Optimal regularity estimates for parabolic p-harmonic equations

In this paper optimal regularity estimates for weak solutions of quasilinear parabolic equations of p-Laplacian type with small BMO coefficients are investigated. Our results improve the known results for such equations using a harmonic analysis free technique.     

Lq‐estimates for the stationary Oseen equations on the exterior of a rotating obstacle

We study the Dirichlet problem for the stationary Oseen equations around a rotating body in an exterior domain. Our main results are the existence and uniqueness of weak and very weak solutions satisfying appropriate L^q‐estimates. The uniqueness of very weak solutions is shown by the method of cut‐off functions with an anisotropic decay. Then our existence result for very weak solutions is deduced by a duality argument from the existence and estimates of strong solutions. From this and interior regularity of very weak solutions, we finally establish the complete D^1,r‐result for weak solutions of the Oseen equations around a rotating body in an exterior domain, where 4/3<r <4. Here, D^1,r is the homogeneous Sobolev space.     

Second‐order elliptic and parabolic equations with BMO coefficients in the whole space

In this paper, we obtain the global regularity estimates in Orlicz spaces for second‐order divergence elliptic and parabolic equations with BMO coefficients in the whole space. In fact, the global result can follow from the local estimates. As a corollary we obtain L^p‐type regularity estimates for such equations. Copyright © 2011 John Wiley & Sons, Ltd.     

Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊2

Given a compact Riemannian manifold, we study the regularity of the optimal transport map between two probability measures with cost given by the squared Riemannian distance. Our strategy is to define a new form of the so‐called Ma‐Trudinger‐Wang condition and to show that this condition, together with the strict convexity on the nonfocal domains, implies the continuity of the optimal transport map. Moreover, our new condition, again combined with the strict convexity of the nonfocal domains, allows us to prove that all injectivity domains are strictly convex too. These results apply, for instance, on any small C⁴‐deformation of the 2‐sphere. © 2009 Wiley Periodicals, Inc.     

Optimal <Emphasis Type="Italic">W</Emphasis><Superscript>1,<Emphasis Type="Italic">p</Emphasis></Superscript> regularity theory for parabolic equations in divergence form

This work treats L^p regularity theory for weak solutions of parabolic equations in divergence form with discontinuous coefficients on nonsmooth domains. We essentially obtain an optimal condition on the coefficients under which the global W^1,p regularity theory holds. This work was supported by SNU foundation in 2005.     

Universal potential estimates

We prove a class of endpoint pointwise estimates for solutions to quasilinear, possibly degenerate elliptic equations in terms of linear and nonlinear potentials of Wolff type of the source term. Such estimates allow to bound size and oscillations of solutions and their gradients pointwise, and entail in a unified approach virtually all kinds of regularity properties in terms of the given datum and regularity of coefficients. In particular, local estimates in Hölder, Lipschitz, Morrey and fractional spaces, as well as Calderón–Zygmund estimates, follow as a corollary in a unified way. Moreover, estimates for fractional derivatives of solutions by mean of suitable linear and nonlinear potentials are also implied. The classical Wolff potential estimate by Kilpeläinen & Malý and Trudinger & Wang as well as recent Wolff gradient bounds for solutions to quasilinear equations embed in such a class as endpoint cases.     

Stokes‐Fourier and acoustic limits for the Boltzmann equation: Convergence proofs

We establish a Stokes‐Fourier limit for the Boltzmann equation considered over any periodic spatial domain of dimension two or more. Appropriately scaled families of DiPerna‐Lions renormalized solutions are shown to have fluctuations that globally in time converge weakly to a unique limit governed by a solution of Stokes‐Fourier motion and heat equations provided that the fluid moments of their initial fluctuations converge to appropriate L² initial data of the Stokes‐Fourier equations. Both the motion and heat equations are both recovered in the limit by controlling the fluxes and the local conservation defects of the DiPerna‐Lions solutions with dissipation rate estimates. The scaling of the fluctuations with respect to Knudsen number is essentially optimal. The assumptions on the collision kernel are little more than those required for the DiPerna‐Lions theory and that the viscosity and heat conduction are finite. For the acoustic limit, these techniques also remove restrictions to bounded collision kernels and improve the scaling of the fluctuations. Both weak limits become strong when the initial fluctuations converge entropically to appropriate L² initial data. © 2001 John Wiley & Sons, Inc.     

