Showing papers in "Archive for Rational Mechanics and Analysis in 2018"

Global Classical and Weak Solutions to the Three-Dimensional Full Compressible Navier-Stokes System with Vacuum and Large Oscillations

Abstract: For the three-dimensional full compressible Navier–Stokes system describing the motion of a viscous, compressible, heat-conductive, and Newtonian polytropic fluid, we establish the global existence and uniqueness of classical solutions with smooth initial data which are of small energy but possibly large oscillations where the initial density is allowed to vanish. Moreover, for the initial data, which may be discontinuous and contain vacuum states, we also obtain the global existence of weak solutions. These results generalize previous ones on classical and weak solutions for initial density being strictly away from a vacuum, and are the first for global classical and weak solutions which may have large oscillations and can contain vacuum states.

Global Well-Posedness of the Incompressible Magnetohydrodynamics

Abstract: This paper studies the Cauchy problem of the incompressible magnetohydro dynamic systems with or without viscosity ν. Under the assumption that the initial velocity field and the displacement of the initialmagnetic field froma non-zero constant are sufficiently small in certain weighted Sobolev spaces, the Cauchy problem is shown to be globally well-posed for all ν ≧ 0 and all spaces with dimension n ≧ 2. Such a result holds true uniformly in nonnegative viscosity parameters. The proof is based on the inherent strong null structure of the systems introduced by Lei (Commun Pure Appl Math 69(11):2072–2106, 2016) and the ghost weight technique introduced by Alinhac (Invent Math 145(3):597–618, 2001).

Nonlinear Calderón-Zygmund Theory in the Limiting Case

Abstract: We prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems. Specifically, we deal with the limiting case of the classical theory of Calderon and Zygmund in the setting of nonlinear, possibly degenerate equations and we show a complete linearization effect with respect to the differentiability of solutions. A prototype of the results obtained here tells for instance that if $$-{\rm div} \, (|Du|^{p-2}Du)=\mu \quad \mbox{in} \ \Omega\subset\mathbb{R}^n,$$ with $${\mu}$$ being a Borel measure with locally finite mass on the open subset $${\Omega\subset \mathbb{R}^n}$$ and $${p > 2-1/n}$$ , then $$|Du|^{p-2}Du \in W^{\sigma, 1}_{\rm{loc}}(\Omega)\quad \mbox{for \, every} \ \sigma \in (0,1).$$ The case $${\sigma=1}$$ is obviously forbidden already in the classical linear case of the Poisson equation $${-\triangle u=\mu}$$ .

Global Classical Solutions of Three Dimensional Viscous MHD System Without Magnetic Diffusion on Periodic Boxes

Abstract: In this paper, we study the global existence of classical solutions to the three dimensional incompressible viscous magneto-hydrodynamical system without magnetic diffusion on periodic boxes, that is, with periodic boundary conditions. We work in Eulerian coordinates and employ a time-weighted energy estimate to prove the global existence result, under the assumptions that the initial magnetic field is close enough to an equilibrium state and the initial data have some symmetries.

Onsager’s Conjecture for the Incompressible Euler Equations in Bounded Domains

Abstract: The goal of this note is to show that, in a bounded domain $${\Omega \subset \mathbb{R}^n}$$ , with $${\partial \Omega\in C^2}$$ , any weak solution $${(u(x,t),p(x,t))}$$ , of the Euler equations of ideal incompressible fluid in $${\Omega\times (0,T) \subset \mathbb{R}^n\times\mathbb{R}_t}$$ , with the impermeability boundary condition $${u\cdot \vec n =0}$$ on $${\partial\Omega\times(0,T)}$$ , is of constant energy on the interval (0,T), provided the velocity field $${u \in L^3((0,T); C^{0,\alpha}(\overline{\Omega}))}$$ , with $${\alpha > \frac13.}$$

Global Regularity for the Fractional Euler Alignment System

Abstract: We study a pressureless Euler system with a non-linear density-dependent alignment term, originating in the Cucker–Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density dependence is natural: the alignment rate increases in the areas of high density due to species discomfort. The diffusive term has the order of a fractional Laplacian $${(-\partial _{xx})^{\alpha/2}, \alpha \in (0, 1)}$$ . The corresponding Burgers equation with a linear dissipation of this type develops shocks in a finite time. We show that the alignment nonlinearity enhances the dissipation, and the solutions are globally regular for all $${\alpha \in (0, 1)}$$ . To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation.

