arXiv subjects

Ya-Guang Wang

Publications and source records attributed to Ya-Guang Wang.

At least 19 recordsLinked to original sources

arXiv2026

Well-posedness of the two-dimensional unsteady Prandtl system in Sobolev space with degenerate critical points

This paper is devoted to the well-posedness of classical Prandtl equations in a finite order Sobolev space. For a initial data with degenerate critical points and general outflow, we obtain the local-in-time existence and uniqueness of the solution to the Prandtl equations in a Sobolev space, by introducing a new iteration scheme and linear cancelation. This result shows that Oleinik's monotonicity condition is not a necessary condition for the Prandtl equations to be well-posed in Sobolev spaces and provides evidence to demonstrate that zero shear stress does not necessarily lead to boundary layer separation in two-dimensional unsteady boundary layers.

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arXiv2026

Well-posedness of the compressible boundary layer equations with data in the Gevrey class

This paper is devoted to the study of the compressible boundary layer equations in the Gevrey-2 solution space. Compared to the classical Prandtl equation, the additional complexity arises from the strong interaction between viscous layer and thermal layer. By introducing new auxiliary functions and observing the cancellation mechanism to overcome the loss of derivatives, we show the local existence and uniqueness of the solution in the Gevrey-2 space in the tangential variable and Sobolev regularity in the normal variable by using a direct energy method.

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arXiv2026

A note for double Hölder regularity of the hydrodynamic pressure for weak solutions of Euler equations

We give an elementary proof for the interior double Hölder regularity of the hydrodynamic pressure for weak solutions of the Euler Equations in a bounded $C^2$-domain $Ω\subset \mathbb{R}^d$; $d\geq 3$. That is, for velocity $u \in C^{0,γ}(Ω;\mathbb{R}^d)$ with some $0<γ<1/2$, we show that the pressure $p \in C^{0,2γ}_{\rm int}(Ω)$. This is motivated by the studies of turbulence and anomalous dissipation in mathematical hydrodynamics and, recently, has been established in [L. De Rosa, M. Latocca, and G. Stefani, Int. Math. Res. Not. 2024.3 (2024), 2511--2560] over $C^{2,1}$-domains by means of pseudodifferential calculus. Our approach involves only standard elliptic PDE techniques, and relies on a variant of the modified pressure introduced in [C. W. Bardos, D. W. Boutros, and E. S. Titi, Hölder regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains, Arch. Rational Mech. Anal. 249 (2025), 28] and the potential estimates in [L. Silvestre, unpublished notes]. The key novel ingredient of our proof is the introduction of two cutoff functions whose localisation parameters are carefully chosen as a power of the distance to $\partialΩ$.

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arXiv2025

Well-posedness of initial boundary value problems for 2D compressible MHD equations in domains with corners

In this paper, the well-posedness is studied for the initial boundary value problem of the two-dimensional compressible ideal magnetohydrodynamic (MHD) equations in bounded perfectly conducting domains with corners. The presence of corners yields intrinsic analytic obstacles: the lack of smooth tangential vectors to the boundary prevents the use of classical anisotropic Sobolev spaces, and due to the coupling of normal derivatives near corners, one can not follow the usual way to estimate the normal derivatives of solutions from the equations. To overcome these difficulties, a new class of anisotropic Sobolev spaces $H^m_*(Ω)$ is introduced to treat corner geometries. Within this framework, the well-posedness theory is obtained for both linear and nonlinear problems of the compressible ideal MHD equations with the impermeable and perfectly conducting boundary conditions. The associated linearized problem is studied in several steps: first one deduces the existence of weak solutions by using a duality argument in high order tangential spaces, then verifies that it is indeed a strong solution by several smoothing procedures preserving traces to get a weak-strong uniqueness result, afterwards the estimates of normal derivatives are obtained by combining the structure of MHD equations with the Helmholtz-type decomposition for both of velocity and magnetic fields.

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arXiv2025

Well-posedness of the compressible boundary layer equations with analytic initial data

We study the well-posedness of the compressible boundary layer equations with data being analytic in the tangential variable of the boundary. The compressible boundary layer equations, a nonlinear coupled system of degenerate parabolic equations and an elliptic equation, describe the behavior of thermal layer and viscous layer in the small viscosity and heat conductivity limit, for the two-dimensional compressible viscous flow with heat conduction with nonslip and zero heat flux boundary conditions. We use the Littlewood-Paley theory to establish the a priori estimates for solutions of this compressible boundary layer problem, and obtain the local existence and uniqueness of the solution in the space of analytic in the tangential variable and Sobolev in the normal variable.

