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On the local well-posedness of the Prandtl and the hydrostatic Euler equations with multiple monotonicity regions
Igor Kukavica, Nader Masmoudi, Vlad Vicol, Tak Kwong Wong
TL;DR
The paper addresses local well-posedness for Prandtl and hydrostatic Euler data with multiple monotonicity regions, where analytic and monotone estimates present technical challenges. It combines monotonicity and analyticity assumptions with methods for constructing and gluing solutions, and establishes local well-posedness for both systems.
Problem
The paper asks whether local well-posedness holds when initial data are monotone on some regions and analytic on their complement, despite incompatibilities between the estimates used in those regions.
Source: Main results · p. 3Method
The approach combines energy methods for monotone Prandtl regions with analyticity in complementary regions, and uses convex extensions and finite-speed gluing for hydrostatic Euler.
Source: §1.2 · p. 5, §2 · p. 6Results
The paper establishes local well-posedness for Prandtl data with multiple monotonicity regions and analytic complements, and local existence and uniqueness for hydrostatic Euler under corresponding vorticity conditions.
Source: Main results · p. 3, §2.2 · p. 9, §3 · p. 20Takeaways & Limitations
The results extend local existence for hydrostatic Euler to a larger class of initial data than previously considered in.
Source: §1.2 · p. 5In the stated setup, overlaps between analytic and monotonicity regions have a uniform positive minimum length, though the statement can be adjusted for infinitely many intervals.
Source: §2.2 · p. 9
Abstract
from arXiv · showhide
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 $\omega_0=\partial_y u_0$.
1. INTRODUCTION
The paper studies local well-posedness for the Prandtl equations, a system formally derived in boundary-layer analysis. It addresses whether solutions exist uniquely in time and the distinct question of justifying the Prandtl asymptotics.
1.1. The Prandtl Equations.
- The Prandtl equations are formally obtained by retaining leading-order terms in a matched asymptotic expansion near a thin boundary layer.The expansion describes the Navier–Stokes flow within the boundary layer in terms of the boundary-layer velocity and viscosity.Source: §1.1 · p. 1
- The paper addresses local well-posedness for the Prandtl equations, not the separate question of rigorously justifying Prandtl asymptotics.The introduction identifies both as fundamental questions raised by the equations’ formal derivation.Source: §1.1 · p. 2
- Existing well-posedness results covered monotone data, data analytic in x, or non-degenerate monotonicity changes with Gevrey regularity in x.The paper presents its results as a fourth regime.Source: §1.1 · p. 2
Summary of previous results.
Prior Prandtl results establish local well-posedness under monotonicity or analyticity assumptions, and a Gevrey result covers certain non-degenerate monotonicity changes. The equations are also reported ill-posed in Sobolev spaces around certain non-monotone shear flows.
Summary of previous results.
- Monotone initial data yield a unique local solution, while a favorable pressure gradient permits global extension in this setting.The cited works use the Crocco transform or energy methods; a special finite-time blowup solution is also known when U = P = 0.Source: Summary of previous results · p. 2
- Analytic data yield local well-posedness; later results remove analyticity in y and allow algebraic decay of u − U as y → ∞.The later energy-based result assumes real analyticity in x only.Source: Summary of previous results · p. 3
- A compactly supported-in-y initial datum is reported to have both local existence and finite-time blowup results for U = P = 0.The local-existence results cited are, and the blowup result is.Source: Summary of previous results · p. 3
- Certain non-monotone shear flows are ill-posed in Sobolev and C∞ spaces, while a Gevrey 7/4 result covers non-degenerate monotonicity changes.The Gevrey result is described as combining the analytic setting with the nonlinear cancellation available in the monotone setting.Source: Summary of previous results · p. 3
Main results.
The paper extends local well-posedness to Prandtl data with multiple monotonicity regions and analytic complements, and to hydrostatic Euler data with convex and concave regions separated by an analytic complement. It constructs and glues solutions adapted to these regions.
Main results.
- For Prandtl, tangential analyticity on the complement of increasing and decreasing monotonicity regions is a sufficient condition for local well-posedness.The paper poses the problem for the two-dimensional system and gives an example using initial vorticity sin(x)(1 + y^2)^−1.Source: Main results · p. 3
- The Prandtl construction faces x-localization, side-boundary, incompatible-norm, and gluing difficulties between analytic and finite-order Sobolev solutions.Analytic functions of x cannot have compact support, and the monotone-region construction requires side-boundary conditions not handled by the existing energy methods.Source: Main results · p. 3
- The Prandtl method constructs analytic solutions without lateral boundary conditions, extends monotone data globally in x, and glues solutions using finite speed of propagation.When the initial overlap contains [a, b], the solutions agree on [a + Mt, b − Mt] for sufficiently small t.Source: Main results · pp. 3–4