Global Existence Analysis of Cross-Diffusion Population Systems for Multiple Species

Abstract: The existence of global-in-time weak solutions to reaction-cross-diffusion systems for an arbitrary number of competing population species is proved. The equations can be derived from an on-lattice random-walk model with general transition rates. In the case of linear transition rates, it extends the two-species population model of Shigesada, Kawasaki, and Teramoto. The equations are considered in a bounded domain with homogeneous Neumann boundary conditions. The existence proof is based on a refined entropy method and a new approximation scheme. Global existence follows under a detailed balance or weak cross-diffusion condition. The detailed balance condition is related to the symmetry of the mobility matrix, which mirrors Onsager’s principle in thermodynamics. Under detailed balance (and without reaction) the entropy is nonincreasing in time, but counter-examples show that the entropy may increase initially if detailed balance does not hold.

The Cucker–Smale Equation: Singular Communication Weight, Measure-Valued Solutions and Weak-Atomic Uniqueness

Abstract: The Cucker–Smale flocking model belongs to a wide class of kinetic models that describe a collective motion of interacting particles that exhibit some specific tendency, e.g. to aggregate, flock or disperse. This paper examines the kinetic Cucker–Smale equation with a singular communication weight. Given a compactly supported measure as an initial datum we construct a global in time weak measure-valued solution in the space $${C_{weak}(0,\infty;\mathcal{M})}$$ . The solution is defined as a mean-field limit of the empirical distributions of particles, the dynamics of which is governed by the Cucker–Smale particle system. The studied communication weight is $${\psi(s)=|s|^{-\alpha}}$$ with $${\alpha \in \left(0,\frac 12\right)}$$ . This range of singularity admits the sticking of characteristics/trajectories. The second result concerns the weak-atomic uniqueness property stating that a weak solution initiated by a finite sum of atoms, i.e. Dirac deltas in the form $${m_i \delta_{x_i} \otimes \delta_{v_i}}$$ , preserves its atomic structure. Hence these coincide with unique solutions to the system of ODEs associated with the Cucker–Smale particle system.

Second-Order Two-Sided Estimates in Nonlinear Elliptic Problems

Abstract: Best possible second-order regularity is established for solutions to p-Laplacian type equations with $${p \in (1, \infty)}$$ and a square-integrable right-hand side. Our results provide a nonlinear counterpart of the classical L2-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are obtained. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required, although our conclusions are new even for smooth domains. If the domain is convex, no regularity of its boundary is needed at all.

The Riemann Problem for the Multidimensional Isentropic System of Gas Dynamics is Ill-Posed if It Contains a Shock

Abstract: In this paper we consider the isentropic compressible Euler equations in two space dimensions together with particular initial data. This data consists of two constant states, where one state lies in the lower and the other state in the upper half plane. The aim is to investigate whether there exists a unique entropy solution or if the convex integration method produces infinitely many entropy solutions. For some initial states this question has been answered by Feireisl and Kreml (J Hyperbolic Differ Equ 12(3):489–499, 2015), and also Chen and Chen (J Hyperbolic Differ Equ 4(1):105–122, 2007), where there exists a unique entropy solution. For other initial states Chiodaroli and Kreml (Arch Ration Mech Anal 214(3):1019–1049, 2014) and Chiodaroli et al. (Commun Pure Appl Math 68(7):1157–1190, 2015), showed that there are infinitely many entropy solutions. For still other initial states the question on uniqueness remained open and this will be the content of this paper. This paper can be seen as a completion of the aforementioned papers by showing that the solution is non-unique in all cases (except if the solution is smooth).

