[Paper Review] Heat Equation With a Geometric Rough Path Potential in One Space Dimension: Existence and Regularity of Solution

H.-J. Kim, Sergey V. Lototsky|arXiv (Cornell University)|Dec 21, 2017

Advanced Mathematical Modeling in Engineering4 references3 citations

TL;DR

This paper establishes the existence and regularity of solutions to the one-dimensional heat equation with a distribution-valued potential derived from a Hölder continuous function via geometric rough path theory. Using a regularization approach and the Stratonovich-type interpretation of the product $ u\dot{W} $, it proves that the solution inherits Hölder regularity $ C^{1+\alpha} $ in space and $ C^{(1+\alpha)/2} $ in time, matching the expected loss of one derivative in space and half a derivative in time compared to classical parabolic theory.

ABSTRACT

A solution of the heat equation with a distribution-valued potential is constructed by regularization. When the potential is the generalized derivative of a Hölder continuous function, regularity of the resulting solution is in line with the standard parabolic theory.

Motivation & Objective

• To rigorously establish the existence and regularity of solutions to the one-dimensional heat equation with a distribution-valued potential $ \dot{W} $, where $ W $ is Hölder continuous of order $ \alpha \in (0,1) $.
• To resolve the ambiguity in interpreting the product $ u\dot{W} $ when $ \dot{W} $ is a generalized function, by employing geometric rough path (GRP) theory.
• To show that the solution loses one derivative in space and half a derivative in time compared to classical parabolic solutions, achieving regularity $ C^{1+\alpha} $ in space and $ C^{(1+\alpha)/2} $ in time.
• To define and analyze both classical and generalized GRP solutions through approximation of $ W $ by smooth functions $ W^{(\varepsilon)} $, ensuring convergence in appropriate norms.
• To extend the results to more general parabolic equations with variable coefficients and non-zero Dirichlet or general boundary conditions.

Proposed method

• Interprets the product $ u\dot{W} $ using geometric rough path theory, ensuring the validity of classical calculus rules like $ \int_0^x W(s)\dot{W}(s)ds = \frac{1}{2}W^2(x) $ even for irregular $ W $.
• Constructs the solution as the limit of solutions $ u^{(\varepsilon)} $ to regularized equations where $ W^{(\varepsilon)} $ are smooth approximations of $ W $, with $ \dot{W}^{(\varepsilon)} \in L_\infty $.
• Uses variational (generalized) formulation for weak solutions and establishes convergence in $ L^2 $-based norms under $ L^\infty $-convergence of approximations.
• Applies spectral theory to the operator $ \mathbf{L} $, showing it has pure point spectrum with eigenvalues $ \lambda_k \sim k^2 $, enabling construction of the fundamental solution via eigenfunction expansion.
• Derives the fundamental GRP solution as $ \mathfrak{p}(t,x,y) = e^{-\int_0^x W(s)ds} \mathfrak{p}_W(t,x,y) e^{\int_0^y W(s)ds} $, where $ \mathfrak{p}_W $ is the fundamental solution of the regularized equation.
• Applies change of variables $ u(t,x) = v(t,x)\exp\left(-\int_{L_1}^x W(s)ds\right) $ to reduce the rough potential problem to a standard parabolic equation with smooth coefficients.

Experimental results

Research questions

• RQ1 Can a classical solution exist for the heat equation with a rough potential $ \dot{W} $, where $ W $ is Hölder continuous of order $ \alpha \in (0,1) $, and what regularity does it possess?
• RQ2 How should the product $ u\dot{W} $ be interpreted when $ \dot{W} $ is a distribution, especially in the absence of a time parameter?
• RQ3 Does the solution to the heat equation with a geometric rough path potential exhibit regularity loss consistent with the expected one-derivative loss in space and half-derivative in time?
• RQ4 Is the generalized solution constructed via regularization stable under approximation of $ W $, and does it coincide with the classical solution when both exist?
• RQ5 Can the results be extended to equations with variable coefficients, non-zero Dirichlet boundary conditions, or periodic boundary conditions?

Key findings

• The classical GRP solution of the heat equation with potential $ \dot{W} $, where $ W $ is Hölder continuous of order $ \alpha \in (0,1) $, is shown to be in $ C^{1+\alpha} $ in space and $ C^{(1+\alpha)/2} $ in time, matching the expected regularity loss.
• The generalized GRP solution is constructed as the limit of solutions to regularized equations, and convergence holds in $ L^2 $-based norms under $ L^\infty $-convergence of the approximations $ W^{(\varepsilon)} $.
• The fundamental GRP solution is given explicitly by $ \mathfrak{p}(t,x,y) = e^{-\int_0^x W(s)ds} \mathfrak{p}_W(t,x,y) e^{\int_0^y W(s)ds} $, where $ \mathfrak{p}_W $ is the fundamental solution of the regularized equation.
• The eigenvalues $ \lambda_k $ of the operator $ \mathbf{L} $ satisfy $ \lambda_k \sim k^2 $, and the eigenfunctions $ \mathfrak{m}_k $ form an orthonormal basis in $ L^2((0,\pi)) $, enabling spectral representation of the solution.
• A classical GRP solution, if it exists, coincides with the generalized GRP solution, ensuring consistency between the two formulations.
• Extensions to equations with variable coefficients $ a,b,c,f $ and general boundary conditions are possible, though require additional analysis beyond standard parabolic theory.