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arXiv2025

Almost sure existence of global weak solutions for incompressible generalized Navier-Stokes equations

In this paper we consider the initial value problem of the incompressible generalized Navier-Stokes equations in torus $\mathbb{T}^d$ with $d \geq 2$. The generalized Navier-Stokes equations is obtained by replacing the standard Laplacian in the classical Navier-Stokes equations by the fractional order Laplacian $-(-Δ)^\al$ with $\al \in \left( \frac{2}{3},1 \right]$. After an appropriate randomization on the initial data, we obtain the almost sure existence of global weak solutions for initial data being in $\Dot{H}^s(\mathbb{T}^d)$ with $s\in (1-2\al,0)$.

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arXiv2024

Local and global existence for the stochastic Prandtl equation driven by multiplicative noises in two and three dimensions

In this paper, we are concerned with the local and global existence for the stochastic Prandtl equation in two and three dimensions, which governs the velocity field inside the boundary layer that appears in the inviscid limit of the stochastic Navier-Stokes equation with non-slip boundary condition. New problem arises when establishing the well-posedness in the stochastic regime: one can never derive a pathwise control of the energy functional which is used to describe the analytic radius of the solution in the deterministic setting. To this end, we establish higher-order estimates in the conormal Sobolev spaces in order to get rid of the dependence of the analytic radius on the unknown. Three approximate schemes are constructed for solving the stochastic Prandtl equation, and a local well-posedness is obtained in a tangentially analytic and normally Sobolev-type space. Furthermore, we study the stochastic Prandtl equation driven by a tangential random diffusion, and find that the noise regularizes the equation in the sense that in both two and three dimensions, there exists a global Gevrey-2 solution with high probability and the radius growing linearly in time.

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arXiv2024

Small-time global approximate controllability for incompressible MHD with coupled Navier slip boundary conditions

We study the small-time global approximate controllability for incompressible magnetohydrodynamic (MHD) flows in smoothly bounded two- or three-dimensional domains. The controls act on arbitrary nonempty open portions of each connected boundary component, while linearly coupled Navier slip-with-friction conditions are imposed along the uncontrolled parts of the boundary. Some choices for the friction coefficients give rise to interacting velocity and magnetic field boundary layers. We obtain sufficient dissipation properties of these layers by a detailed analysis of the corresponding asymptotic expansions. For certain friction coefficients, or if the obtained controls are not compatible with the induction equation, an additional pressure-like term appears. We show that such a term does not exist for problems defined in planar simply-connected domains and various choices of Navier slip-with-friction boundary conditions.

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arXiv2023

On Kato's conditions for the inviscid limit of the two-dimensional stochastic Navier-Stokes equation

We study the asymtotic behavior of solutions to the two-dimensional stochasitc Navier-Stokes (SNS) equation in the small viscosity limit. The SNS equation is supplemented with no-slip boundary condition, in which a strong boundary layer shall appear in the limit due to the mismatch of the boundary conditions of the SNS equation and the corresponding limit problem. Several equivalent dissipation conditions are derived to ensure the convergence hold in the energy space. One novelty of this work is that we do not assume any smallness for the noise.

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arXiv2021

Global exact controllability of ideal incompressible magnetohydrodynamic flows through a planar duct

This article is concerned with the global exact controllability for ideal incompressible magnetohydrodynamics in a rectangular domain where the controls are situated in both vertical walls. First, global exact controllability via boundary controls is established for a related Elsässer type system by applying the return method, introduced in [Coron J.M., Math. Control Signals Systems, 5(3) (1992) 295--312]. Similar results are then inferred for the original magnetohydrodynamics system with the help of a special pressure-like corrector in the induction equation. Overall, the main difficulties stem from the nonlinear coupling between the fluid velocity and the magnetic field in combination with the aim of exactly controlling the system. In order to overcome some of the obstacles, we introduce ad-hoc constructions, such as suitable initial data extensions outside of the physical part of the domain and a certain weighted space.

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arXiv2020

On the interaction problem between a compressible viscous fluid and a nonlinear thermoelastic plate

In this paper we study the interaction problem between a nonlinear thermoelastic plate and a compressible viscous fluid with the adiabatic constant $γ>12/7$. The existence of a weak solution for this problem is obtained by constructing a time-continuous operator splitting scheme that decouples the fluid and the structure. The fluid sub-problem is given on a fixed reference domain in the arbitrary Lagrangian-Eulerian (ALE) formulation, and the continuity equation is damped on this domain as well. This allows the majority of the analysis to be performed on the fixed reference domain, while the convergence of the approximate pressure is obtained on the physical domain.

