The Experts below are selected from a list of 174 Experts worldwide ranked by ideXlab platform
Zhifei Zhang - One of the best experts on this subject based on the ideXlab platform.
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global c regularity of the steady Prandtl Equation with favorable pressure gradient
Annales de l'Institut Henri Poincaré C Analyse non linéaire, 2021Co-Authors: Yue Wang, Zhifei ZhangAbstract:Abstract In the case of favorable pressure gradient, Oleinik obtained the global-in-x solutions to the steady Prandtl Equations with low regularity (see Oleinik and Samokhin [9] , P.21, Theorem 2.1.1). Due to the degeneracy of the Equation near the boundary, the question of higher regularity of Oleinik's solutions remains open. See the local-in-x higher regularity established by Guo and Iyer [5] . In this paper, we prove that Oleinik's solutions are smooth up to the boundary y = 0 for any x > 0 , using further maximum principle techniques. Moreover, since Oleinik only assumed low regularity on the data prescribed at x = 0 , our result implies instant smoothness (in the steady case, x = 0 is often considered as initial time).
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Global C∞ regularity of the steady Prandtl Equation with favorable pressure gradient
Annales de l'Institut Henri Poincaré C Analyse non linéaire, 2021Co-Authors: Yue Wang, Zhifei ZhangAbstract:Abstract In the case of favorable pressure gradient, Oleinik obtained the global-in-x solutions to the steady Prandtl Equations with low regularity (see Oleinik and Samokhin [9] , P.21, Theorem 2.1.1). Due to the degeneracy of the Equation near the boundary, the question of higher regularity of Oleinik's solutions remains open. See the local-in-x higher regularity established by Guo and Iyer [5] . In this paper, we prove that Oleinik's solutions are smooth up to the boundary y = 0 for any x > 0 , using further maximum principle techniques. Moreover, since Oleinik only assumed low regularity on the data prescribed at x = 0 , our result implies instant smoothness (in the steady case, x = 0 is often considered as initial time).
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Blow-up criterion for the 2-D Prandtl Equation
arXiv: Analysis of PDEs, 2020Co-Authors: Yue Wang, Zhifei ZhangAbstract:In this paper, we consider the 2-D Prandtl Equation with constant outer flow and monotonic data. We prove that if the curvature of the velocity distribution(i.e., $\partial_y^2u$) is bounded near the boundary, then the solution can not develop the singularity.
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Global $C^\infty$ Regularity of the Steady Prandtl Equation
arXiv: Analysis of PDEs, 2019Co-Authors: Yue Wang, Zhifei ZhangAbstract:In this paper, we prove the global $C^\infty$ regularity of the Oleinik's solution for the steady Prandtl Equation with favorable pressure.
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global c infty regularity of the steady Prandtl Equation
arXiv: Analysis of PDEs, 2019Co-Authors: Yue Wang, Zhifei ZhangAbstract:In this paper, we prove the global $C^\infty$ regularity of the Oleinik's solution for the steady Prandtl Equation with favorable pressure.
Anne-laure Dalibard - One of the best experts on this subject based on the ideXlab platform.
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Separation for the stationary Prandtl Equation
Publications mathématiques de l'IHÉS, 2019Co-Authors: Anne-laure Dalibard, Nader MasmoudiAbstract:In this paper, we prove that separation occurs for the stationary Prandtl Equation, in the case of adverse pressure gradient, for a large class of boundary data at x = 0 $x=0$ . We justify the Goldstein singularity: more precisely, we prove that under suitable assumptions on the boundary data at x = 0 $x=0$ , there exists x ∗ > 0 $x^{*}>0$ such that ∂ y u | y = 0 ( x ) ∼ C x ∗ − x $\partial _{y} u_{|y=0}(x) \sim C \sqrt{x^{*} -x}$ as x → x ∗ $x\to x^{*}$ for some positive constant C $C$ , where u $u$ is the solution of the stationary Prandtl Equation in the domain { 0 < x < x ∗ , y > 0 } $\{0< x< x^{*},\ y> 0\}$ . Our proof relies on three main ingredients: the computation of a “stable” approximate solution, using modulation theory arguments; a new formulation of the Prandtl Equation, for which we derive energy estimates, relying heavily on the structure of the Equation; and maximum principle and comparison principle techniques to handle some of the nonlinear terms.
