Global renormalized solutions for hard potential non-cutoff Boltzmann equation without defect measure

1. Problem Statement

The paper addresses the global existence of renormalized solutions to the spatially inhomogeneous Boltzmann equation with long-range interactions (non-cutoff) in the whole space $\mathbb{R}^3$. The equation is given by:
$$\partial_t f + v \cdot \nabla_x f = Q(f, f)$$
where the collision kernel $B(v-v_*, \sigma)$ exhibits an angular singularity (non-integrable at $\theta = 0$) and a velocity dependence of the form $|v-v_*|^\gamma$.

The specific focus is on the hard potential regime where $0 \le \gamma \le 1$ (covering Maxwell molecules $\gamma=0$ and hard spheres $\gamma=1$).

The Core Challenge:
Previous work by Alexandre and Villani (2002) established the existence of renormalized solutions for non-cutoff equations but required the introduction of a non-negative defect measure. This defect measure arises because the standard Fatou's lemma argument yields only a lower semicontinuity inequality for the collision term, rather than an equality. The authors aim to prove that for hard potentials, this defect measure vanishes, thereby establishing the existence of renormalized solutions in the standard sense (satisfying the equation exactly, not just as an inequality).

2. Methodology

The authors employ a rigorous approximation and compactness strategy, refining the techniques of Alexandre and Villani to exploit the specific properties of hard potentials.

A. Approximation Scheme

1. Smooth Approximation: They construct a sequence of smooth approximate solutions $f_n$ to a regularized problem.
   • The initial data $f_0$ is approximated by Schwartz functions $f_{0,n}$ with Gaussian lower bounds.
   • The collision kernel $B$ is truncated to remove grazing collisions ($\theta \approx 0$), resulting in $B_n$.
   • A nonlinear damping factor $(1 + \frac{1}{n}\int f_n dv)^{-1}$ is introduced to ensure global existence and uniqueness of smooth solutions $f_n$ for the approximate equation.

B. Key Technical Innovations

To remove the defect measure, the authors establish two critical estimates that were not available or sufficient in the soft potential case:

1. Improved Operator Bounds (Regularity of Test Functions):

   • In the Alexandre-Villani framework, the bilinear term $(R_2)$ of the renormalized collision operator was controlled using test functions in $W^{2,\infty}$. For hard potentials, the growth of the kernel at infinity ($|v-v_*|^\gamma$) prevents the necessary decay.
   • Innovation: The authors utilize test functions in the Hölder space $C^\alpha_c(\mathbb{R}^3_v)$ with $\nu < \alpha < 1$. They prove that the associated linear operator $T$ maps $C^\alpha_c$ to $L^\infty_{loc}$ with a bound involving $|v-v_*|^{\alpha+\gamma}$. This allows them to control the singularity using the available a priori bounds on mass and energy, which is crucial for the hard potential case.

2. Average Time-Velocity Estimate:

   • The defect measure typically arises from the term $(R_3)$, which is non-negative. To prove its convergence, the authors derive a specific average time-velocity estimate for $(R_3)_n$.
   • By expanding the squared term in the entropy dissipation and utilizing the cancellation lemma, they bound the integral of $(R_3)_n$ by the entropy dissipation and collision integrals.
   • Crucial Step: For hard potentials ($0 \le \gamma \le 1$), the collision integral $\int f_n f_{n*} |v-v_*|^\gamma$ is controllable by the second moment (energy). This allows the derivation of a uniform bound on $(R_3)_n$ that prevents the "loss" of mass/energy to a defect measure.

3. Strong Compactness via Entropy Dissipation:

   • They establish a functional inequality showing that the high-frequency components of $\sqrt{f_n}$ in velocity space are controlled by the entropy dissipation.
   • Combining this with the averaged velocity lemma and the M. Riesz-Fréchet-Kolmogorov theorem, they prove that $f_n$ converges strongly in $L^1$ to a limit $f$. This strong convergence is essential for passing to the limit in the nonlinear collision terms.

C. Decomposition Strategy for $(R_3)_n$

To handle the angular singularity at $\theta=0$, the authors decompose the collision operator into:

• Non-singular part: Converges via standard dominated convergence and strong convergence of $f_n$.
• Singular part: They prove a uniform smallness estimate for small angular deflections. This estimate shows that the contribution of the singular part vanishes uniformly as the cutoff angle goes to zero, relying on the fractional Sobolev regularity ($H^{\nu/2}$) of $\sqrt{f_n}$ derived from the entropy dissipation.

3. Key Contributions

1. Elimination of the Defect Measure: The primary contribution is proving that for hard potentials ($0 \le \gamma \le 1$), the defect measure in the renormalized formulation is identically zero. This upgrades the solution concept from an inequality (Alexandre-Villani) to an equality (standard renormalized solution).
2. Refined Regularity Analysis: The introduction of Hölder continuous test functions ($C^\alpha$) instead of Sobolev spaces ($W^{2,\infty}$) to handle the growth of the collision kernel at infinity for hard potentials.
3. Counterexample for Soft Potentials: The authors provide a rigorous counterexample (Example 9.1) demonstrating that their approach fails for soft potentials ($-3 < \gamma < 0$). They construct a distribution with finite mass, energy, and entropy where the collision integral $\int f f_* |v-v_*|^\gamma$ diverges. This explains why the defect measure cannot be removed for soft potentials using current methods.

4. Main Results

Theorem 1.1:
Let $0 \le \gamma \le 1$. Given initial data $f_0$ with finite mass, energy, and entropy, the Cauchy problem for the non-cutoff Boltzmann equation admits a global renormalized solution $f$ in the standard sense (without defect measure).
The solution satisfies:

• Continuity: $f \in C(\mathbb{R}_+; w\text{-}L^1(\mathbb{R}^3_x; B^{-\alpha}_{1,1}(\mathbb{R}^3_v)))$.
• Uniform Bounds: Finite mass, energy, and entropy uniformly in time.
• Entropy Inequality: The solution satisfies the exact entropy inequality:
  $$H(f)(t) + \frac{1}{4} \int_0^t \int B(f'f'_* - ff_*) \log\left(\frac{f'f'_*}{ff_*}\right) \le H(f_0)$$
  (Note: The inequality becomes an equality in the limit due to the vanishing defect measure).

5. Significance

• Theoretical Advancement: This work resolves a long-standing open problem regarding the necessity of defect measures for hard potentials. It confirms that the "loss" of information in the limit is an artifact of the soft potential regime or insufficient estimates, not an intrinsic feature of hard potential non-cutoff equations.
• Methodological Impact: The techniques developed, particularly the use of Hölder spaces for test functions and the specific time-velocity averaging estimates, provide a new toolkit for analyzing kinetic equations with singular kernels.
• Clarification of Regimes: By explicitly constructing a counterexample for soft potentials, the paper delineates the boundary between regimes where standard renormalized solutions exist and those where defect measures are unavoidable, guiding future research in soft potential models.

In summary, the paper successfully proves that for hard potentials, the non-cutoff Boltzmann equation possesses global renormalized solutions that satisfy the equation exactly, removing the need for the defect measure introduced in previous foundational works.