[Paper Review] On the local well-posedness of the Prandtl and the hydrostatic Euler equations with multiple monotonicity regions

Igor Kukavica, Nader Masmoudi+2 more|arXiv (Cornell University)|Feb 9, 2014

Navier-Stokes equation solutions46 references4 citations

TL;DR

This paper establishes local well-posedness for the Prandtl and hydrostatic Euler equations in a new class of initial data: functions with multiple monotonicity regions in the normal direction, combined with analyticity in the complement. The key result is that local existence and uniqueness hold under these generalized monotonicity and analyticity conditions, extending prior well-posedness regimes beyond analytic or single-monotonicity settings.

ABSTRACT

We find a new class of data for which the Prandtl boundary layer equations and the hydrostatic Euler equations are locally in time well-posed. In the case of the Prandtl equations, we assume that the initial datum $u_0$ is monotone on a number of intervals (on some strictly increasing on some strictly decreasing) and analytic on the complement and show that the local existence and uniqueness hold. The same is true for the hydrostatic Euler equations except that we assume this for the vorticity $ω_0=\partial_y u_0$.

Motivation & Objective

• To extend the local well-posedness theory of the Prandtl and hydrostatic Euler equations beyond existing regimes of analyticity or single monotonicity.
• To address the fundamental challenge of ill-posedness in Sobolev spaces by introducing a new class of initial data with piecewise monotonic behavior.
• To prove local existence and uniqueness for the Prandtl equations when the initial velocity is monotone on finitely many intervals and analytic elsewhere.
• To extend the same well-posedness result to the hydrostatic Euler equations, assuming the vorticity (not velocity) has multiple monotonicity regions and is analytic in the complement.
• To provide a new framework for analyzing boundary layer and hydrostatic flows under less restrictive assumptions than previous results.

Proposed method

• Introduce a new class of initial data for the Prandtl equations: velocity $ u_0 $ monotone on finitely many intervals in $ y $, and real-analytic on the complement.
• For the hydrostatic Euler equations, assume the vorticity $ abla imes u_0 = ho_0 $ is monotone on finitely many intervals and analytic elsewhere.
• Use a weighted energy estimate involving $ rac{ ho^2}{ ho_y} $, where $ ho $ is the vorticity, to control the evolution of the solution in time.
• Define a time-dependent interval $ I_t = (Mt, 1 - Mt) $ to localize the analysis and avoid boundary layer singularities.
• Derive a differential inequality for the weighted energy $ X(t) = rac{1}{2} igint_{I_t imes (0,1)} rac{ ho^2}{ ho_y} dx dy $, showing $ X'(t) o 0 $ for small time.
• Apply integration by parts and energy estimates to control nonlinear terms and boundary contributions, ensuring the energy remains bounded for small time $ t o 0 $.

Experimental results

Research questions

• RQ1 Can the Prandtl equations be locally well-posed when the initial velocity has multiple monotonicity regions rather than a single monotonicity interval?
• RQ2 Does the well-posedness result extend to the hydrostatic Euler equations under similar assumptions on the vorticity?
• RQ3 Can the standard analyticity requirement in the $ x $-variable be relaxed while preserving local well-posedness?
• RQ4 Is it possible to construct a solution framework that combines monotonicity in $ y $ with analyticity in the complement, avoiding the need for full analyticity?
• RQ5 What is the role of the Crocco-type transformation and weighted energy estimates in stabilizing solutions with multiple monotonicity regions?

Key findings

• The Prandtl equations are locally well-posed for initial data $ u_0 $ that are monotone on finitely many intervals in $ y $, and analytic on the complement.
• The hydrostatic Euler equations are locally well-posed when the initial vorticity $ ho_0 = abla imes u_0 $ is monotone on finitely many intervals and analytic elsewhere.
• A weighted energy functional $ X(t) = rac{1}{2} igint_{I_t imes (0,1)} rac{ ho^2}{ ho_y} dx dy $ is used to control the solution, and it satisfies $ X'(t) o 0 $ for small time $ t $.
• The method ensures that $ X(t) o 0 $ for $ t o 0 $, implying the solution remains bounded and unique in a small time interval.
• The proof relies on careful integration by parts and boundary term control, particularly at $ x = Mt $ and $ x = 1 - Mt $, to maintain energy estimates.
• Uniqueness is established via the same energy method, showing that any two solutions must agree in the time interval $ (0, t_0) $ with $ t_0 = 1/C_0 $.