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Separation for the stationary Prandtl Equation
2018Co-Authors: Anne-laure Dalibard, Nader MasmoudiAbstract:In this paper, we prove that separation occurs for the stationary Prandtl Equation, in the case of adverse pressure gradient, for a large class of boundary data at $x=0$. We justify the Goldstein singularity: more precisely, we prove that under suitable assumptions on the boundary data at $x=0$, there exists $x^*>0$ such that $\p_y u_{y=0}(x)\sim C \sqrt{x^* -x}$ as $x\to x^*$ for some positive constant $C$, where $u$ is the solution of the stationary Prandtl Equation in the domain $\{0
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Separation for the stationary Prandtl Equation
arXiv: Analysis of PDEs, 2018Co-Authors: Anne-laure Dalibard, Nader MasmoudiAbstract:In this paper, we prove that separation occurs for the stationary Prandtl Equation, in the case of adverse pressure gradient, for a large class of boundary data at $x=0$.We justify the Goldstein singularity: more precisely, we prove that under suitable assumptions on the boundary data at $x=0$, there exists $x^*>0$ such that $\p\_y u\_{y=0}(x)\sim C \sqrt{x^* -x}$ as $x\to x^*$ for some positive constant $C$, where $u$ is the solution of the stationary Prandtl Equation in the domain $\{0 0\}$. Our proof relies on three main ingredients: the computation of a "stable" approximate solution, using modulation theory arguments, a new formulation of the Prandtl Equation, for which we derive energy estimates, relying heavily on the structure of the Equation, and maximum principle techniques to handle nonlinear terms.
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High frequency analysis of the unsteady Interactive Boundary Layer model
SIAM Journal on Mathematical Analysis, 2018Co-Authors: Anne-laure Dalibard, Helge Dietert, David Gérard-varet, Frédéric MarbachAbstract:The present paper is about a famous extension of the Prandtl Equation, the so-called Interactive Boundary Layer model (IBL). This model has been used intensively in the numerics of steady boundary layer flows, and compares favorably to the Prandtl one, especially past separation. We consider here the unsteady version of the IBL, and study its linear well-posedness, namely the linear stability of shear flow solutions to high frequency perturbations. We show that the IBL model exhibits strong unrealistic instabilities, that are in particular distinct from the Tollmien-Schlichting waves. We also exhibit similar instabilities for a Prescribed Displacement Thickness model (PDT), which is one of the building blocks of numerical implementations of the IBL model.
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An existence result for the steady rotating Prandtl Equation
arXiv: Analysis of PDEs, 2016Co-Authors: Anne-laure Dalibard, Matthew PaddickAbstract:We consider a steady, geophysical 2D fluid in a domain, and focus on its western boundary layer, which is formally governed by a variant of the Prandtl Equation. By using the von Mises change of variables, we show that this Equation is well-posed under the assumption that the trace of the interior stream function has large variations, and that the variations in the coastline profile are moderate.
Nader Masmoudi - One of the best experts on this subject based on the ideXlab platform.
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Separation for the stationary Prandtl Equation
Publications mathématiques de l'IHÉS, 2019Co-Authors: Anne-laure Dalibard, Nader MasmoudiAbstract:In this paper, we prove that separation occurs for the stationary Prandtl Equation, in the case of adverse pressure gradient, for a large class of boundary data at x = 0 $x=0$ . We justify the Goldstein singularity: more precisely, we prove that under suitable assumptions on the boundary data at x = 0 $x=0$ , there exists x ∗ > 0 $x^{*}>0$ such that ∂ y u | y = 0 ( x ) ∼ C x ∗ − x $\partial _{y} u_{|y=0}(x) \sim C \sqrt{x^{*} -x}$ as x → x ∗ $x\to x^{*}$ for some positive constant C $C$ , where u $u$ is the solution of the stationary Prandtl Equation in the domain { 0 < x < x ∗ , y > 0 } $\{0< x< x^{*},\ y> 0\}$ . Our proof relies on three main ingredients: the computation of a “stable” approximate solution, using modulation theory arguments; a new formulation of the Prandtl Equation, for which we derive energy estimates, relying heavily on the structure of the Equation; and maximum principle and comparison principle techniques to handle some of the nonlinear terms.
