On the symmetry of deformable crystals

We prove existence, uniqueness, and regularity properties for a solution u of the Bellman-Dirichlet equation of dynamic programming: (1) $$\left\{ \begin{gathered} \max {\text{ }}\{ L^i u + f^i = 0{\text{ in }}\Omega \hfill \\ i{\text{ = 1,2 }} \hfill \\ u{\text{ = 0 on }}\partial \Omega , \hfill \\ \end{gathered} \right.$$ where L ¹ and L ² are two second order, uniformly elliptic operators. The method of proof is to rephrase (1) as a variational inequality for the operator K=L ²(L ¹)^?1 in L ²(Ω) and to invoke known existence theorems. For sufficiently nice f ¹ and f ² we prove in addition that u is in H ³(Ω)?C ^2,α(Ω) (for some 0<α<1) and hence is a classical solution of (1).     

A stabilized finite element method for the Stokes problem based on polynomial pressure projections

A new stabilized finite element method for the Stokes problem is presented. The method is obtained by modification of the mixed variational equation by using local L² polynomial pressure projections. Our stabilization approach is motivated by the inherent inconsistency of equal‐order approximations for the Stokes equations, which leads to an unstable mixed finite element method. Application of pressure projections in conjunction with minimization of the pressure–velocity mismatch eliminates this inconsistency and leads to a stable variational formulation. Unlike other stabilization methods, the present approach does not require specification of a stabilization parameter or calculation of higher‐order derivatives, and always leads to a symmetric linear system. The new method can be implemented at the element level and for affine families of finite elements on simplicial grids it reduces to a simple modification of the weak continuity equation. Numerical results are presented for a variety of equal‐order continuous velocity and pressure elements in two and three dimensions. Copyright © 2004 John Wiley & Sons, Ltd.     

Turbulence in a skewed three‐dimensional wall‐bounded shear flow: effect of mean vorticity on structure modification

Mean‐flow three‐dimensionalities affect both the turbulence level and the coherent flow structures in wall‐bounded shear flows. A tailor‐made flow configuration was designed to enable a thorough investigation of moderately and severely skewed channel flows. A unidirectional shear‐driven plane Couette flow was skewed by means of an imposed spanwise pressure gradient. Three different cases with 8°, 34°and 52°skewing were simulated numerically and the results compared with data from a purely two‐dimensional plane Couette flow. The resulting three‐dimensional flow field became statistically stationary and homogeneous in the streamwise and spanwise directions while the mean velocity vector V and the mean vorticity vector Ω remained parallel with the walls. Mean flow profiles were presented together with all components of the Reynolds stress tensor. The mean shear rate in the core region gradually increased with increasing skewing whereas the velocity fluctuations were enhanced in the spanwise direction and reduced in the streamwise direction. The Reynolds shear stress is known to be closely related to the coherent flow structures in the near‐wall region. The instantaneous and ensemble‐averaged flow structures were turned by the skewed mean flow. We demonstrated for the medium‐skewed case that the coherent structures should be examined in a coordinate system aligned with V to enable a sound interpretation of 3D effects. The conventional symmetry between Case 1 and Case 2 vortices was broken and Case 1 vortices turned out to be stronger than Case 2. This observation is in conflict with the common understanding on the basis of the spanwise (secondary) mean shear rate. A refined model was proposed to interpret the structure modifications in three‐dimensional wall‐flows. What matters is the orientation of the mean vorticity vector Ω relative to the vortex vorticity vector ω _v, that is, the sign of Ω · ω _v. In the present situation, Ω · ω _v > 0 for the Case 1 vortices causing a strengthening relative to the Case 2 vortices. Copyright © 2011 John Wiley & Sons, Ltd.     

Towards a Unified Field Theory for Classical Electrodynamics

In this paper we introduce a model which describes the relation of matter and the electromagnetic field from a unitarian standpoint in the spirit of ideas of Born and Infeld. In this model, based on a semilinear perturbation of Maxwell equations, the particles are finite-energy solitary waves due to the presence of the nonlinearity. In this respect the matter and the electromagnetic field have the same nature. Finite energy means that particles have finite mass and this makes electrodynamics consistent with the special relativity. We analyze the invariants of the motion of the semilinear Maxwell equations (SME) and their static solutions. In the magnetostatic case (i.e., when the electric field E = 0 and the magnetic field H does not depend on time) SME are reduced to the semilinear equation where × denotes the curloperator, f is the gradient of a strictly convex smooth function f:R³R and A:R³R³ is the gauge potential related to the magnetic field H (H = × A). Due to the presence of the curl operator, (1) is a strongly degenerate elliptic equation. Moreover, physical considerations impel f to be flat at zero (f(0)=0) and this fact leads us to study the problem in a functional setting related to the Orlicz space L^p+L^q. The existence of a nontrivial finite- energy solution of (1) is proved under suitable growth conditions on f. The proof is carried out by using a suitable variational framework related to the Hodge splitting of the vector field A.We thank Marino Badiale and Charles Stuart for their useful suggestions.     

