Prandtl Equations Research Articles

Gevrey well-posedness for the Prandtl equation

Gevrey well-posedness for the Prandtl equation

Global Tangentially Analytical Solutions of the 3D Axially Symmetric Prandtl Equations

Global Tangentially Analytical Solutions of the 3D Axially Symmetric Prandtl Equations

Nonuniqueness of Generalised Weak Solutions to the Primitive and Prandtl Equations

We develop a convex integration scheme for constructing nonunique weak solutions to the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics) in both two and three dimensions. We also develop such a scheme for the construction of nonunique weak solutions to the three-dimensional viscous primitive equations, as well as the two-dimensional Prandtl equations. While in Boutros et al. (Calc Var Partial Differ Equ 62(8):219, 2023) the classical notion of weak solution to the hydrostatic Euler equations was generalised, we introduce here a further generalisation. For such generalised weak solutions, we show the existence and nonuniqueness for a large class of initial data. Moreover, we construct infinitely many examples of generalised weak solutions which do not conserve energy. The barotropic and baroclinic modes of solutions to the hydrostatic Euler equations (which are the average and the fluctuation of the horizontal velocity in the z-coordinate, respectively) that are constructed have different regularities.

Global Gevrey-2 solutions of the 3D axially symmetric Prandtl equations

In this paper, we prove the global existence of small Gevrey-2 solutions to the 3D axially symmetric Prandtl equations. The index 2 is the optimal index for well-posedness result in smooth Gevrey function spaces for data without monotonic assumptions. The novelty of our paper lies in two aspects: one is the tangentially weighted energy construction to match the [Formula: see text] weight in the incompressibility and the other is introducing of the new linearly good unknowns to obtain the fast decay of the lower order Gevrey-2 norms of the solutions and auxiliary functions.

Navier–Stokes Equations in the Half Space with Non Compatible Data

This paper considers the Navier–Stokes equations in the half plane with Euler-type initial conditions, i.e., initial conditions with a non-zero tangential component at the boundary. Under analyticity assumptions for the data, we prove that the solution exists for a short time independent of the viscosity. We construct the Navier–Stokes solution through a composite asymptotic expansion involving solutions of the Euler and Prandtl equations plus an error term. The norm of the error goes to zero with the square root of the viscosity. The Prandtl solution contains a singular term, which influences the regularity of the error. The error term is the sum of a first-order Euler correction, a first-order Prandtl correction, and a further error term. The use of an analytic setting is mainly due to the Prandtl equation.

Quantitative aspects on the ill-posedness of the Prandtl and hyperbolic Prandtl equations

We address the Prandtl equations and a physically meaningful extension known as hyperbolic Prandtl equations. For the extension, we show that the linearised model around a non-monotonic shear flow is ill-posed in any Sobolev spaces. Indeed, shortly in time, we generate solutions that experience a dispersion relation of order k3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\root 3 \\of {k}$$\\end{document} in the frequencies of the tangential direction, akin the pioneering result of Gérard-Varet, D., Dormy, E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc., 23(2), 591–609 (2010) for Prandtl (where the dispersion was of order k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqrt{k}$$\\end{document}). We emphasise, however, that this growth rate does not imply (a-priori) ill-posedness in Gevrey-class m, with m>3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m>3$$\\end{document}. We relate these aspects to the original Prandtl equations in Gevrey-class m, with m>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m>2$$\\end{document}: We show that the result in Gérard-Varet, D., Dormy, E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc., 23(2), 591–609 (2010) determines a dispersion relation of order k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqrt{k}$$\\end{document} for a short time proportional to ln(k)/k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ln (\\sqrt{k})/\\sqrt{k}$$\\end{document}. Therefore, the ill-posedness in Gérard-Varet, D., Dormy, E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc., 23(2), 591–609 (2010) in its generality is momentarily constrained to Sobolev spaces rather than extending to the Gevrey classes.

Global existence and decay estimates of solutions for the compressible Prandtl type equations with small analytic data

This paper aims to address the issues of the global existence and the large-time decay estimates of strong solutions for the two-dimensional compressible Prandtl equations with small initial data, which is analytical in the tangential variable. We investigate a more complicated system, which contains more physics than the incompressible system. Not only the loss of the horizontal derivative in the estimate of nonlinear terms, but also the failure to satisfy the divergence-free condition, may create significant difficulties when closing energy estimates. Motivated by Paicu and Zhang (2021) [36] for the incompressible case, we find a new quantity G=defu+y2〈t〉φ and derive a sufficiently fast decay-in-time estimate of some weighted analytic norm to G. Together with a linear combination of the tangential velocity u with its primitive one φ, which controls the evolution of the analytic radius to solutions ultimately. Another interesting feature of this paper is to take into account the influence of the density, however, the density is independent of the y variable. In order to avoid these obstacles, the approaches we take are some delicate treatments, including the special cut-off function and new inequalities in the process of energy estimates. This paper can be considered as a first global-in-time Cauchy–Kowalevsakya result for the compressible Prandtl equations with small analytic data.

