Uniform Regularity Research Articles

On a supersonic-sonic patch arising from the two-dimensional Riemann problem of the compressible Euler equations

We are interested in the two-dimensional four-constant Riemann problem to the isentropic compressible Euler equations. In terms of the self-similar variables, the governing system is of nonlinear mixed-type and the solution configuration typically contains transonic and small-scale structures. We construct a supersonic-sonic patch along a pseudo-streamline from the supersonic part to a sonic point. This kind of patch appears frequently in the two-dimensional Riemann problem and is a building block for constructing a global solution. To overcome the difficulty caused by the sonic degeneracy, we apply the characteristic decomposition technique to handle the problem in a partial hodograph plane. We establish a regular supersonic solution for the original problem by showing the global one-to-one property of the partial hodograph transformation. The uniform regularity of the solution and the regularity of an associated sonic curve are also discussed.

Schauder estimates for parabolic equations with degenerate or singular weights

We establish some C0,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{0,\\alpha }$$\\end{document} and C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\alpha }$$\\end{document} regularity estimates for a class of weighted parabolic problems in divergence form. The main novelty is that the weights may vanish or explode on a characteristic hyperplane Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document} as a power a>-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a > -1$$\\end{document} of the distance to Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document}. The estimates we obtain are sharp with respect to the assumptions on coefficients and data. Our methods rely on a regularization of the equation and some uniform regularity estimates combined with a Liouville theorem and an approximation argument. As a corollary of our main result, we obtain similar C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\alpha }$$\\end{document} estimates when the degeneracy/singularity of the weight occurs on a regular hypersurface of cylindrical type.

A linearly implicit finite element full-discretization scheme for SPDEs with nonglobally Lipschitz coefficients

Abstract The present article deals with strong approximations of additive noise driven stochastic partial differential equations (SPDEs) with nonglobally Lipschitz nonlinearity in a bounded domain $ \mathcal{D} \in{\mathbb{R}}^{d}$, $ d \leq 3$. As the first contribution, we establish the well-posedness and regularity of the considered SPDEs in space dimension $d \le 3$, under more relaxed assumptions on the stochastic convolution. This improves relevant results in the literature and covers both the space-time white noise ($d=1$) and the trace-class noises ($\text{Tr} (Q) < \infty $) in multiple dimensions $d=2,3$. Such an improvement is achieved based on a key perturbation estimate for a perturbed PDE, with the aid of which we prove the convergence and uniform regularity of a spectral approximation of the SPDEs and thus get the improved regularity results. The second contribution of the paper is to propose and analyze a spatio-temporal discretization of the SPDEs, by incorporating a standard finite element method in space and a linearly implicit nonlinearity-tamed Euler method for the temporal discretization. The proposed time-stepping scheme is linearly implicit and does not suffer from solving nonlinear algebra equations as the backward Euler scheme does. Based on the improved regularity results, we recover the expected strong convergence rates of the fully discrete scheme and reveal how the convergence rates rely on the regularity of the noise process. In particular, a classical convergence rate of order $O(h^{2} +\tau )$ can be obtained even in high dimension $d=3$, as the driven noise is of trace class and satisfies certain regularity assumptions. The optimal error estimates turn out to be challenging and face some essential difficulties when the tamed time-stepping scheme meets the finite element spatial discretization, particularly in the context of low regularity and multiple dimensions $d \le 3$. Some highly nontrivial arguments are introduced to overcome the difficulties. Finally, numerical examples corroborate the claimed strong orders of convergence.

Uniform regularity for an Ericksen–Leslie’s parabolic–hyperbolic liquid crystals model

In this work, we prove the uniform regularity of smooth solutions to an Ericksen–Leslie’s parabolic–hyperbolic liquid crystals model in a three-dimensional torus.