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arXiv2020

Remarks on Exponential Stability for a Coupled System of Elasticity and Thermoelasticity with Second Sound

We study the large time behavior of solutions to a linear transmission problem in one space dimension. The problem at hand models a thermoelastic material with second sound confined by a purely elastic one. We shall characterize all equilibrium states of the considered system and prove that every solution approaches one designated equilibrium state with an exponential rate as time goes to infinity. Hereto, we apply methods from the theory of strongly continuous semigroups. In particular, we obtain uniform resolvent bounds for the underlying generator. This removes the largeness assumption of elastic wave speeds imposed in [Y.P. Meng and Y.G. Wang, Anal. Appl. (Singap.) 13 (2015)] for having an exponential energy decay rate when the problem only has the trivial equilibrium. In an appendix we provide a similar exponential stability result for the case where heat conduction is modeled using Fourier's law.

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arXiv2019

Separation of the two-dimensional unsteady Prandtl boundary layers under an adverse pressure gradient

In this paper, we study the back flow of the two-dimensional unsteady Prandtl boundary layer under an adverse pressure gradient. The occurrence of back flow is an important physical event in the evolution of boundary layer, which eventually leads to separation. For the two-dimensional unsteady Prandtl boundary layer equations, when the initial tangential velocity is strictly monotonic with respect to the normal variable, and the pressure gradient of the outer flow is adverse, we obtain that the first critical point of the tangential velocity profile with respect to the normal variable, if exists when the boundary layer evolves in time, must appear on the boundary. Moreover, we give a condition on the growth rate of the initial tangential velocity such that there is a back flow point of the Prandtl boundary layer under the adverse pressure gradient. In the appendix, we introduce two examples showing that back flow occurs either when the flow distance is long in the streamwise direction for a given initial monotonic tangential velocity field, or when the initial tangential velocity grows slowly in a large neighborhood of the boundary for a fixed flow distance in the streamwise direction.

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arXiv2019

Existence of a weak solution to the fluid-structure interaction problem in 3D

We study a nonlinear fluid-structure interaction problem in which the fluid is described by the three-dimensional incompressible Navier-Stokes equations, and the elastic structure is modeled by the nonlinear plate equation which includes a generalization of Kirchhoff, von Kármán and Berger plate models. The fluid and the structure are fully coupled via kinematic and dynamic boundary conditions. The existence of a weak solution is obtained by designing a hybrid approximation scheme that successfully deals with the nonlinearities of the system. We combine time-discretization and operator splitting to create two sub-problems, one piece-wise stationary for the fluid and one in the Galerkin basis for the plate. To guarantee the convergence of approximate solutions to a weak solution, a sufficient condition is given on the number of time discretization sub-intervals in every step in a form of dependence with number of the Galerkin basis functions and nonlinearity order of the plate equation.

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arXiv2019

Well-posedness and blowup of the geophysical boundary layer problem

Under the assumption that the initial velocity and outflow velocity are analytic in the horizontal variable, the local well-posedness of the geophysical boundary layer problem is obtained by using energy method in the weighted Chemin-Lerner spaces. Moreover, when the initial velocity and outflow velocity satisfy certain condition on a transversal plane, it is proved that the $W^{1,\infty}-$norm of any smooth solution decaying exponentially in the normal variable to the geophysical boundary layer problem blows up in a finite time.

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arXiv2019

Blowup of solutions to the thermal boundary layer problem in two-dimensional incompressible heat conducting flow

In this paper, we study the formation of finite time singularities for the solution of the boundary layer equations in the two-dimensional incompressible heat conducting flow. We obtain that the first spacial derivative of the solution blows up in a finite time for the thermal boundary layer problem, for a kind of data which are analytic in the tangential variable but do not satisfy the Oleinik monotonicity condition, by constructing a Lyapunov functional. Moreover, it is observed that the buoyancy coming from the temperature difference in the flow may destabilize the thermal boundary layer.

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arXiv2016

Well-posedness of thermal layer equations for inviscid compressible flows

A semi-explicit formula of solution to the boundary layer system for thermal layer derived from the compressible Navier-Stokes equations with the non-slip boundary condition when the viscosity coefficients vanish is given, in particular in three space dimension. In contrast to the inviscid Prandtl system studied by [7] in two space dimension, the main difficulty comes from the coupling of the velocity field and the temperature field through a degenerate parabolic equation. The convergence of these boundary layer equations to the inviscid Prandtl system is justified when the initial temperature goes to a constant. Moreover, the time asymptotic stability of the linearized system around a shear flow is given, and in particular, it shows that in three space dimension, the asymptotic stability depends on whether the direction of tangential velocity field of the shear flow is invariant in the normal direction respective to the boundary.

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arXiv2015

Global existence of weak solutions to the three-dimensional Prandtl equations with A special structure

The global existence of weak solutions to the three space dimensional Prandtl equations is studied under some constraint on its structure. This is a continuation of our recent study on the local existence of classical solutions with the same structure condition. It reveals the sufficiency of the monotonicity condition on one component of the tangential velocity field and the favorable condition on pressure in the same direction that leads to global existence of weak solutions. This generalizes the result obtained by Xin-Zhang on the two-dimensional Prandtl equations to the three-dimensional setting.

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