Regular and irregular solutions for a class of elliptic systems in the critical dimension

We study regularity properties of weak solutions in the Sobolev space ${W^{1,n}_0}$ to inhomogeneous elliptic systems under a natural growth condition and on bounded Lipschitz domains in ${\mathbb{R}^n}$ , i. e. we investigate weak solutions in the limiting situation of the Sobolev embedding. Several counterexamples of irregular solutions are constructed in cases, where additional structure conditions might have led to regularity. Among others we present both bounded irregular and unbounded weak solutions to elliptic systems obeying a one-sided condition, and we further construct unbounded extremals of two-dimensional variational problems. These counterexamples do not exclude the existence of a regular solution. In fact, we establish the existence of regular solutions—under standard assumptions on the principal part and the aforementioned one-sided condition on the inhomogeneity. This extends previous works for n = 2 to more general cases, including arbitrary dimensions. Moreover, this result is achieved by a simplified proof invoking modern techniques.     

Long-Time Behavior and Critical Limit of Subcritical SQG Equations in Scale-Invariant Sobolev Spaces

We consider the subcritical SQG equation in its natural scale-invariant Sobolev space and prove the existence of a global attractor of optimal regularity. The proof is based on a new energy estimate in Sobolev spaces to bootstrap the regularity to the optimal level, derived by means of nonlinear lower bounds on the fractional Laplacian. This estimate appears to be new in the literature and allows a sharp use of the subcritical nature of the \(L^\infty \) bounds for this problem. As a by-product, we obtain attractors for weak solutions as well. Moreover, we study the critical limit of the attractors and prove their stability and upper semicontinuity with respect to the strength of the diffusion.     

Everywhere regularity of certain nonlinear diffusion systems

In this paper I discuss nonlinear parabolic systems that are generalizations of scalar diffusion equations. More precisely, I consider systems of the form $$\mathbf{u}_t -\Delta\left[ \mathbf{\nabla}\Phi(\mathbf{u})\right] = 0,$$ where ${\Phi(z)}$ is a strictly convex function. I show that when ${\Phi}$ is a function only of the norm of u, then bounded weak solutions of these parabolic systems are everywhere Hölder continuous and thus everywhere smooth. I also show that the method used to prove this result can be easily adopted to simplify the proof of the result due to Wiegner (Math Ann 292(4):711–727, 1992) on everywhere regularity of bounded weak solutions of strongly coupled parabolic systems.     

On a Regularized Family of Models for Homogeneous Incompressible Two-Phase Flows

We consider a general family of regularized models for incompressible two-phase flows based on the Allen–Cahn formulation in \(n\) -dimensional compact Riemannian manifolds for \(n=2,3\) . The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes- \(\alpha \) -like model, the Leray- \(\alpha \) model, the modified Leray- \(\alpha \) model, the simplified Bardina model, the Navier–Stokes–Voight model, and the Navier–Stokes model) for the fluid velocity \(u\) suitably coupled with a convective Allen–Cahn equation for the order (phase) parameter \(\phi \) . We give a unified analysis of the entire three-parameter family of two-phase models using only abstract mapping properties of the principal dissipation and smoothing operators and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We establish existence, stability, and regularity results and some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the \(\alpha \rightarrow 0\) limit in \(\alpha \) models. Then we show the existence of a global attractor and exponential attractor for our general model and establish precise conditions under which each trajectory \(\left( u,\phi \right) \) converges to a single equilibrium by means of a Lojasiewicz–Simon inequality. We also derive new results on the existence of global and exponential attractors for the regularized family of Navier–Stokes equations and magnetohydrodynamics models that improve and complement the results of Holst et al. (J Nonlinear Sci 20(5):523–567, 2010). Finally, our analysis is applied to certain regularized Ericksen–Leslie models for the hydrodynamics of liquid crystals in \(n\) -dimensional compact Riemannian manifolds.     

Existence of solutions for a nonlinear system of parabolic equations with gradient flow structure

We prove global existence of weak solutions of a variant of the parabolic-parabolic Keller–Segel model for chemotaxis on the whole space \({{\mathbb {R}}^d}\) for \(d\ge 3\) with a supercritical porous-medium diffusion exponent and an external drift. The structure of the equations allow the chemotactic drift to be seen both as attraction and repulsion. The method of proof relies on the inherent gradient flow structure of this system with respect to a coupled Wasserstein- \(L^2\) metric. Additional regularity estimates are derived from the dissipation of an entropy functional.     

Summability for solutions to some quasilinear elliptic systems

We prove local regularity in Lebesgue spaces for weak solutions \(u\) of quasilinear elliptic systems whose off-diagonal coefficients are small when \(|u|\) is large: the faster off-diagonal coefficients decay, the higher integrability of \(u\) becomes.     