The Cauchy problem for an inviscid and non-diffusive Oldroyd-B model in two dimensions

A two-dimensional inviscid and diffusive Oldroyd-B model was investigated by Elgindi and Rousset (2015) where the global existence and uniqueness of the strong solution were established for arbitrarily large initial data. As pointed out by Bhave et al. (1991), since the diffusion coefficient is significantly smaller than other effects, it is interesting to study the non-diffusive model. In the present work, we obtain the global-in-time existence and uniqueness of the strong solution to the non-diffusive model with small initial data by deriving some uniform regularity estimates and taking vanishing diffusion limits. In addition, the large time behavior of the solution is studied and the optimal time-decay rates for each order of spatial derivatives are obtained. The main challenges focus on the lack of dissipation and regularity effects of the system and on the slower decay in the two-dimensional settings. A combination of the spectral analysis and the Fourier splitting method is adopted.

Uniform regularity and zero capillarity-viscosity limit for an inhomogeneous incompressible fluid model of Korteweg type in half-space

In this paper, we study the uniform regularity and zero capillarity-viscosity limit for an inhomogeneous incompressible fluid model of Korteweg type in the half-space R+3 . We consider the Navier-slip boundary condition for velocity and the Dirichlet boundary condition for the gradient of density. By establishing the conormal energy estimates, we prove that there exists a unique strong solution of the model in a finite time interval [0,T0] , where T 0 is independent of the capillary and viscosity coefficients, and the solution is uniformly bounded in a conormal Sobolev space. Based on the aforementioned uniform estimates, we further show that there exists a constant 0<T1⩽T0 , such that the solutions of this model converge to the solution of the inhomogeneous incompressible Euler equations with the rates of convergence in L∞(0,T1;L2(R+3)) and L∞(0,T1;H1(R+3)) , as the capillary and viscosity coefficients tend to zero simultaneously.

Uniform Regularity Estimates for Three Dimensional Nonhomogeneous Incompressible Heat Conducting Navier–Stokes Flows

Uniform Regularity Estimates for Three Dimensional Nonhomogeneous Incompressible Heat Conducting Navier–Stokes Flows

Uniformly continuous extension in L-uniform convergence tower spaces

Using an extension theorem for continuous mappings between quantale-valued convergence tower spaces that we obtained in a previous paper, we prove an extension theorem for uniformly continuous mappings between quantale-valued uniform convergence spaces. To this end, we define and study suitable uniform diagonal axioms and uniform regularity for quantale-valued uniform convergence tower spaces.

Uniqueness and non-degeneracy of ground states for Choquard equations with fractional Laplacian

In this paper, we study the following Choquard equations with fractional Laplacian{(−Δ)su+u=(Iα⁎|u|p)|u|p−2uinRN,lim|x|→∞⁡u(x)=0,u∈Hs(RN), where (−Δ)s is the fractional Laplacian, Iα is the Riesz potential, s∈(0,1), 2s<N∈N, α∈(0,N) and p∈(N+αN,N+αN−2s). Via studying limiting profiles of ground states of the above problem, we establish the uniqueness and non-degeneracy of positive ground states as α is close to 0 and α is close to N respectively. As a by-product, some uniform regularity and decay estimates for the solutions to the fractional Choquard equation, which are also of interest and importance independently, are given by taking full advantage of the Bessel kernel and employing an iterative process.

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

Incompressible limit for the free surface Navier-Stokes system

We establish uniform regularity estimates with respect to the Mach number for the three-dimensional free surface compressible Navier-Stokes system in the case of slightly well-prepared initial data in the sense that the acoustic components like the divergence of the velocity field are of size $$\sqrt{\varepsilon }$$ , $$\varepsilon $$ being the Mach number. These estimates allow us to justify the convergence towards the free surface incompressible Navier-Stokes system in the low Mach number limit. One of the main difficulties is the control of the regularity of the surface in presence of boundary layers with fast oscillations.

Existence and regularity of solutions of a supersonic-sonic patch arising in axisymmetric relativistic transonic flow with general equation of state

In this article, we prove the existence and regularity of a smooth solution for a supersonic-sonic patch arising in a modified Frankl problem in the study of three-dimensional axisymmetric steady isentropic relativistic transonic flows over a symmetric airfoil. We consider a general convex equation of state which makes this problem complicated as well as interesting in the context of the general theory for transonic flows. Such type of patches appear in many transonic flows over an airfoil and flow near the nozzle throat. Here the main difficulty is the coupling of nonhomogeneous terms due to axisymmetry and the sonic degeneracy for the relativistic flow. However, using the well-received characteristic decompositions of angle variables and a partial hodograph transformation we prove the existence and regularity of solution in the partial hodograph plane first. Further, by using an inverse transformation we construct a smooth solution in the physical plane and discuss the uniform regularity of solution up to the associated sonic curve. Finally, we also discuss the uniform regularity of the sonic curve.