A Weak-L p Prodi–Serrin Type Regularity Criterion for the Navier–Stokes Equations

We give simple proofs that a weak solution u of the Navier–Stokes equations with H ¹ initial data remains strong on the time interval [0, T] if it satisfies the Prodi–Serrin type condition u ∈ L ^s (0, T;L ^r,∞(Ω)) or if its L ^s,∞(0, T;L ^r,∞(Ω)) norm is sufficiently small, where 3 < r ≤ ∞ and (3/r) + (2/s) = 1.     

Velocities, Stresses and Vector Bundle Valued Chains

A mathematical framework for the fundamental objects of continuum mechanics is presented. In the geometric setting of general differentiable manifolds, velocity fields over bodies, modeled as sections of a vector bundle W, are generalized using notions of homological integration theory such as flat chains and cochains. The class of bodies includes fractal sets whose irregular boundaries may have infinite measures. Stresses, initially modeled as smooth differential forms valued in the dual of the jet bundle of W, are generalized to cochains represented by L ^??-sections whose weak divergences are also L ^??. The divergence of a stress field, defined in an earlier work, is generalized to apply to stress cochains. The co-divergence of a velocity field is a weak form of the jet extension mapping and it is the counterpart of the boundary operator for real valued flat chains.     

Optimal Decay Rate of the Compressible Navier–Stokes–Poisson System in {\mathbb {R}^3}

The compressible Navier–Stokes–Poisson (NSP) system is considered in ${\mathbb {R}^3}The compressible Navier–Stokes–Poisson (NSP) system is considered in \mathbb R³{\mathbb {R}^3} in the present paper, and the influences of the electric field of the internal electrostatic potential force governed by the self-consistent Poisson equation on the qualitative behaviors of solutions is analyzed. It is observed that the rotating effect of electric field affects the dispersion of fluids and reduces the time decay rate of solutions. Indeed, we show that the density of the NSP system converges to its equilibrium state at the same L ²-rate (1+t)^{-\frac 34}{(1+t)^{-\frac {3}{4}}} or L ^∞-rate (1 + t)^−3/2 respectively as the compressible Navier–Stokes system, but the momentum of the NSP system decays at the L ²-rate (1+t)^{-\frac 14}{(1+t)^{-\frac {1}{4}}} or L ^∞-rate (1 + t)⁻¹ respectively, which is slower than the L ²-rate (1+t)^{-\frac 34}{(1+t)^{-\frac {3}{4}}} or L ^∞-rate (1 + t)^−3/2 for compressible Navier–Stokes system [Duan et al., in Math Models Methods Appl Sci 17:737–758, 2007; Liu and Wang, in Comm Math Phys 196:145–173, 1998; Matsumura and Nishida, in J Math Kyoto Univ 20:67–104, 1980] and the L ^∞-rate (1 + t)^−p with p ? (1, 3/2){p \in (1, 3/2)} for irrotational Euler–Poisson system [Guo, in Comm Math Phys 195:249–265, 1998]. These convergence rates are shown to be optimal for the compressible NSP system.     

Existence and Uniqueness of Solutions¶of an Asymptotic Equation¶Arising from a Variational Wave Equation¶with General Data

We establish here the global existence and uniqueness of admissible (both dissipative and conservative) weak solutions to a canonical asymptotic equation () for weakly nonlinear solutions of a class of nonlinear variational wave equations with any L ²(ℝ) initial datum. We use the method of Young measures and mollification techniques. Accepted April 25, 2000?Published online November 16, 2000     

Regularity for Energy-Minimizing Area-Preserving Deformations

In this paper we establish the square integrability of the nonnegative hydrostatic pressure p, that emerges in the minimization problem $$\inf_{\mathcal{K}}\int_{\varOmega}|\nabla \textbf {v}|^2, \quad\varOmega\subset \mathbb {R}^2 $$ as the Lagrange multiplier corresponding to the incompressibility constraint det?v=1 a.e. in Ω. Our method employs the Euler-Lagrange equation for the mollified Cauchy stress C satisfied in the image domain Ω ^?=u(Ω). This allows to construct a convex function ψ, defined in the image domain, such that the measure of the normal mapping of ψ controls the L ² norm of the pressure. As a by-product we conclude that $\textbf {u}\in C^{\frac{1}{2}}_{\textrm {loc}}(\varOmega)$ if the dual pressure (introduced in Karakhanyan, Manuscr. Math. 138:463, 2012) is nonnegative.     