Approximation of a Brittle Fracture Energy with a Constraint of Non-interpenetration

Abstract: Linear fracture mechanics (or at least the initiation part of that theory) can be framed in a variational context as a minimization problem over a SBD type space. The corresponding functional can in turn be approximated in the sense of Γ-convergence by a sequence of functionals involving a phase field as well as the displacement field. We show that a similar approximation persists if additionally imposing a non-interpenetration constraint in the minimization, namely that only nonnegative normal jumps should be permissible. 2010 Mathematics subject classification: 26A45

Enhanced Dissipation and Axisymmetrization of Two-Dimensional Viscous Vortices

Abstract: This paper is devoted to the stability analysis of the Lamb–Oseen vortex in the regime of high circulation Reynolds numbers. When strongly localized perturbations are applied, it is shown that the vortex relaxes to axisymmetry in a time proportional to $${Re^{2/3}}$$ , which is substantially shorter than the diffusion time scale given by the viscosity. This enhanced dissipation effect is due to the differential rotation inside the vortex core. Our result relies on a recent work by Li et al. (Pseudospectral and spectral bounds for the Oseen vortices operator, 2017, arXiv:1701.06269 ), where optimal resolvent estimates for the linearized operator at Oseen’s vortex are established. A comparison is made with the predictions that can be found in the physical literature, and with the rigorous results that were obtained for shear flows using different techniques.

Data-Driven Problems in Elasticity

Abstract: We consider a new class of problems in elasticity, referred to as Data-Driven problems, defined on the space of strain-stress field pairs, or phase space. The problem consists of minimizing the distance between a given material data set and the subspace of compatible strain fields and stress fields in equilibrium. We find that the classical solutions are recovered in the case of linear elasticity. We identify conditions for convergence of Data-Driven solutions corresponding to sequences of approximating material data sets. Specialization to constant material data set sequences in turn establishes an appropriate notion of relaxation. We find that relaxation within this Data-Driven framework is fundamentally different from the classical relaxation of energy functions. For instance, we show that in the Data-Driven framework the relaxation of a bistable material leads to material data sets that are not graphs.

Acoustic Scattering from Corners, Edges and Circular Cones

Abstract: Consider the time-harmonic acoustic scattering from a bounded penetrable obstacle imbedded in an isotropic homogeneous medium. The obstacle is supposed to possess a circular conic point or an edge point on the boundary in three dimensions and a planar corner point in two dimensions. The opening angles of cones and edges are allowed to be any number in $${(0,2\pi)\backslash\{\pi\}}$$ . We prove that such an obstacle scatters any incoming wave non-trivially (that is, the far field patterns cannot vanish identically), leading to the absence of real non-scattering wavenumbers. Local and global uniqueness results for the inverse problem of recovering the shape of penetrable scatterers are also obtained using a single incoming wave. Our approach relies on the singularity analysis of the inhomogeneous Laplace equation in a cone.

The Method of Reflections, Homogenization and Screening for Poisson and Stokes Equations in Perforated Domains

Abstract: We study the convergence of the method of reflections for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains which exist in the exterior of balls. We prove that the method converges if the balls are contained in a bounded region and the density of the electrostatic capacity of the balls is sufficiently small. If the capacity density is too large or the balls extend to the whole space, the method diverges, but we provide a suitable modification of the method that converges to the solution of the Dirichlet problem also in this case. We give new proofs of classical homogenization results for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains using the (modified) method of reflections.

On the Homogenization of the Stokes Problem in a Perforated Domain

Abstract: We consider the Stokes equations on a bounded perforated domain completed with non-zero constant boundary conditions on the holes. We investigate configurations for which the holes are identical spheres and their number N goes to infinity while their radius aN tends to zero. Under the assumption that aN scales like a/N and that there is no concentration in the distribution of holes, we prove that the solution is well approximated asymptotically by solving a Stokes–Brinkman problem.