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Separation for the stationary Prandtl Equation
arXiv: Analysis of PDEs, 2018Co-Authors: Anne-laure Dalibard, Nader MasmoudiAbstract:In this paper, we prove that separation occurs for the stationary Prandtl Equation, in the case of adverse pressure gradient, for a large class of boundary data at $x=0$.We justify the Goldstein singularity: more precisely, we prove that under suitable assumptions on the boundary data at $x=0$, there exists $x^*>0$ such that $\p\_y u\_{y=0}(x)\sim C \sqrt{x^* -x}$ as $x\to x^*$ for some positive constant $C$, where $u$ is the solution of the stationary Prandtl Equation in the domain $\{0 0\}$. Our proof relies on three main ingredients: the computation of a "stable" approximate solution, using modulation theory arguments, a new formulation of the Prandtl Equation, for which we derive energy estimates, relying heavily on the structure of the Equation, and maximum principle techniques to handle nonlinear terms.
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Separation for the stationary Prandtl Equation
2018Co-Authors: Anne-laure Dalibard, Nader MasmoudiAbstract:In this paper, we prove that separation occurs for the stationary Prandtl Equation, in the case of adverse pressure gradient, for a large class of boundary data at $x=0$. We justify the Goldstein singularity: more precisely, we prove that under suitable assumptions on the boundary data at $x=0$, there exists $x^*>0$ such that $\p_y u_{y=0}(x)\sim C \sqrt{x^* -x}$ as $x\to x^*$ for some positive constant $C$, where $u$ is the solution of the stationary Prandtl Equation in the domain $\{0
Toan T. Nguyen - One of the best experts on this subject based on the ideXlab platform.
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A note on Prandtl boundary layers
Communications on Pure and Applied Mathematics, 2011Co-Authors: Yan Guo, Toan T. NguyenAbstract:This note concerns nonlinear ill-posedness of the Prandtl Equation and an invalidity of asymptotic boundary layer expansions of incompressible fluid flows near a solid boundary. Our analysis is built upon recent remarkable linear illposedness results established by Gerard-Varet and Dormy and an analysis by Guo and Tice. We show that the asymptotic boundary layer expansion is not valid for nonmonotonic shear layer flows in Sobolev spaces. We also introduce a notion of weak well-posedness and prove that the nonlinear Prandtl Equation is not well-posed in this sense near nonstationary and nonmonotonic shear flows. On the other hand, we are able to verify that Oleinik's monotonic solutions are well-posed. © 2011 Wiley Periodicals, Inc.
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A note on the Prandtl layers
2010Co-Authors: Yan Guo, Toan T. NguyenAbstract:This note concerns a nonlinear ill-posedness of the Prandtl Equation and an invalidity of asymptotic boundary-layer expansions of incompressible fluid flows near a solid boundary. Our analysis is built upon recent remarkable linear ill-posedness results established by Gerard-Varet and Dormy [2], and an analysis in Guo and Tice [5]. We show that the asymptotic boundary-layer expansion is not valid for non-monotonic shear layer flows in Sobolev spaces. We also introduce a notion of weak well-posedness and prove that the nonlinear Prandtl Equation is not well-posed in this sense near non-stationary and non-monotonic shear flows. On the other hand, we are able to verify that Oleinik's monotonic solutions are well-posed.