Steady Flows in Unsaturated Soils Are Stable

The stability of steady, vertically upward and downward flow of water in a homogeneous layer of soil is analyzed. Three equivalent dimensionless forms of the Richards equation are introduced, namely the pressure head, saturation, and matric flux potential forms. To illustrate general results and derive special results, use is made of several representative classes of soils. For all classes of soils with a Lipschitz continuous relationship between the hydraulic conductivity and the matric flux potential, steady flows are shown to be unique. In addition, linear stability of these steady flows is proved. To this end, use is made of the energy method, in which one considers (weighted) L ²-norms of the perturbations of the steady flows. This gives a general restriction of the dependence of the hydraulic conductivity upon the matric flux potential, yielding linear stability and exponential decay with time of a specific weighted L ²-norm. It is shown that for other norms the ultimate decay towards the steady-solution is preceded by transient growth. An extension of the Richards equation to take into account dynamic memory effects is also considered. It is shown that the stability condition for the standard Richards equation implies linear stability of the steady solution of the extended model.     

A Semilinear Schrödinger Equation in the Presence of a Magnetic Field

We consider the semilinear stationary Schrödinger equation in a magnetic field: (–i+A)² u+V(x)u=g(x,|u|)u in ^N , where V is the scalar (or electric) potential and A is the vector (or magnetic) potential. We study the existence of nontrivial solutions both in the critical and in the subcritical case (respectively g(x,|u|)=|u|^{2 * –2} and |g(x,|u|)|c(1+|u| ^{p –2}), where 2<p<2^*). The results are obtained by variational methods. For g critical we use constrained minimization and for subcritical g we employ a minimax-type argument. In the latter case we also study the existence of infinitely many geometrically distinct solutions.     

Discontinuous Solutions of the Navier-Stokes Equations for Multidimensional Flows of Heat-Conducting Fluids

We prove the global existence of weak solutions of the Navier-Stokes equations for compressible, heat-conducting fluids in two and three space dimensions when the initial density is close to a constant in L ²∩L ^∞, the initial temperature is close to a constant in L ², and the initial velocity is small in H ^s ∩L ⁴, where s=0 when n=2 and when n=3. (The L ^p norms must be weighted slightly when n=2.) In particular, the initial data may be discontinuous across a hypersurface of ⁿ . A great deal of qualitative information about the solution is obtained. For example, we show that the velocity, vorticity, and temperature are relatively smooth in positive time, as is the “effective viscous flux”F, which is the divergence of the velocity minus a certain multiple of the pressure. We find that F plays a central role in the entire analysis, particularly in closing the required energy estimates and in understanding rates of regularization near the initial layer. Moreover, F is precisely the quantity through which the hyperbolicity of the corresponding equations for inviscid fluids shows itself, an effect which is crucial for obtaining time-independent pointwise bounds for the density. (Accepted June 13, 1996)     

Global Low-Energy Weak Solutions of the Equations of Three-Dimensional Compressible Magnetohydrodynamics

We prove the global-in-time existence of weak solutions of the equations of compressible magnetohydrodynamics in three space dimensions with initial data small in L ² and initial density positive and essentially bounded. A great deal of information concerning partial regularity is obtained: velocity, vorticity, and magnetic field become relatively smooth in positive time (H ¹ but not H ²) and singularities in the pressure cancel those in a certain multiple of the divergence of the velocity, thus giving concrete expression to conclusions obtained formally from the Rankine–Hugoniot conditions.     

Stability and Asymptotic Behavior of Periodic Traveling Wave Solutions of Viscous Conservation Laws in Several Dimensions

Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem, we determine sharp L ^p estimates on the linearized solution operator about a multidimensional planar periodic wave of a system of conservation laws with viscosity, yielding linearized L ¹ ∩ L ^p → L ^p stability for all p \geqq 2{p \geqq 2} and dimensions d \geqq 1{d \geqq 1} and nonlinear L ¹ ∩ H ^s → L ^p ∩ H ^s stability and L ²-asymptotic behavior for p\geqq 2{p\geqq 2} and d\geqq 3{d\geqq 3} . The behavior can in general be rather complicated, involving both convective (that is, wave-like) and diffusive effects.     

On a Crystalline Variational Problem, Part I:¶First Variation and Global L∞ Regularity

Let φ:ℝ ⁿ → [0,+∞[ be a given positively one-homogeneous convex function, and let ?_φ≔{φ≤ 1 }. Pursuing our interest in motion by crystalline mean curvature in three dimensions, we introduce and study the class ?_φ (ℝ ⁿ ) of “smooth” boundaries in the relative geometry induced by the ambient Banach space (ℝ ⁿ , φ). It can be seen that, even when ?_φ is a polytope, ?_φ(ℝ ⁿ ) cannot be reduced to the class of polyhedral boundaries (locally resembling ∂?_φ). Curved portions must be necessarily included and this fact (as well as the nonsmoothness of ∂?_φ) is the source of several technical difficulties related to the geometry of Lipschitz manifolds. Given a boundary δE in the class ?_φ(ℝ ⁿ ), we rigorously compute the first variation of the corresponding anisotropic perimeter, which leads to a variational problem on vector fields defined on δE. It turns out that the minimizers have a uniquely determined (intrinsic) tangential divergence on δE. We define such a divergence to be the φ-mean curvature κ_φ of δE; the function κ_φ is expected to be the initial velocity of δE, whenever δE is considered as the initial datum for the corresponding anisotropic mean curvature flow. We prove that κ_φ is bounded on δE and that its sublevel sets are characterized through a variational inequality.     

