here, the authors construct a specific class of solutions that conserve energy at the lowest possible regularity where such conservation is analytically tractable.

The authors emphasize that while weak solutions at this regularity are not unique (as shown by convex integration methods), their construction provides a canonical class of solutions that preserve the physical energy invariant.