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A note on the Prandtl boundary layers
arXiv: Analysis of PDEs, 2010Co-Authors: Yan Guo, Toan T. NguyenAbstract:This note concerns a nonlinear ill-posedness of the Prandtl Equation and an invalidity of asymptotic boundary-layer expansions of incompressible fluid flows near a solid boundary. Our analysis is built upon recent remarkable linear ill-posedness results established by G\'erard-Varet and Dormy [2], and an analysis in Guo and Tice [5]. We show that the asymptotic boundary-layer expansion is not valid for non-monotonic shear layer flows in Sobolev spaces. We also introduce a notion of weak well-posedness and prove that the nonlinear Prandtl Equation is not well-posed in this sense near non-stationary and non-monotonic shear flows. On the other hand, we are able to verify that Oleinik's monotonic solutions are well-posed.
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Remarks on the ill-posedness of the Prandtl Equation
arXiv: Analysis of PDEs, 2010Co-Authors: David Gérard-varet, Toan T. NguyenAbstract:In the lines of a recent paper by Gerard-Varet and Dormy, we establish various ill-posedness results for the Prandtl Equation. By considering perturbations of stationary shear flows, we show that for some linearizations of the Prandtl Equation and some $C^\infty$ initial data, local in time $C^\infty$ solutions do not exist. At the nonlinear level, we prove that if a flow exists in the Sobolev setting, it cannot be Lipschitz continuous. Besides ill-posedness in time, we also establish some ill-posedness in space, that casts some light on the results obtained by Oleinik for monotonic data.
Yue Wang - One of the best experts on this subject based on the ideXlab platform.
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global c regularity of the steady Prandtl Equation with favorable pressure gradient
Annales de l'Institut Henri Poincaré C Analyse non linéaire, 2021Co-Authors: Yue Wang, Zhifei ZhangAbstract:Abstract In the case of favorable pressure gradient, Oleinik obtained the global-in-x solutions to the steady Prandtl Equations with low regularity (see Oleinik and Samokhin [9] , P.21, Theorem 2.1.1). Due to the degeneracy of the Equation near the boundary, the question of higher regularity of Oleinik's solutions remains open. See the local-in-x higher regularity established by Guo and Iyer [5] . In this paper, we prove that Oleinik's solutions are smooth up to the boundary y = 0 for any x > 0 , using further maximum principle techniques. Moreover, since Oleinik only assumed low regularity on the data prescribed at x = 0 , our result implies instant smoothness (in the steady case, x = 0 is often considered as initial time).
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Global C∞ regularity of the steady Prandtl Equation with favorable pressure gradient
Annales de l'Institut Henri Poincaré C Analyse non linéaire, 2021Co-Authors: Yue Wang, Zhifei ZhangAbstract:Abstract In the case of favorable pressure gradient, Oleinik obtained the global-in-x solutions to the steady Prandtl Equations with low regularity (see Oleinik and Samokhin [9] , P.21, Theorem 2.1.1). Due to the degeneracy of the Equation near the boundary, the question of higher regularity of Oleinik's solutions remains open. See the local-in-x higher regularity established by Guo and Iyer [5] . In this paper, we prove that Oleinik's solutions are smooth up to the boundary y = 0 for any x > 0 , using further maximum principle techniques. Moreover, since Oleinik only assumed low regularity on the data prescribed at x = 0 , our result implies instant smoothness (in the steady case, x = 0 is often considered as initial time).
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Blow-up criterion for the 2-D Prandtl Equation
arXiv: Analysis of PDEs, 2020Co-Authors: Yue Wang, Zhifei ZhangAbstract:In this paper, we consider the 2-D Prandtl Equation with constant outer flow and monotonic data. We prove that if the curvature of the velocity distribution(i.e., $\partial_y^2u$) is bounded near the boundary, then the solution can not develop the singularity.
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Global $C^\infty$ Regularity of the Steady Prandtl Equation
arXiv: Analysis of PDEs, 2019Co-Authors: Yue Wang, Zhifei ZhangAbstract:In this paper, we prove the global $C^\infty$ regularity of the Oleinik's solution for the steady Prandtl Equation with favorable pressure.
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global c infty regularity of the steady Prandtl Equation
arXiv: Analysis of PDEs, 2019Co-Authors: Yue Wang, Zhifei ZhangAbstract:In this paper, we prove the global $C^\infty$ regularity of the Oleinik's solution for the steady Prandtl Equation with favorable pressure.