Compressible Fluids Interacting with a Linear-Elastic Shell

Abstract: We study the Navier–Stokes equations governing the motion of an isentropic compressible fluid in three dimensions interacting with a flexible shell of Koiter type. The latter one constitutes a moving part of the boundary of the physical domain. Its deformation is modeled by a linearized version of Koiter’s elastic energy. We show the existence of weak solutions to the corresponding system of PDEs provided the adiabatic exponent satisfies \({\gamma > \frac{12}{7}}\) (\({\gamma >1 }\) in two dimensions). The solution exists until the moving boundary approaches a self-intersection. This provides a compressible counterpart of the results in Lengeler and Růžickaka (Arch Ration Mech Anal 211(1):205–255, 2014) on incompressible Navier–Stokes equations.

Regularity for Fully Nonlinear Elliptic Equations with Oblique Boundary Conditions

Abstract: In this paper, we obtain a series of regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique derivative boundary conditions. In particular, we derive the pointwise C α, C 1,α and C 2,α regularity. As byproducts, we also prove the A–B–P maximum principle, Harnack inequality, uniqueness and solvability of the equations.

Large Time Behavior of Solutions to 3-D MHD System with Initial Data Near Equilibrium

Abstract: Califano and Chiuderi (Phys Rev E 60 (PartB):4701–4707, 1999) conjectured that the energy of an incompressible Magnetic hydrodynamical system is dissipated at a rate that is independent of the ohmic resistivity. The goal of this paper is to mathematically justify this conjecture in three space dimensions provided that the initial magnetic field and velocity is a small perturbation of the equilibrium state (e3, 0). In particular, we prove that for such data, a 3-D incompressible MHD system without magnetic diffusion has a unique global solution. Furthermore, the velocity field and the difference between the magnetic field and e3 decay to zero in both L∞ and L2 norms with explicit rates. We point out that the decay rate in the L2 norm is optimal in sense that this rate coincides with that of the linear system. The main idea of the proof is to exploit Hormander’s version of the Nash–Moser iteration scheme, which is very much motivated by the seminar papers by Klainerman (Commun Pure Appl Math 33:43–101, 1980, Arch Ration Mech Anal 78:73–98, 1982, Long time behaviour of solutions to nonlinear wave equations. PWN, Warsaw, pp 1209–1215, 1984) on the long time behavior to the evolution equations.

Time-Domain Analysis of an Acoustic–Elastic Interaction Problem

Abstract: Consider the scattering of a time-domain acoustic plane wave by a bounded elastic obstacle which is immersed in a homogeneous air or fluid. This paper concerns the mathematical analysis of such a time-domain acoustic–elastic interaction problem. An exact transparent boundary condition (TBC) is developed to reduce the scattering problem from an open domain into an initial-boundary value problem in a bounded domain. The well-posedness and stability are established for the reduced problem. A priori estimates with explicit time dependence are achieved for the pressure of the acoustic wave field and the displacement of the elastic wave field. Our proof is based on the method of energy, the Lax–Milgram lemma, and the inversion theorem of the Laplace transform. In addition, a time-domain absorbing perfectly matched layer (PML) method is introduced to replace the nonlocal TBC by a Dirichlet boundary condition. A first order symmetric hyperbolic system is derived for the truncated PML problem. The well-posedness and stability are proved. The time-domain PML results are expected to be useful in the computational air/fluid–solid interaction problems.

Sharp Interface Limit for a Stokes/Allen–Cahn System

Abstract: We consider the sharp interface limit of a coupled Stokes/Allen–Cahn system, when a parameter $${\epsilon > 0}$$ that is proportional to the thickness of the diffuse interface tends to zero, in a two dimensional bounded domain. For sufficiently small times we prove convergence of the solutions of the Stokes/Allen–Cahn system to solutions of a sharp interface model, where the interface evolution is given by the mean curvature equation with an additional convection term coupled to a two-phase Stokes system with an additional contribution to the stress tensor, which describes the capillary stress. To this end we construct a suitable approximation of the solution of the Stokes/Allen–Cahn system, using three levels of the terms in the formally matched asymptotic calculations, and estimate the difference with the aid of a suitable refinement of a spectral estimate of the linearized Allen–Cahn operator. Moreover, a careful treatment of the coupling terms is needed.