L∞ Variational Problems and Aronsson Equations

In this paper, we prove that viscosity solutions of Aronsson equations are absolute minimizers in certain L ^∞ variational problems.     

Pore structures and transport properties of sandstone

To quantitatively analyze the macroscopic properties of the flow in porous media by means of the continuum approach, detailed information (velocity and pressure fields) on the microscopic scale is necessary. In this paper, the numerical solution for incompressible, Newtonian flow in a diverging-converging representative unit cell (RUC) is presented. A new solution procedure for the problem is introduced. A review of the accuracy of the computational method is given.Nomenclature A _ff ^* area of entrance and exit of RUC - A _fs ^* interfacial area between the fluid and solid phases - d throat diameter of RUC (m) - D pore diameter of RUC (m) - i, j unit vector for RUC - L ^* wave length of a unit cell - L _p pore length of RUC (m) - L _t throat length of RUC (m) - n unit outwardly directed vector for the fluid phase - p ^* fluid pressure - ^* cross-sectional mean pressure - _en ^* entrance cross-sectional mean pressure - Re _d Reynolds number - x ^*, r^* cylindrical coordinates - u ^*, v^* velocity - u _cl ^* centerline velocity - _d mean velocity at the throat of RUC (m/s) - _D mean velocity at the large segment of RUC (m/s) Greek viscosity coefficient (Ns/m²) - _p excess momentum loss factor defined in (4.1) - fluid density (kg/m³) - ^* stream function - ^* vorticity - dimensionless circulation defined in (2.7) Symbols - the mean value - * dimensionless quantities     

Phragmén-Lindelöf theorems for nonlinear elliptic equations

We obtain theorems of Phragmén-Lindelöf type for the following classes of elliptic partial differential inequalities in an arbitrary unbounded domain \(\Omega \subseteq \mathbb{R}^n ,{\text{ }}n \geqq 2\) (A.1) $$\sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial x_i }}\left( {a_{ij} 9(x)\frac{{\partial u}}{{\partial xj}}} \right)} + \sum\limits_{i = 1}^n {b_i (x,{\text{ }}u,{\text{ }}\nabla u)\frac{\partial }{{\partial x_i }}} \geqq f(x,{\text{ }}u)$$ where a _ij are elliptic in Ω and b _i ε L^∞(Ω) and where also a _ij are uniformly elliptic and Holder continuous at infinity and b _i = O(|x|⁺¹) as x → ∞; (A.2) $${\text{(A}}{\text{.2) }}\sum\limits_{i,j = 1}^n {a_{ij} (x,{\text{ }}u,{\text{ }}\nabla u)\frac{{\partial ^2 u}}{{\partial x_i \partial x_j }}} + \sum\limits_{i = 1}^n {b_i (x,{\text{ }}u,{\text{ }}\nabla u)\frac{\partial }{{\partial x_i }}} \geqq f(x,{\text{ }}u)$$ where a_ijare uniformly elliptic in Ω and b _iε L^∞(Ω); and finally (A.3) $${\text{div(}}\nabla u^p \nabla u {\text{)}} \geqq f{\text{(}}u{\text{), }}p > - 1,$$ where the operator on the left is the so-called P-Laplacian. The function f is always supposed positive and continuous. Moreover u is assumed throughout to be in the natural weak Sobolev space corresponding to the particular inequality under consideration, namely u ε. W _loc ^1,2 (Ω) ∩L _loc ^t8 (Ω) for (A.I), W _loc ^2,n(Ω) for (A.2), and W _loc ^1,p+2 (Ω) ∩ L _loc ^t8 (Ω) for (A.3). As a consequence of our results we obtain both non-existence and Liouville theorems, as well as existence theorems for (A.1).     

L 2-Stability Theory of the Boltzmann Equation near a Global Maxwellian

We present three a priori L ²-stability estimates for classical solutions to the Boltzmann equation with a cut-off inverse power law potential, when initial datum is a perturbation of a global Maxwellian. We show that L ²-stability estimates of classical solutions depend on Strichartz type estimates of perturbations and the non-positive definiteness of the linearized collision operator. Several well known classical solutions to the Boltzmann equation fit our L ²-stability framework.