[Paper Review] Oleinik type estimates for the Ostrovsky-Hunter eequation

Giuseppe Maria Coclite, Lorenzo di Ruvo|arXiv (Cornell University)|Mar 10, 2014

Advanced Mathematical Physics Problems13 references3 citations

TL;DR

This paper establishes the existence and uniqueness of entropy solutions for the Ostrovsky-Hunter equation using an Oleïnik-type estimate in place of Kruźkov-type entropy inequalities. By introducing a nonlocal adjoint problem, the authors prove that shock waves are admissible only if they jump downward in value—mirroring the inviscid Burgers equation—under suitable initial and flux conditions, ensuring bounded, globally defined solutions with decay and regularity properties.

ABSTRACT

The Ostrovsky-Hunter equation provides a model for small-amplitude long waves in a rotating fluid of finite depth. It is a nonlinear evolution equation. In this paper we study the well-posedness for the Cauchy problem associated to this equation within a class of bounded discontinuous solutions. We show that we can replace the Kruzkov-type entropy inequalities by an Oleinik-type estimate and prove uniqueness via a nonlocal adjoint problem. An implication is that a shock wave in an entropy weak solution to the Ostrovsky-Hunter equation is admissible only if it jumps down in value (like the inviscid Burgers equation).

Motivation & Objective

• To establish well-posedness for the Cauchy problem of the Ostrovsky-Hunter equation in the space of bounded discontinuous solutions.
• To replace the standard Kruźkov-type entropy inequalities with an Oleïnik-type estimate for proving uniqueness.
• To demonstrate that shock waves in entropy solutions are admissible only if they exhibit a downward jump in value, analogous to the inviscid Burgers equation.
• To ensure global existence and boundedness of solutions under initial data with $ L^1 \cap L^∞ $ and $ P_0 \in L^2 $, with zero total mass.
• To develop a nonlocal adjoint problem framework to overcome limitations of the doubling of variables method in this nonlocal setting.

Proposed method

• Use of an Oleïnik-type estimate as a substitute for Kruźkov-type entropy inequalities to enforce uniqueness.
• Formulation of the Ostrovsky-Hunter equation in integro-differential form: $ \partial_t u + u\partial_x u = \gamma \int_{-\infty}^x u(t,y)dy $, with $ P = \int_{-\infty}^x u $.
• Introduction of a nonlocal adjoint problem to analyze the dual formulation and derive uniqueness via energy estimates.
• Application of compensated compactness arguments based on the strict convexity of the flux $ f $, ensuring weak convergence and existence.
• Use of mollified solutions $ \phi_\varepsilon $, $ \psi_\varepsilon $, and regularization via $ b_\varepsilon $ to approximate the solution and pass to the limit.
• Employment of $ L^2 $ and $ H^1 $ estimates on $ \psi_\varepsilon $, combined with $ L^\infty $ and $ L^1 $ convergence, to pass to the limit in the weak formulation.

Experimental results

Research questions

• RQ1 Can the Kruźkov-type entropy inequality framework be replaced by an Oleïnik-type estimate in the analysis of the Ostrovsky-Hunter equation?
• RQ2 What conditions ensure the existence and uniqueness of bounded entropy solutions for the Ostrovsky-Hunter equation with nonlocal source terms?
• RQ3 Under what conditions is a shock wave admissible in an entropy solution of the Ostrovsky-Hunter equation?
• RQ4 How does the nonlocal structure of the equation affect the applicability of standard methods like doubling of variables?
• RQ5 What role does the strict convexity of the flux $ f $ play in deriving regularity and uniqueness via compensated compactness?

Key findings

• The initial value problem for the Ostrovsky-Hunter equation admits a unique entropy solution in $ L^\infty((0,T)\times\mathbb{R}) $ under the stated assumptions on $ u_0 $, $ f $, and $ P_0 $.
• The Oleïnik-type estimate $ \frac{u(t,x) - u(t,y)}{x - y} \leq C(T)\left(\frac{1}{t} + 1\right) $ is equivalent to the entropy solution condition and ensures uniqueness.
• Shock waves in entropy solutions are admissible only if they satisfy a downward jump condition, similar to the inviscid Burgers equation.
• The nonlocal adjoint problem technique successfully replaces the doubling of variables method, which fails for this formulation.
• The solution remains bounded and globally defined due to the subquadratic growth and decay assumptions on $ u_0 $ and $ P_0 $, ensuring $ P_0 \in L^2 $.
• The equivalence between the entropy solution and the Oleïnik-type estimate is established via a limiting argument on regularized solutions and convergence in $ L^1 $ and $ H^2 $.