Convergence of global attractors of a 2D non-Newtonian system to the global attractor of the 2D Navier-Stokes system

This paper discusses the relation between the long-time dynamics of solutions of the two-dimensional(2D) incompressible non-Newtonian fluid system and the 2D Navier-Stokes system.We first show that the solutions of the non-Newtonian fluid system converge to the solutions of the Navier-Stokes system in the energy norm.Then we establish that the global attractors {AHε} 0<ε≤1 of the non-Newtonian fluid system converge to the global attractor AH0 of the Navier-Stokes system as → 0.We also construct the minimal limit AHmin of the global attractors {AHε}0<ε≤1 as ε→ 0 and prove that AHmin is a strictly invariant and connected set.     

A De Giorgi–Nash type theorem for time fractional diffusion equations

We study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order $\alpha \in (0,1)$ . The coefficients are merely assumed to be bounded and measurable, and they satisfy a uniform parabolicity condition. Our main result is a De Giorgi–Nash type theorem, which gives an interior Hölder estimate for bounded weak solutions in terms of the data and the $L_\infty $ -bound of the solution. The proof relies on new a priori estimates for time fractional problems and uses De Giorgi’s technique and the method of non-local growth lemmas, which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional Laplacian.     

Weak solutions of the stationary Navier–Stokes equations for a viscous incompressible fluid past an obstacle

Consider the stationary Navier–Stokes equations in an exterior domain $\varOmega \subset \mathbb{R }^3 $ with smooth boundary. For every prescribed constant vector $u_{\infty } \ne 0$ and every external force $f \in \dot{H}_2^{-1} (\varOmega )$ , Leray (J. Math. Pures. Appl., 9:1–82, 1933) constructed a weak solution $u $ with $\nabla u \in L_2 (\varOmega )$ and $u - u_{\infty } \in L_6(\varOmega )$ . Here $\dot{H}^{-1}_2 (\varOmega )$ denotes the dual space of the homogeneous Sobolev space $\dot{H}^1_{2}(\varOmega ) $ . We prove that the weak solution $u$ fulfills the additional regularity property $u- u_{\infty } \in L_4(\varOmega )$ and $u_\infty \cdot \nabla u \in \dot{H}_2^{-1} (\varOmega )$ without any restriction on $f$ except for $f \in \dot{H}_2^{-1} (\varOmega )$ . As a consequence, it turns out that every weak solution necessarily satisfies the generalized energy equality. Moreover, we obtain a sharp a priori estimate and uniqueness result for weak solutions assuming only that $\Vert f\Vert _{\dot{H}^{-1}_2(\varOmega )}$ and $|u_{\infty }|$ are suitably small. Our results give final affirmative answers to open questions left by Leray (J. Math. Pures. Appl., 9:1–82, 1933) about energy equality and uniqueness of weak solutions. Finally we investigate the convergence of weak solutions as $u_{\infty } \rightarrow 0$ in the strong norm topology, while the limiting weak solution exhibits a completely different behavior from that in the case $u_{\infty } \ne 0$ .     

Global solutions for the Navier-Stokes equations in the rotational framework

The existence of global unique solutions to the Navier-Stokes equations with the Coriolis force is established in the homogeneous Sobolev spaces $\dot{H}^s (\mathbb R ^3)^3$ for $1/2 < s < 3/4$ if the speed of rotation is sufficiently large. This phenomenon is so-called the global regularity. The relationship between the size of initial datum and the speed of rotation is also derived. The proof is based on the space time estimates of the Strichartz type for the semigroup associated with the linearized equations. In the scaling critical space $\dot{H}^{\frac{1}{2}} (\mathbb R ^3)^3$ , the global regularity is also shown.     

Local {W^{2,m(\cdot)}_{loc}}regularity for p(.)-Laplace equations

We study ${W^{2,m(\cdot)}_{loc}}$ regularity for local weak solutions of p(·)-Laplace equations where ${p\in C^1(\Omega) \cap C(\overline{\Omega})}$ and ${\min_{x\in \overline{\Omega}} p(x) > 1}$ .     

Singular perturbation method for inhomogeneous nonlinear free boundary problems

In this paper, we study solutions of one phase inhomogeneous singular perturbation problems of the type: $ F(D^2u,x)=\beta _{\varepsilon }(u) + f_{\varepsilon }(x) $ and $ \Delta _{p}u=\beta _{\varepsilon }(u) + f_{\varepsilon }(x)$ , where $\beta _{\varepsilon }$ approaches Dirac $\delta _{0}$ as $\varepsilon \rightarrow 0$ and $f_{\varepsilon }$ has a uniform control in $L^{q}, q>N.$ Uniform local Lipschitz regularity is obtained for these solutions. The existence theory for variational (minimizers) and non variational (least supersolutions) solutions for these problems is developed. Uniform linear growth rate with respect to the distance from the $\varepsilon -$ level surfaces are established for these variational and nonvaritional solutions. Finally, letting $\varepsilon \rightarrow 0$ basic properties such as local Lipschitz regularity and non-degeneracy property are proven for the limit and a Hausdorff measure estimate for its free boundary is obtained.     

