On an inhomogeneous coagulation model with a differential sedimentation kernel

Technical Summary of "On an inhomogeneous coagulation model with a differential sedimentation kernel"

Problem Statement
The paper investigates the local existence of mass-conserving solutions for an inhomogeneous coagulation equation that includes a transport term modeling the sedimentation of particle clusters. The specific equation under study is:
$$\partial_t f(x, v, t) + v^\alpha \partial_x f(x, v, t) = \frac{1}{2} \int_0^v K(v-v', v') f(x, v-v', t) f(x, v', t) dv' - \int_0^\infty K(v, v') f(x, v, t) f(x, v', t) dv'$$
where $x \in \mathbb{R}$ represents the spatial variable, $v \in (0, \infty)$ is the particle volume, and $\alpha \in (0, 1)$ governs the sedimentation velocity. The coagulation kernel $K$ is assumed to be symmetric, continuous, and satisfy a growth condition of the form $K(v, v') \leq K_1(v^\gamma + v'^\gamma)$ with $\gamma \in [0, 1+\alpha)$.

The central challenge addressed is that for the spatially homogeneous case (where the transport term vanishes), coagulation kernels with homogeneity $\gamma > 1$ lead to "instantaneous gelation," a phenomenon where mass is lost immediately, and solutions in $L^1$ do not exist. The paper specifically targets kernels such as the differential sedimentation kernel $K(v, v') = |v^{2/3} - v'^{2/3}|(v^{1/3} + v'^{1/3})^2$ (where $\gamma = 5/3 > 1$) and sum-type kernels $v^\gamma + v'^\gamma$ with $\gamma > 1$.

Methodology
The authors employ an iterative scheme based on a linear reformulation of the model to construct a solution. The core of the methodology involves the following steps:

1. Formal Approximation: The authors approximate the coagulation term for large particles by a transport term. By assuming a specific decay rate for the first moment of the solution, they derive a linear transport equation with a variable coefficient that approximates the behavior of the full non-linear system.
2. Construction of a Supersolution: A crucial component of the proof is the construction of a suitable supersolution, denoted as $G(x, v, t)$. This function is built upon the solution $G_L$ of a linear transport equation:
   $$\partial_t G_L + v^\alpha \partial_x G_L + \frac{L v^\gamma}{1 + |x|^{m-d}} \xi_R(v) \partial_v G_L = 0$$
   The authors analyze the backward characteristics of this linear equation to establish precise bounds on the solution's behavior, particularly regarding its decay in space and volume.
3. Iterative Scheme and Convergence: An inductive sequence $\{f_n\}$ is defined. Using the constructed supersolution $G$, the authors prove that $f_n \leq G$ for all $n$. They then establish uniform bounds and moment estimates for the sequence.
4. Compactness and Limit: By deriving uniform estimates independent of the iteration index, the authors demonstrate the compactness of the sequence $\{f_n\}$ in the appropriate function space. This allows them to pass to the limit and show that the limit function satisfies the mild formulation of the original equation.
5. Mass Conservation: Mass conservation is verified by testing the equation with the weight $v$ and integrating, relying on the sufficiently fast decay of the solution in both $x$ and $v$ (specifically, decay rates involving $|x|^m$ and $v^p$).

Key Contributions and Results

• Local Existence for $\gamma > 1$: The primary result (Theorem 1.4) establishes the local existence of mass-conserving mild solutions for inhomogeneous coagulation equations where the kernel homogeneity $\gamma$ satisfies $1 < \gamma < 1 + \alpha$. This range includes kernels for which the spatially homogeneous problem is ill-posed due to instantaneous gelation.
• Specific Kernel Coverage: The result applies to the differential sedimentation kernel (relevant for rain onset and air bubbles) and sum-type kernels $v^\gamma + v'^\gamma$ with $\gamma > 1$.
• Mass Conservation: Unlike previous works on discrete models with similar kernels, this paper rigorously proves that the constructed solutions conserve total mass for a short time interval.
• Technical Estimates: The paper provides detailed estimates for the characteristics of the transport equation, including bounds on the derivatives of the characteristic curves with respect to the volume variable, which are essential for controlling the non-linear coagulation terms.

Significance
The paper claims to provide the first result of existence for mass-conserving solutions involving sum-type kernels with homogeneity $\gamma > 1$ in the context of space-dependent coagulation models. While previous literature (e.g., [23]) has addressed discrete versions of such models, this work extends the theory to the continuous case and explicitly ensures mass conservation, a property not established in prior discrete studies for these specific kernel classes.

The authors note that their approach relies on the transport term $v^\alpha \partial_x f$ dominating the coagulation term's contribution, which is why the condition $\gamma < 1 + \alpha$ is required. They acknowledge that extending these results to $\alpha \geq 1$ would require a different approach to extracting information from the characteristics. Furthermore, the paper distinguishes itself from existing multi-dimensional coagulation models (which typically assume $\gamma < 1$ or product-type kernels) by handling the super-linear growth case $\gamma > 1$ in a spatially inhomogeneous setting.