1.2. The Hydrostatic Euler Equations.
- For hydrostatic Euler, convex and concave regions are permitted when the complement is uniformly real analytic; the resulting solution is locally smooth, Sobolev on the monotonicity domain, and analytic on its complement.The theorem assumes a uniform Rayleigh condition on the monotonicity domain and a compatibility condition for the initial velocity.Source: §1.2 · p. 5
- The hydrostatic Euler construction extends convex data across the strip, uses interior analytic estimates, and glues the solutions by finite speed of propagation.The authors state that nonuniformity of the Rayleigh condition prevents the needed integration by parts in x on the original domain.Source: §1.2 · p. 5
2. THE MAIN RESULTS FOR THE PRANDTL EQUATIONS
The section establishes local existence and uniqueness for Prandtl solutions under Oleinik monotonicity on unbounded x-domains and for data combining monotone and analytic regions. It also constructs a global-in-x, local-in-time solution by patching real-analytic and monotone solutions, using finite speed of propagation.
2.1. Local existence for Oleinik solutions in unbounded domains.
- Under the Oleinik monotonicity condition, local solutions exist by energy methods even on x-unbounded domains, extending the periodic result in [MW12a] via finite propagation.Theorem 2.1 gives uniqueness on R or T, and on bounded intervals uniqueness holds on the shrinking interior region I_t.Source: §2 · p. 6, §2.1 · p. 7
2.2. Local existence with multiple monotonicity regions.
- A smooth solution exists on [0, T], is monotone with respect to y in I± and analytic with respect to x in Jj, and is unique in this class.Source: §2.2 · p. 9
2.2.1. Construction of the analytic solution.
- The multiple-region proof constructs an analytic solution on the analytic region, a uniformly local monotone solution on monotone regions, then glues them using finite propagation.The analytic solution is obtained locally by the Cauchy–Kowalewski theorem; analytic estimates avoid integration by parts in x by decreasing the analyticity radius to compensate for derivative loss.Source: §2.2 · p. 9, §2.2.1 · pp. 10–11
- Picard iteration constructs a real-analytic solution; uniqueness requires estimating solution differences in an analytic norm because lateral boundary conditions are absent.The real-analytic solution also has higher regularity in y, used later to glue it to the monotone solutions.Source: §2.2.1 · p. 12
2.2.2. Construction of the monotone solution.
- The initial data admit positive and negative extensions whose corresponding monotone Prandtl solutions exist on a short time interval by Theorem 2.1.The extensions preserve the data on the monotonicity intervals and can be made with at most doubled uniform local norm.Source: §2.2.2 · p. 14
2.2.3. Glueing the analytic and monotone solutions.
- Patching the analytic and monotone solutions across their overlap yields a global-in-x solution of the Prandtl system for a local time interval.The solutions agree on a shrinking overlap, with finite speed of propagation in x enabling the gluing.Source: §2.2.3 · pp. 15, 17
2.2.4. Uniqueness.
- Uniqueness holds for two solutions in the theorem’s class, by applying the gluing argument on shrinking monotonicity and analyticity intervals for sufficiently small time.The intervals overlap initially by τ0/2, and their shrinking speed is finite.Source: §2.2.4 · p. 17
3. THE MAIN RESULT FOR THE HYDROSTATIC EULER EQUATIONS
The hydrostatic Euler equations have a unique local smooth solution when the initial vorticity is Rayleigh-monotone on one region and analytic on another, subject to compatibility of the initial velocity. This extends to mixed increasing and decreasing monotonicity regions, including infinitely many intervals.
Theorem 3.2
- Under a uniform Rayleigh condition and Sobolev regularity on the monotonicity domain, analytic vorticity on its complement, and velocity compatibility, there is a unique local smooth solution with zero vertical mean.The solution lies in X_{τ0/2}(Da) ∩ H^s_{σ,uloc}(Dm) for s ≥ 4.Source: §3 · p. 20
Remark 3.3
- The result extends to regions where vorticity is strictly increasing or decreasing, with analytic regions covering the remainder; the number of intervals may be infinite.On each monotonicity interval either ω0 or −ω0 satisfies the stated Sobolev condition, while analytic intervals require radius at least τ0.Source: §3 · p. 20
Proof
- The proof constructs a solution from a globally convex extension of the monotone-region data, using countably many periodic solutions that share a time interval and uniform Sobolev bounds.Those periodic solutions are then glued together.Source: §3 · p. 20
- A weighted difference estimate makes the monotone and analytic solutions agree on an overlap, allowing them to be stitched into a solution on the full domain.The construction uses a sufficiently large M and a sufficiently short time interval; uniqueness follows similarly.Source: §3 · p. 22