Square Function Characterization of Weak Hardy Spaces

We obtain a new square function characterization of the weak Hardy space \(H^{p,\infty }\) for all \(p\in (0,\infty )\) . This space consists of all tempered distributions whose smooth maximal function lies in weak \(L^p\) . Our proof is based on interpolation between \(H^p\) spaces. The main difficulty we overcome is the lack of a good dense subspace of \(H^{p,\infty }\) which forces us to work with general \(H^{p,\infty }\) distributions.     

A smooth global branch of solutions for a semilinear elliptic equation on {\mathbb{R}^N}

The existence of a global branch of positive spherically symmetric solutions ${\{(\lambda,u(\lambda)):\lambda\in(0,\infty)\}}$ of the semilinear elliptic equation $$\Delta u - \lambda u + V(x)|u|^{p-1}u = 0 \quad \text{in}\,\mathbb{R}^N\,\text{with}\,N\geq3$$ is proved for ${1 < p < 1+\frac{4-2b}{N-2}}$ , where ${b\in(0,2)}$ is such that the radial function V vanishes at infinity like |x|^?b . V is allowed to be singular at the origin but not worse than |x|^?b . The mapping ${\lambda\mapsto u(\lambda)}$ is of class ${C^r((0,\infty),H^1(\mathbb{R}^N))}$ if ${V\in C^r(\mathbb{R}^N\setminus\{0\},\mathbb{R})}$ , for r = 0, 1. Further properties of regularity and decay at infinity of solutions are also established. This work is a natural continuation of previous results by Stuart and the author, concerning the existence of a local branch of solutions of the same equation for values of the bifurcation parameter λ in a right neighbourhood of λ = 0. The variational structure of the equation is deeply exploited and the global continuation is obtained via an implicit function theorem.     

A regularized Newton method without line search for unconstrained optimization

In this paper, we propose a regularized Newton method without line search. The proposed method controls a regularization parameter instead of a step size in order to guarantee the global convergence. We show that the proposed algorithm has the following convergence properties. (a) The proposed algorithm has global convergence under appropriate conditions. (b) It has superlinear rate of convergence under the local error bound condition. (c) An upper bound of the number of iterations required to obtain an approximate solution \(x\) satisfying \(\Vert \nabla f(x) \Vert \le \varepsilon \) is \(O(\varepsilon ^{-2})\) , where \(f\) is the objective function and \(\varepsilon \) is a given positive constant.     

Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part II: Convergence analysis and numerical results

In the first part of this work, we presented a global optimization algorithm, Branch-and-Sandwich, for optimistic bilevel programming problems that satisfy a regularity condition in the inner problem (Kleniati and Adjiman in J Glob Optim, 2014). The proposed approach can be interpreted as the exploration of two solution spaces (corresponding to the inner and the outer problems) using a single branch-and-bound tree, where two pairs of lower and upper bounds are computed: one for the outer optimal objective value and the other for the inner value function. In the present paper, the theoretical properties of the proposed algorithm are investigated and finite \(\varepsilon \) -convergence to a global solution of the bilevel problem is proved. Thirty-four problems from the literature are tackled successfully.     

Bounds for graph regularity and removal lemmas

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck ²/log^* k pairs of parts which are not ${\epsilon}$ -regular, where ${c,\epsilon >0 }$ are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemerédi’s regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter ${\epsilon}$ may require as many as ${2^{\Omega}(\epsilon^{-2})}$ parts. This is tight up to the implied constant and solves a problem studied by Lovász and Szegedy.     

Generalized nonlinear complementarity problems with order P_0 and R_0 properties

We generalize the \(P, P_0, R_0\) properties for a nonlinear function associated with the standard nonlinear complementarity problem to the setting of generalized nonlinear complementarity problem (GNCP). We prove that if a continuous function has the order \(P_0\) and \(R_0\) properties then all the associated GNCPs have solutions.     

The gradient and heavy ball with friction dynamical systems: the quasiconvex case

We consider the gradient system $\dot x(t)+\nabla\phi(x(t))=0$ and the so-called heavy ball with friction dynamical system $\ddot x(t) +\lambda\dot x(t)+\nabla\phi(x(t))=0$ , as well as an implicit discrete (proximal) version of it, and study the asymptotic behavior of their solutions in the case of a smooth and quasiconvex objective function Φ. Minimization properties of trajectories are obtained under various additional assumptions. We finally show a minimizing property of the heavy ball method which is not shared by the gradient method: the genericity of the convergence of each trajectory, at least when Φ is a Morse function, towards local minimum of Φ.