Smoothing of Transport Plans with Fixed Marginals and Rigorous Semiclassical Limit of the Hohenberg–Kohn Functional

Abstract: We prove rigorously that the exact N-electron Hohenberg–Kohn density functional converges in the strongly interacting limit to the strictly correlated electrons (SCE) functional, and that the absolute value squared of the associated constrained search wavefunction tends weakly in the sense of probability measures to a minimizer of the multi-marginal optimal transport problem with Coulomb cost associated to the SCE functional. This extends our previous work for N = 2 (Cotar etal. in Commun Pure Appl Math 66:548–599, 2013). The correct limit problem has been derived in the physics literature by Seidl (Phys Rev A 60 4387–4395, 1999) and Seidl, Gorigiorgi and Savin (Phys Rev A 75:042511 1-12, 2007); in these papers the lack of a rigorous proofwas pointed out.We also give amathematical counterexample to this type of result, by replacing the constraint of given one-body density—an infinite dimensional quadratic expression in the wavefunction—by an infinite-dimensional quadratic expression in the wavefunction and its gradient. Connections with the Lawrentiev phenomenon in the calculus of variations are indicated.

Rigorous Numerics for ill-posed PDEs: Periodic Orbits in the Boussinesq Equation

Abstract: In this paper, we develop computer-assisted techniques for the analysis of periodic orbits of ill-posed partial differential equations. As a case study, our proposed method is applied to the Boussinesq equation, which has been investigated extensively because of its role in the theory of shallow water waves. The idea is to use the symmetry of the solutions and a Newton–Kantorovich type argument (the radii polynomial approach) to obtain rigorous proofs of existence of the periodic orbits in a weighted l1 Banach space of space-time Fourier coefficients with exponential decay. We present several computer-assisted proofs of the existence of periodic orbits at different parameter values.

On the Measure and the Structure of the Free Boundary of the Lower Dimensional Obstacle Problem

Abstract: We provide a thorough description of the free boundary for the lower dimensional obstacle problem in $${\mathbb{R}^{n+1}}$$ up to sets of null $${\mathcal{H}^{n-1}}$$ measure. In particular, we prove

Doubly Nonlinear Equations of Porous Medium Type

Abstract: In this paper we prove the existence of solutions to doubly nonlinear equations whose prototype is given by $$\partial_t u^m- {\rm div}\, D_{\xi}\, f(x,Du) =0,$$ with $${m\in (0,\infty )}$$ , or more generally with an increasing and piecewise C1 nonlinearity b and a function f depending on u $$\partial_{t}b(u) - {\rm div}\, D_{\xi}\, f(x,u,Du)= -D_u f(x,u,Du).$$ For the function f we merely assume convexity and coercivity, so that, for instance, $${f(x,u,\xi)=\alpha(x)|\xi|^p + \beta(x)|\xi|^q}$$ with 1 < p < q and non-negative coefficients α, β with $${\alpha(x)+\beta(x)\geqq u > 0}$$ , and $${f(\xi)=\exp(\tfrac12|\xi|^2)}$$ are covered. Thus, for functions $${f(x,u,\xi )}$$ satisfying only a coercivity assumption from below but very general growth conditions from above, we prove the existence of variational solutions; mean while, if $${f(x,u,\xi )}$$ grows naturally when $${\left\vert \xi \right\vert \rightarrow +\infty }$$ as a polynomial of order p (for instance in the case of the p-Laplacian operator), then we obtain the existence of solutions in the sense of distributions as well as the existence of weak solutions. Our technique is purely variational and we treat both the cases of bounded and unbounded domains. We introduce a nonlinear version of the minimizing movement approach that could also be useful for the numerics of doubly nonlinear equations.