Local existence of solutions to the 2D MHD boundary layer equations without monotonicity in Sobolev space

<abstract><p>In this work, we investigated the local existence of the solutions to the 2D magnetohy-drodynamic (MHD) boundary layer equations on the half plane by energy methods in weighted Sobolev space. Compared to the existence of solutions to the classical Prandtl equations where the monotonicity condition of the tangential velocity plays an important role, we used the initial tangential magnetic field with a lower bound $ \delta > 0 $ instead of the monotonicity condition of the tangential velocity.</p></abstract>

On the Analytical Determination of the Contour of Well - Styeamlined Bodies

The problem is solved with the help of a modified Prandtl equation applied to the case under study. This is a two-dimensional problem of flowing around a flat body when the essential factor is to take into account the limitation of its dimensions in the longitudinal and transverse directions. Thanks to the above Prandtl equation it was possible to reduce the problem to a self-similar equation. An analytical solution has been found. Thanks to this solution the shape of the body is analytically determined when the resistance is at its lowest. An analysis of the solution of the problem for different Reynolds numbers is carried out. The resulting equation is solved numerically for different values of its included parameters. With the help of a graphic illustration the different shapes of such contours are shown.

Gevrey-Class-3 Regularity of the Linearised Hyperbolic Prandtl System on a Strip

In the present paper, we address a physically-meaningful extension of the linearised Prandtl equations around a shear flow. Without any structural assumption, it is well-known that the optimal regularity of Prandtl is given by the class Gevrey 2 along the horizontal direction. The goal of this paper is to overcome this barrier, by dealing with the linearisation of the so-called hyperbolic Prandtl equations in a strip domain. We prove that the local well-posedness around a general shear flow U_{textrm{sh}}in W^{3, infty }(0,1) holds true, with solutions that are Gevrey class 3 in the horizontal direction.

Boundary layer separation for the steady compressible Prandtl equation

In this paper, we investigate the boundary layer separation for the steady compressible Prandtl equation in the case of adverse pressure gradient. When the pressure satisfies appropriate conditions, we find that the singularity of the solution and the separation point occurs simultaneously. This confirms the conclusion of [41] where Stewartson found that if the heat transfer in the boundary layer disappears, the singularity will be the same as that in the incompressible case. We also investigate the separation rate and the local behavior near the separation point.

Example of a solution for Dorodnitzyn’s limit formula

In the present paper, we show an example of a solution for Dorodnitzyn’s gaseous boundary layer limit formula. Oleinik’s no back-flow condition ensures the existence and uniqueness of solutions for the Prandtl equations in a rectangular domain R⊂R2. It also allowed us to find a limit formula for Dorodnitzyn’s stationary compressible boundary layer with constant total energy on a bounded convex domain in the plane R2. Under the same assumption, we can give an approximate solution u for the limit formula if |u|<<<1 such that: u(z)≅δ∗c∗z+625⋅12i0⋅4U23z4+o(z5),that corresponds to an approximate horizontal velocity component when a small parameter ϵ given by the quotient of the maximum height of the domain divided by its length tends to zero. Here, c>0, δ is the boundary layer’s height in Dorodnitzyn’s coordinates, U is the free-stream velocity at the upper boundary of the domain, and T0 is the absolute surface temperature.

A numerical method for solving systems of hypersingular integro-differential equations

This paper is concerned with a collocation-quadrature method for solving systems of Prandtl's integro-differential equations based on de la Vallée Poussin filtered interpolation at Chebyshev nodes. We prove stability and convergence in Hölder-Zygmund spaces of locally continuous functions. Some numerical tests are presented to examine the method's efficacy.