Inviscid limit of compressible viscoelastic equations with the no-slip boundary condition

The inviscid limit for the two-dimensional compressible viscoelastic equations in the half plane is considered under the no-slip boundary condition. When the initial deformation tensor is a perturbation of the identity matrix and the initial density is near a positive constant, we establish the uniform estimates of solutions to the compressible viscoelastic flows in the conormal Sobolev spaces. It is well-known that for the corresponding inviscid limit of the compressible Navier-Stokes equations with the no-slip boundary condition, one does not expect the uniform energy estimates of solutions due to the appearance of strong boundary layers. However, when the deformation tensor effect is taken into account, our results show that the deformation tensor plays an important role in the vanishing viscosity process and can surprisingly prevent the formation of strong boundary layers. As a result we are able to justify the inviscid limit of solutions for the compressible viscous flows under the no-slip boundary condition governed by the viscoelastic equations, based on the uniform conormal regularity estimates achieved in this paper.

Uniform regularity for the nonisentropic MHD system

In this work, we prove the uniform regularity of smooth solutions to the full compressible MHD system in \(\mathbb{T}^3\). Here our result is obtained by using the bilinear commutator and product estimates.

Uniform regularity estimates and inviscid limit for the compressible non-resistive magnetohydrodynamics system

We are concerned with the uniform regularity estimates of solutions to the two dimensional compressible non-resistive magnetohydrodynamics (MHD) equations with the no-slip boundary condition on velocity in the half plane. Under the assumption that the initial magnetic field is transverse to the boundary, the uniform conormal energy estimates are established for the solutions to compressible MHD equations with respect to the small viscosity coefficient. As a direct consequence, we proved the inviscid limit of solutions from viscous MHD systems to the ideal MHD systems in sense by some compact arguments. Our results show that the transverse magnetic field near the boundary can prevent the strong boundary layers from occurring.

Uniform regularity for a two-phase model with magnetic field

Uniform regularity for a two-phase model with magnetic field

Vanishing viscosity limits for the free boundary problem of compressible viscoelastic fluids with surface tension

We consider the free boundary problem of compressible isentropic neo-Hookean viscoelastic fluid equations with surface tension. Under the physical kinetic and dynamic conditions proposed on the free boundary, we investigate the regularity of classical solutions to viscoelastic fluid equations in Sobolev spaces which are uniform in viscosity and justify the corresponding vanishing viscosity limits. The key ingredient of our proof is that the deformation gradient tensor in Lagrangian coordinates can be represented as a parameter in terms of the flow map so that the inherent structure of the elastic term improves the uniform regularity of normal derivatives in the limit of vanishing viscosity. This result indicates that the boundary layer does not appear in the free boundary problem of compressible viscoelastic fluids, which is different from the case studied by Mei et al. (2018) for the free boundary compressible Navier-Stokes system.

Uniform regularity for the flow of a chemically reacting gaseous mixture

Uniform regularity plays an important role in the global existence of strong solutions and large time behavior of global solutions. In this work, we prove the uniform regularity of smooth solutions to the compressible flow of a chemically reacting gaseous mixture in \mathbb{T}^3 .

Uniform regularity and vanishing viscosity limit for the incompressible non-resistive magneto-micropolar equations

In this paper, we consider the vanishing viscosity limit for the incompressible non-resistive magneto-micropolar equations on the half-space with no-slip boundary condition (3). We prove that the vanishing viscosity limit is uniform over a time interval, which indicates that the incompressible non-resistive magneto-micropolar equations with the no-slip boundary condition have a strong solution and the solution is uniformly bounded in both the conormal Sobolev norm and norm. As a direct result, we obtain the vanishing viscosity limit for the incompressible non-resistive magneto-micropolar equations by a strong compactness argument.

Uniform regularity for a density-dependent incompressible Hall-MHD system

This paper proves uniform regularity for a density-dependent incompressible Hall-MHD system with positive density.