Congested aggregation via Newtonian interaction

Abstract: We consider a congested aggregation model that describes the evolution of a density through the competing effects of nonlocal Newtonian attraction and a hard height constraint. This provides a counterpoint to existing literature on repulsive–attractive nonlocal interaction models, where the repulsive effects instead arise from an interaction kernel or the addition of diffusion. We formulate our model as the Wasserstein gradient flow of an interaction energy, with a penalization to enforce the constraint on the height of the density. From this perspective, the problem can be seen as a singular limit of the Keller–Segel equation with degenerate diffusion. Two key properties distinguish our problem from previous work on height constrained equations: nonconvexity of the interaction kernel (which places the model outside the scope of classical gradient flow theory) and nonlocal dependence of the velocity field on the density (which causes the problem to lack a comparison principle). To overcome these obstacles, we combine recent results on gradient flows of nonconvex energies with viscosity solution theory. We characterize the dynamics of patch solutions in terms of a Hele-Shaw type free boundary problem and, using this characterization, show that in two dimensions patch solutions converge to a characteristic function of a disk in the long-time limit, with an explicit rate on the decay of the energy. We believe that a key contribution of the present work is our blended approach, combining energy methods with viscosity solution theory.

Bubbling with $L^2$-almost constant mean curvature and an Alexandrov-type theorem for crystals

Abstract: A compactness theorem for volume-constrained almost-critical points of elliptic integrands is proven. The result is new even for the area functional, as almost-criticality is measured in an integral rather than in a uniform sense. Two main applications of the compactness theorem are discussed. First, we obtain a description of critical points/local minimizers of elliptic energies interacting with a confinement potential. Second, we prove an Alexandrov-type theorem for crystalline isoperimetric problems.

Global Well-posedness of the Spatially Homogeneous Kolmogorov–Vicsek Model as a Gradient Flow

Abstract: We consider the so-called spatially homogenous Kolmogorov–Vicsek model, a non-linear Fokker–Planck equation of self-driven stochastic particles with orientation interaction under the space-homogeneity. We prove the global existence and uniqueness of weak solutions to the equation. We also show that weak solutions exponentially converge to a steady state, which has the form of the Fisher-von Mises distribution.

A Note on Weak Solutions of Conservation Laws and Energy/Entropy Conservation

Abstract: A common feature of systems of conservation laws of continuum physics is that they are endowed with natural companion laws which are in such cases most often related to the second law of thermodynamics. This observation easily generalizes to any symmetrizable system of conservation laws; they are endowed with nontrivial companion conservation laws, which are immediately satisfied by classical solutions. Not surprisingly, weak solutions may fail to satisfy companion laws, which are then often relaxed from equality to inequality and overtake the role of physical admissibility conditions for weak solutions. We want to answer the question: what is a critical regularity of weak solutions to a general system of conservation laws to satisfy an associated companion law as an equality? An archetypal example of such a result was derived for the incompressible Euler system in the context of Onsager’s conjecture in the early nineties. This general result can serve as a simple criterion to numerous systems of mathematical physics to prescribe the regularity of solutions needed for an appropriate companion law to be satisfied.

The Inviscid Limit of Navier–Stokes Equations for Analytic Data on the Half-Space

Abstract: In their classical work, Sammartino and Caflisch (Commun Math Phys 192(2):433–461, 1998a; Commun Math Phys 192(2):463–491, 1998b) proved the inviscid limit of the incompressible Navier–Stokes equations for well-prepared data with analytic regularity in the half-space. Their proof is based on the detailed construction of Prandtl’s boundary layer asymptotic expansions. In this paper, we give a direct proof of the inviscid limit for general analytic data without having to construct Prandtl’s boundary layer correctors. Our analysis makes use of the boundary vorticity formulation and the abstract Cauchy–Kovalevskaya theorem on analytic boundary layer function spaces that capture unbounded vorticity.