Gelfand-type problem for turbulent jets: sharp $L^\infty$ bound on extremal solutions

Technical Summary: Sharp $L^\infty$ Bounds on Extremal Solutions for Gelfand-Type Problems in Turbulent Jets

Problem Statement
The paper investigates a boundary value problem modeling thermal explosion in reactive turbulent jets, originally derived in [GHH]. The problem is formulated on the unit disk $B \subset \mathbb{R}^2$ and belongs to the class of Gelfand-type problems. The governing equation is:
$$\begin{cases} -\Delta u - \alpha r \phi(r) \frac{\partial u}{\partial r} = \lambda \psi(r) f(u) & \text{in } B, \\ u > 0 & \text{in } B, \\ u = 0 & \text{on } \partial B, \end{cases}$$
where $u$ represents normalized temperature, $\lambda$ is the Frank-Kamenetskii parameter, and $\alpha$ is the injection velocity. The functions $\phi$ and $\psi$ describe flow velocity and reactive component profiles, respectively, while $f$ is a positive, increasing, convex reaction rate satisfying the integrability condition $\int_0^\infty ds/f(s) < \infty$.

Previous work by [GMN] established the existence of a critical parameter $\lambda^*(\alpha)$ and an extremal solution $u^*_\alpha$ (the limit of minimal solutions as $\lambda \nearrow \lambda^*$). While [GMN] proved that $u^*_\alpha(0) \to \infty$ as $\alpha \to \infty$, they could only establish an upper bound on $u^*_\alpha(0)$ under additional assumptions regarding the nonlinearity $f$. The central objective of this paper is to close this gap by deriving matching upper and lower bounds for $u^*_\alpha(0)$ in the asymptotic limit of strong flows ($\alpha \to \infty$) without imposing extra conditions on $f$.

Methodology
The authors exploit the radial symmetry of the extremal solution to reduce the partial differential equation to an ordinary differential equation. By introducing the weight function $\mu(r) = \exp(\alpha \int_0^r s\phi(s)ds)$ and performing a change of variables $z = \alpha r^2$, the problem is transformed into an integral equation for the rescaled solution $\bar{u}(z)$:
$$\bar{u}'(z) = -\frac{1}{z\bar{\mu}(z)} \int_0^z \bar{\mu}(s)\bar{\psi}(s)F(\bar{u}(s))ds,$$
where $F = \frac{\lambda^*}{4\alpha}f$.

The core of the analysis relies on the semi-stability condition satisfied by the extremal solution. In the rescaled coordinates, this condition takes the form of a weighted Poincaré-type inequality:
$$\int_0^\alpha z (\bar{\eta}')^2 \bar{\mu} dz \geq \int_0^\alpha \bar{\eta}^2 \bar{\psi} \bar{\mu} F'(\bar{u}) dz,$$
for all $\eta \in H^1_0(B)$.

The proof strategy involves two main components:

1. Upper Bound (Theorem 1.1): The authors construct a specific test function $\bar{\eta}_z$ (piecewise constant/logarithmic) to utilize the semi-stability inequality. This allows them to bound the integral of the derivative of the reaction term, $\int F'(\bar{u})$, by a logarithmic term. Combining this with the integral form of the ODE and the convexity of $f$, they derive an inequality relating $f'(u(0))$ to $\log \alpha$.
2. Lower Bound (Theorem 1.2): The authors employ a contradiction argument. They assume the lower bound fails, which implies the existence of a perturbation $v > 0$ such that $\bar{u} + v$ remains a solution to the equation. By analyzing the linearized operator and utilizing a kernel estimate (Lemma 4.2) derived from the properties of $\mu$ and $\psi$, they show that for sufficiently large $\alpha$, such a positive perturbation cannot exist without violating the extremality of $\lambda^*$. Specifically, they demonstrate that if the bound were false, one could construct a solution corresponding to a parameter $\lambda > \lambda^*$, which contradicts the definition of the critical parameter.

Key Contributions and Results
The paper establishes the following sharp asymptotic bounds for the extremal solution at the origin, $u^*_\alpha(0)$, as $\alpha \to \infty$:

• Theorem 1.1 (Upper Bound): There exist constants $\alpha_0, c, C > 0$ such that for all $\alpha \geq \alpha_0$,
  $$u^*_\alpha(0) \leq c A,$$
  where $A$ is the unique solution to $f'(A) = C \log \alpha$.

• Theorem 1.2 (Lower Bound): There exist constants $\tilde{\alpha}_0, \tilde{c}, \tilde{C} > 0$ such that for all $\alpha \geq \tilde{\alpha}_0$,
  $$u^*_\alpha(0) \geq \tilde{c} \tilde{A},$$
  where $\tilde{A}$ is the unique solution to $f'(\tilde{A}) = \tilde{C} \log \alpha$.

These results confirm that the growth of the extremal solution's peak temperature is governed by the inverse of the derivative of the reaction rate evaluated at a logarithmic scale of the flow velocity.

Significance
The paper claims to resolve a partially open question left by [GMN] regarding the $L^\infty$ behavior of extremal solutions in turbulent jet models. The primary significance lies in the removal of "additional assumptions on the nonlinear reaction rate" that were previously necessary to obtain the upper bound. By proving that the matching upper and lower bounds hold for the general class of convex reaction rates satisfying the standard integrability condition, the authors provide a complete characterization of the blow-up behavior of the extremal solution in the strong flow limit. This result solidifies the theoretical understanding of thermal explosion limits in reactive turbulent jets within the Gelfand framework.