Global existence of weak‐type solutions for models of monotone type in the theory of inelastic deformations

This article introduces the notion of weak‐type solutions for systems of equations from the theory of inelastic deformations, assuming that the considered model is of monotone type (for the definition see [Lecture Notes in Mathematics, 1998, vol. 1682]). For the boundary data associated with the initial‐boundary value problem and satisfying the safe‐load condition the existence of global in time weak‐type solutions is proved assuming that the monotone model is rate‐independent or of gradient type. Moreover, for models possessing an additional regularity property (see Section 5) the existence of global solutions in the sense of measures, defined by Temam in Archives for Rational Mechanics and Analysis, 95 : 137, is obtained, too. Copyright © 2002 John Wiley & Sons, Ltd.     

Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes

A new weak Galerkin (WG) finite element method is introduced and analyzed in this article for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter‐free. Optimal order error estimates in a discrete H² norm is established for the corresponding WG finite element solutions. Error estimates in the usual L² norm are also derived, yielding a suboptimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1003–1029, 2014     

Global higher integrability for the parabolic equations in Reifenberg domains

In this paper we generalize global L^p‐type gradient estimates to Orlicz spaces for weak solutions of the parabolic equations with small BMO coefficients in Reifenberg flat domains (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stabilized mixed finite element methods for the Navier‐Stokes equations with damping

In this paper, the stabilized mixed finite element methods are presented for the Navier‐Stokes equations with damping. The existence and uniqueness of the weak solutions are proven by use of the Brouwer fixed‐point theorem. Then, optimal error estimates for the H¹‐norm and L²‐norm of the velocity and the L²‐norm of the pressure are derived. Moreover, on the basis of the optimal L²‐norm error estimate of the velocity, a stabilized two‐step method is proposed, which is more efficient than the usual stabilized methods. Finally, two numerical examples are implemented to confirm the theoretical analysis.     

Nonlinear Fourier Transforms and the mKdV Equation in the Quarter Plane

The unified transform method introduced by Fokas can be used to analyze initial‐boundary value problems for integrable evolution equations. The method involves several steps, including the definition of spectral functions via nonlinear Fourier transforms and the formulation of a Riemann‐Hilbert problem. We provide a rigorous implementation of these steps in the case of the mKdV equation in the quarter plane under limited regularity and decay assumptions. We give detailed estimates for the relevant nonlinear Fourier transforms. Using the theory of L²‐RH problems, we consider the construction of quarter plane solutions which are C¹ in time and C³ in space.     

Sharp regularity results on second derivatives of solutions to the Monge‐Ampère equation with VMO type data

We establish interior estimates for L^p‐norms, Orlicz norms, and mean oscillation of second derivatives of solutions to the Monge‐Ampère equation det D²u = f(x) with zero boundary value, where f(x) is strictly positive, bounded, and satisfies a VMO‐type condition. These estimates develop the regularity theory of the Monge‐Ampère equation in VMO‐type spaces. Our Orlicz estimates also sharpen Caffarelli's celebrated W^{2, p}‐estimates. © 2008 Wiley Periodicals, Inc.     

On the Dirichlet problem for Monge-Ampère type equations

In this paper, we prove second derivative estimates together with classical solvability for the Dirichlet problem of certain Monge-Ampére type equations under sharp hypotheses. In particular we assume that the matrix function in the augmented Hessian is regular in the sense used by Trudinger and Wang in Ann. Scoula Norm. Sup. Pisa Cl. Sci. VIII, 143–174 2009 in their study of global regularity in optimal transportation as well as the existence of a smooth subsolution. The latter hypothesis replaces a barrier condition also used in their work. The applications to optimal transportation and prescribed Jacobian equations are also indicated.     

Decay property of Timoshenko system in thermoelasticity

We investigate the decay property of a Timoshenko system of thermoelasticity in the whole space for both Fourier and Cattaneo laws of heat conduction. We point out that although the paradox of infinite propagation speed inherent in the Fourier law is removed by changing to the Cattaneo law, the latter always leads to a solution with the decay property of the regularity‐loss type. The main tool used to prove our results is the energy method in the Fourier space together with some integral estimates. We derive L² decay estimates of solutions and observe that for the Fourier law the decay structure of solutions is of the regularity‐loss type if the wave speeds of the first and the second equations in the system are different. For the Cattaneo law, decay property of the regularity‐loss type occurs no matter what the wave speeds are. In addition, by restricting the initial data to with a suitably large s and γ ∈ [0,1], we can derive faster decay estimates with the decay rate improvement by a factor of t^?γ/2. Copyright © 2011 John Wiley & Sons, Ltd.     