Global regularity of solutions to the 2D steady compressible Prandtl equations

&lt;abstract&gt;&lt;p&gt;In this paper, we study the global $ C^{\infty} $ regularity of solutions to the boundary layer equations for two-dimensional steady compressible flow under the favorable pressure gradient. To our knowledge, the difficulty of the proof is the degeneracy near the boundary. By using the regularity theory and maximum principles of parabolic equations together with the von Mises transformation, we give a positive answer to it. When the outer flow and the initial data satisfied appropriate conditions, we prove that Oleinik type solutions smooth up the boundary $ y = 0 $ for any $ x &amp;gt; 0 $.&lt;/p&gt;&lt;/abstract&gt;

Asymptotic behavior of the steady Prandtl equation

We study the asymptotic behavior of the Oleinik’s solution to the steady Prandtl equation when the outer flow \(U(x)=1\). Serrin proved that the Oleinik’s solution converges to the famous Blasius solution \({\bar{u}}\) in \(L^\infty _y\) sense as \(x\rightarrow +\infty \). The explicit decay estimates of \(u-{\bar{u}}\) and its derivatives were proved by Iyer (ARMA 237,2020) when the initial data is a small localized perturbation of the Blasius profile. In this paper, we prove the explicit decay estimate of \(u-\bar{u}\) for general initial data with exponential decay. We also prove the decay estimates of its derivatives when the initial data has an additional concave assumption. Our proof is based on the maximum principle techniques. The key ingredient is to find a series of barrier functions.

Global well-posedness of a Prandtl model from MHD in Gevrey function spaces

We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer. A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index 2. The proof is based on a cancellation mechanism through some auxiliary functions from the study of the Prandtl equation and an observation about the structure of the loss of one order tangential derivatives through twice operations of the Prandtl operator.

Slipping flows and their breaking

The process of breaking of inviscid incompressible flows along a rigid body with slipping boundary conditions is studied. Such slipping flows may be considered compressible on the rigid surface, where the normal velocity vanishes. It is the main reason for the formation of a singularity for the gradient of the velocity component parallel to rigid border. Slipping flows are studied analytically in the framework of two- and three-dimensional inviscid Prandtl equations. Criteria for a gradient catastrophe are found in both cases. For 2D Prandtl equations breaking takes place both for the parallel velocity along the boundary and for the vorticity gradient. For three-dimensional Prandtl flows, breaking, i.e. the formation of a fold in a finite time, occurs for the symmetric part of the velocity gradient tensor, as well as for the antisymmetric part — vorticity. The problem of the formation of velocity gradients for flows between two parallel plates is studied numerically in the framework of two-dimensional Euler equations. It is shown that the maximum velocity gradient grows exponentially with time on a rigid boundary with a simultaneous increase in the vorticity gradient according to a double exponential law. Careful analysis shows that this process is nothing more than the folding, with a power-law relationship between the maximum velocity gradient and its width: max|ux|∝ℓ−2/3.

A Numerical Algorithm for Solving the Prandtl Equations with Induced Pressure in the Periodic Case

A Numerical Algorithm for Solving the Prandtl Equations with Induced Pressure in the Periodic Case

On the Ill-Posedness of the Triple Deck Model

We analyze the stability properties of the so-called triple deck model, a classical refinement of the Prandtl equation to describe boundary layer separation. Combining the methodology introduced in [A.-L. Dalibard et al., SIAM J. Math. Anal., 50 (2018), pp. 4203--4245], based on complex analysis tools, and stability estimates inspired from Dietert and Gérard-Varet [Anal. PDE, 5 (2019), 8], we exhibit unstable linearizations of the triple deck equation. The growth rates of the corresponding unstable eigenmodes scale linearly with the tangential frequency. This shows that the recent result of Iyer and Vicol [Comm. Pure Appl. Math., 74 (2021), pp. 1641--1684] of local well-posedness for analytic data is essentially optimal.

INVESTIGATION OF A TWO-DIMENSIONAL UNSTEADY LAMINAR BOUNDARY LAYER OF AN INCOMPRESSIBLE FLUID FLOW

A system of Navier-Stokes equations for two-dimensional unsteady flows of a viscous incompressible fluid in a boundary layer is considered. The physical quantities of the longitudinal and transverse components of velocity, pressure are decomposed into an asymptotic series by degrees of the small parameter ε. The estimation of the individual terms of the equations from the point of order of their magnitude has been preliminarily performed. Output to a recurrent system of partial differential equations of the second order with respect to the first approximation of physical quantities, a system of Prandtl equations is obtained, which is the main one in the study of incompressible fluid flows in the boundary layer, the following approximations are linear partial differential equations of the second order with variable coefficients. Precisely asymptotic solutions of the equation of hydrodynamic boundary layer on a flat plate have been found. Integrate a nonlinear ordinary third-order differential equation with certain current functions distributed by current speeds. The conditions for the separation of the boundary layer and the friction resistance have been established.