[Paper Review] On the Dirichlet Problem for First Order Linear Hyperbolic PDEs on Bounded Domains with Mere Inflow Boundary

Thomas März|arXiv (Cornell University)|Aug 20, 2010

Advanced Mathematical Modeling in Engineering11 references3 citations

TL;DR

This paper establishes existence, uniqueness, and continuous dependence of solutions to first-order linear and quasi-linear hyperbolic PDEs on bounded 2D domains with an interior outflow set Σ and a mere inflow boundary, using a Lyapunov function to ensure causality and constructing solutions in the space of functions of bounded variation (BV), enabling compactness for fixed-point arguments in the quasi-linear case.

ABSTRACT

Here we study the Dirichlet problem for first order linear and quasi-linear hyperbolic PDEs on a simply connected bounded domain of $\R^2$, where the domain has an interior outflow set and a mere inflow boundary. By means of a Lyapunov function we show the existence of a unique solution in the space of functions of bounded variation and its continuous dependence on all the data of the linear problem. Finally, we conclude the existence of a solution to the quasi-linear case by utilizing the Schauder fixed point theorem. This type of problems considered here appears in applications such as transport based image inpainting.

Motivation & Objective

• To establish the existence and uniqueness of solutions to first-order linear hyperbolic PDEs on bounded 2D domains with a mere inflow boundary and an interior outflow set Σ.
• To develop a framework for handling shocks that form on Σ by ensuring the solution belongs to BV(Ω), not just BV(Ω\Σ), thereby closing the solution gap on Σ.
• To extend the linear theory to the quasi-linear case by leveraging compactness and continuous dependence in BV(Ω), enabling application of the Schauder fixed point theorem.
• To provide a rigorous mathematical foundation for transport-based image inpainting, where discontinuities naturally arise on internal stop sets.
• To generalize the use of time functions T with complex topologies (e.g., saddle nodes) in the BV framework, allowing for multi-stage solution construction.

Proposed method

• Utilizes a global Lyapunov function T: Ω → ℝ satisfying ⟨c(x), ∇T(x)⟩ ≥ β|∇T(x)| with β > 0, ensuring causality and that all boundary points are inflow.
• Constructs solutions via the method of characteristics, with characteristics originating from the boundary and terminating at the interior set Σ.
• Defines the solution in the space BV(Ω) ∩ L∞(Ω), ensuring the solution is well-defined and compactly representable even across Σ.
• Establishes continuous dependence on data (c, f, u₀) in the BV(Ω) framework, crucial for stability and fixed-point arguments.
• For the quasi-linear case, fixes the functional dependence of coefficients c[u] and f[u] on a test function v, reducing the problem to a linear PDE with solution operator U[v].
• Applies the Schauder fixed point theorem to the operator U: X → X (with X ⊂ BV(Ω)) by exploiting weak* compactness of BV(Ω) and continuous dependence, proving existence of a fixed point u = U[u] solving the quasi-linear problem.

Experimental results

Research questions

• RQ1 Can a unique solution be constructed for a first-order linear hyperbolic PDE on a bounded 2D domain with an interior outflow set Σ and a mere inflow boundary, even when shocks form on Σ?
• RQ2 How can the solution be defined on the entire domain Ω, including Σ, to ensure compactness and functional analytic properties necessary for nonlinear problems?
• RQ3 What conditions on the transport field c and the time function T guarantee global existence and uniqueness of solutions in BV(Ω)?
• RQ4 Can the Schauder fixed point theorem be applied to the quasi-linear case by leveraging the compactness of BV(Ω) and continuous dependence in the linear case?
• RQ5 How does the use of a Lyapunov function with non-trivial topology (e.g., saddle nodes) affect the solution structure and the possibility of multi-stage solution construction?

Key findings

• The linear hyperbolic Dirichlet problem admits a unique solution in BV(Ω) ∩ L∞(Ω) under the existence of a global Lyapunov function T satisfying the causality condition ⟨c, ∇T⟩ ≥ β|∇T| with β > 0.
• The solution is continuously dependent on all data (c, f, u₀), ensuring robustness and stability under perturbations.
• The solution is well-defined on the entire domain Ω, including the interior outflow set Σ, thereby closing the gap between Ω\Σ and Ω and enabling compactness in BV(Ω).
• The compactness of the solution operator U in the BV(Ω) weak* topology, combined with continuous dependence, allows the application of the Schauder fixed point theorem to prove existence of a solution to the quasi-linear problem.
• For a class of quasi-linear problems with coefficients depending on the entire function u (not just pointwise values), a fixed point u = U[u] exists, and explicit examples show multiple fixed points are possible.
• The framework supports complex time functions T with saddle nodes, enabling decomposition into decoupled problems on subdomains, with shocks forming at the saddle node even for smooth boundary data.