Convexity of solutions and estimates for fully nonlinear elliptic equations

The starting point of this work is a paper by Alvarez, Lasry and Lions (1997) concerning the convexity and the partial convexity of solutions of fully nonlinear degenerate elliptic equations. We extend their results in two directions. First, we deal with possibly sublinear (but epi-pointed) solutions instead of 1-coercive ones; secondly, the partial convexity of C² solutions is extended to the class of continuous viscosity solutions. A third contribution of this paper concerns C^1,1 estimates for convex viscosity solutions of strictly elliptic nonlinear equations. To finish with, all the tools and techniques introduced here permit us to give a new proof of the Alexandroff estimate obtained by Trudinger (1988) and Caffarelli (1989).     

Low‐curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes

We consider a class of fourth‐order nonlinear diffusion equations motivated by Tumblin and Turk's “low‐curvature image simplifiers” for image denoising and segmentation. The PDE for the image intensity u is of the form where g(s) = k²/(k² + s²) is a “curvature” threshold and λ denotes a fidelity‐matching parameter. We derive a priori bounds for Δu that allow us to prove global regularity of smooth solutions in one space dimension, and a geometric constraint for finite‐time singularities from smooth initial data in two space dimensions. This is in sharp contrast to the second‐order Perona‐Malik equation (an ill‐posed problem), on which the original LCIS method is modeled. The estimates also allow us to design a finite difference scheme that satisfies discrete versions of the estimates, in particular, a priori bounds on the smoothness estimator in both one and two space dimensions. We present computational results that show the effectiveness of such algorithms. Our results are connected to recent results for fourth‐order lubrication‐type equations and the design of positivity‐preserving schemes for such equations. This connection also has relevance for other related fourth‐order imaging equations. © 2004 Wiley Periodicals, Inc.     

On the regularity of flows with Ladyzhenskaya Shear‐dependent viscosity and slip or non‐slip boundary conditions

Navier‐Stokes equations with shear dependent viscosity under the classical non‐slip boundary condition have been introduced and studied, in the sixties, by O. A. Ladyzhenskaya and, in the case of gradient dependent viscosity, by J.‐L. Lions. A particular case is the well known Smagorinsky turbulence model. This is nowadays a central subject of investigation. On the other hand, boundary conditions of slip type seems to be more realistic in some situations, in particular in numerical applications. They are a main research subject. The existence of weak solutions u to the above problems, with slip (or non‐slip) type boundary conditions, is well known in many cases. However, regularity up to the boundary still presents many open questions. In what follows we present some regularity results, in the stationary case, for weak solutions to this kind of problems; see Theorems 3.1 and 3.2. The evolution problem is studied in the forthcoming paper [6]; see the remark at the end of the introduction. © 2004 Wiley Periodicals, Inc.     

Vanishing viscosity limit of the Navier‐Stokes equations to the euler equations for compressible fluid flow

We establish the vanishing viscosity limit of the Navier‐Stokes equations to the isentropic Euler equations for one‐dimensional compressible fluid flow. For the Navier‐Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup‐norm of solutions with respect to the physical viscosity coefficient may not be directly controllable. Furthermore, convex entropy‐entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy‐type estimates with respect to the viscosity coefficient for solutions of the Navier‐Stokes equations and establish the existence of measure‐valued solutions of the isentropic Euler equations generated by the Navier‐Stokes equations. Based on the uniform energy‐type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier‐Stokes equations for weak entropy‐entropy flux pairs, generated by compactly supported C² test functions, are confined in a compact set in H^?1, which leads to the existence of measure‐valued solutions that are confined by the Tartar‐Murat commutator relation. A careful characterization of the unbounded support of the measure‐valued solution confined by the commutator relation yields the reduction of the measurevalued solution to a Dirac mass, which leads to the convergence of solutions of the Navier‐Stokes equations to a finite‐energy entropy solution of the isentropic Euler equations with finite‐energy initial data, relative to the different end‐states at infinity. © 2010 Wiley Periodicals, Inc.     

Second-Order Elliptic Equations of Nondivergence Form with Small BMO Coefficients in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper we obtain the global regularity estimates in Orlicz spaces for second-order elliptic equations of nondivergence form with small BMO coefficients in ℝ ⁿ . As a corollary we obtain L ^p -type regularity for such equations. Our results improve the known results for such problems.