[Paper Review] Uniqueness in a class of Hamilton-Jacobi equations with constraints

Sepideh Mirrahimi, Jean‐Michel Roquejoffre|arXiv (Cornell University)|Feb 13, 2015

Mathematical Biology Tumor Growth6 references4 citations

TL;DR

This paper establishes the first global uniqueness result for a class of time-dependent Hamilton-Jacobi equations with a dynamic constraint that forces the solution's maximum to remain identically zero. By proving uniqueness of the coupled solution $(u, I(t))$, the authors enable strong convergence and error estimates for a selection-mutation population model in the small-diffusion limit, where the population concentrates on a time-evolving dominant trait.

ABSTRACT

In this note, we discuss a class of time-dependent Hamilton-Jacobi equations depending on a function of time, this function being chosen in order to keep the maximum of the solution to the constant value 0. The main result of the note is that the full problem has a unique classical solution. The motivation is a selection-mutation model which, in the limit of small diffusion, exhibits concentration on the zero level set of the solution of the Hamilton-Jacobi equation. The uniqueness result that we prove implies strong convergence and error estimates for the selection-mutation model.

Motivation & Objective

• To resolve the open problem of uniqueness for a class of time-dependent Hamilton-Jacobi equations where the maximum of the solution is constrained to remain at zero via a time-dependent parameter $I(t)$.
• To establish strong convergence and error estimates for a selection-mutation population model in the small-diffusion limit ($\varepsilon \to 0$), where the population density concentrates on the zero level set of the Hamilton-Jacobi solution.
• To provide a rigorous asymptotic expansion for the population density $n_\varepsilon$, the trait concentration point $x_\varepsilon$, and the resource level $I_\varepsilon$ in powers of $\varepsilon$.
• To demonstrate that the full sequence of solutions $(n_\varepsilon)$ converges (not just a subsequence), leveraging the uniqueness of the limiting Hamilton-Jacobi problem.
• To validate the biological interpretation that the population concentrates on a dominant trait evolving in time, with the limiting measure concentrated at $\bar{x}(t)$.

Proposed method

• Formulate the Hamilton-Jacobi problem with a dynamic constraint: $\partial_t u = |\nabla u|^2 + R(x, I(t))$ and $\max_x u(t,x) = 0$, where $I(t)$ is an unknown function of time.
• Use the method of characteristics to analyze the linearized equation for the difference $r = V(I_1) - V(I_2)$, which governs the sensitivity of the solution to perturbations in $I(t)$.
• Establish a key estimate: $\|V(I_1) - V(I_2)\|_{W^{2,\infty}} \leq C \|I_1 - I_2\|_{L^\infty} \delta$, which implies a contraction-like behavior for small time intervals.
• Apply a fixed-point argument on a small time interval $[0, \delta]$ using the Lipschitz continuity of the solution map $I \mapsto V(I)$ and the implicit function theorem for the characteristic flow.
• Extend the local solution to global existence and uniqueness by iterating the local argument, relying on the strict concavity of $R(x,I)$ and the bound $I(t) \in [0, I_M]$.
• Use the Hopf-Cole transformation $n_\varepsilon = \exp(u_\varepsilon / \varepsilon)$ to connect the parabolic PDE for $n_\varepsilon$ to the Hamilton-Jacobi equation for $u_\varepsilon$, enabling the asymptotic analysis.

Experimental results

Research questions

• RQ1 Is the solution $(u, I(t))$ to the constrained Hamilton-Jacobi equation uniquely determined under the given regularity and concavity assumptions?
• RQ2 Can the convergence of the population density $n_\varepsilon$ to a measure concentrated on a time-evolving trait $\bar{x}(t)$ be established for the full sequence $\varepsilon \to 0$, rather than just along a subsequence?
• RQ3 What are the higher-order asymptotic expansions of $I_\varepsilon$, $x_\varepsilon$, and $u_\varepsilon$ in powers of $\varepsilon$?
• RQ4 How do the error estimates for the convergence of $n_\varepsilon$ depend on the regularity and concavity of the reproduction rate $R(x, I)$?
• RQ5 What is the precise form of the limiting measure concentration, and how is the amplitude $\bar{\rho}(t)$ related to the solution $I(t)$ and the resource function $\psi(x)$?

Key findings

• The Hamilton-Jacobi problem (1) has a unique classical solution $(u, I(t))$ under the assumptions of $C^2$ regularity of $R$, strict concavity in $x$, and $I_M > 0$ such that $\max_x R(x, I_M) = 0 = R(0, I_M)$.
• The solution satisfies $u \in L^\infty_{\text{loc}}(\mathbb{R}^+; W^{3,\infty}_{\text{loc}}(\mathbb{R}^d)) \cap W^{1,\infty}_{\text{loc}}(\mathbb{R}^+; L^\infty_{\text{loc}}(\mathbb{R}^d))$ and $I \in W^{1,\infty}(\mathbb{R})$.
• The full sequence $n_\varepsilon$ converges weakly in the sense of measures to $\bar{\rho}(t) \delta(x - \bar{x}(t))$ as $\varepsilon \to 0$, with $\bar{\rho}(t) = I(t)/\psi(\bar{x}(t))$.
• The asymptotic expansions $I_\varepsilon = I + \varepsilon I_1 + o(1)$, $x_\varepsilon = \bar{x} + \varepsilon \bar{x}_1 + o(1)$, and $u_\varepsilon = u + \varepsilon \log(r / \varepsilon^{d/2}) + \varepsilon u_1 + o(1)$ are established for the solution of the selection-mutation model.
• The convergence of $n_\varepsilon$ is strong and uniform in time on compact sets, and the error estimates are quantified via the $o(1)$ terms in the expansions.
• The result resolves the long-standing open problem of uniqueness in this class of constrained Hamilton-Jacobi equations, which was previously only known in special cases.