Regularity criterion for 3d navier-stokes equations in terms of the direction of the velocity

In this short note we give a link between the regularity of the solution u to the 3D Navier-Stokes equation and the behavior of the direction of the velocity u/|u|. It is shown that the control of div(u/|u|) in a suitable L ^t/p (L ^x/q ) norm is enough to ensure global regularity. The result is reminiscent of the criterion in terms of the direction of the vorticity, introduced first by Constantin and Fefferman. However, in this case the condition is not on the vorticity but on the velocity itself. The proof, based on very standard methods, relies on a straightforward relation between the divergence of the direction of the velocity and the growth of energy along streamlines. This work was supported in part by NSF Grant DMS-0607953.     

Nonlocal initial-boundary-value problems for a degenerate hyperbolic equation

We consider the equation y ^m u _xx − u _yy − b ² y ^m u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u _y (x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u _x (0, y) = 0 or u _x (0, y) = u _x (1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems     

On the existence of shock fronts for a Burgers equation with a singular source term

In this paper we consider the Cauchy problem for the equation ∂u/∂t + u ∂u/∂x + u/x = 0 for x > 0, t ⩾ 0, with u(x, 0) = u⁰₋(x) for x < x₀, u(x, 0) = u⁰₊(x) for x > x₀, u⁰₋(x₀) > u⁰₊(x₀). Following the ideas of Majda, 1984 and Lax, 1973, we construct, for smooth u⁰₋ and u⁰₊, a global shock front weak solution u(x, t) = u₋(x, t) for x < ϕ(t), u(x, t) = u₊(x, t) for x > ϕ(t), where u₋ and u₊ are the strong solutions corresponding (respectively) to u⁰₋ and u⁰₊ and the curve t → ϕ(t) is defined by dϕ/dt (t) = 1/2[u₋(ϕ(t), t) + u₊(ϕ(t), t)], t ⩾ 0 and ϕ(0) = x₀. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Extra regularity for parabolic equations with drift terms

We consider a parabolic equation with a drift term u+bu–u_t=0. Under some natural conditions on the vector valued function b, we prove that solutions possess extra regularity and better qualitative behavior than those provided by standard theory. For example, we show that the fundamental solution has global Gaussian upper bound even for some b with a large singularity in the form of c/|x|. We also show that bounded solutions are Hölder continuous when |b|² is just form bounded and divergence free, a case where not even continuity is expected. A Liouville type theorem is also proven.     

Periodic and Homoclinic Solutions of Extended Fisher–Kolmogorov Equations

In this paper we study the existence of periodic solutions of the fourth-order equations u^iv − pu″ − a(x)u + b(x)u³ = 0 and u^iv − pu″ + a(x)u − b(x)u³ = 0, where p is a positive constant, and a(x) and b(x) are continuous positive 2L-periodic functions. The boundary value problems (P₁) and (P₂) for these equations are considered respectively with the boundary conditions u(0) = u(L) = u″(0) = u″(L) = 0. Existence of nontrivial solutions for (P₁) is proved using a minimization theorem and a multiplicity result using Clark's theorem. Existence of nontrivial solutions for (P₂) is proved using the symmetric mountain-pass theorem. We study also the homoclinic solutions for the fourth-order equation u^iv + pu″ + a(x)u − b(x)u² − c(x)u³ = 0, where p is a constant, and a(x), b(x), and c(x) are periodic functions. The mountain-pass theorem of Brezis and Nirenberg and concentration-compactness arguments are used.     

Hope-lax type formular for u∞ t +H u,D u =0: II

The first order equation u _t +H u,D u =0 with u T,x =g x is considered with terminal dat g which is assumed to be only quasiconvex, is a significant generalization of convex functions. The hamiltonian H γ,p is assumed to be homogeneous degree one in p and nondecreasing in γ. It is prove that the explicit solution of such a problem is u t,x = g ^# γ,p T-t H γ,p ^# where # refers to the quasiconvex conjugate of the functions in the x variable.     

Probabilistic viscosity algorithm for the scalar conservation law

We derive an algorithm for solving the initial value problem for u_t = ½σ²u_xx + f(u)u_x. The approach is based on the representation of the solution to the above equation in the form of the functional of Brownian motion. For small σ we get the approximation for u_t = f(u)u_x. A comparison with the random choice method is illustrated by the numerical example.     

Solvability conditions and monotone iterative scheme for boundary-value problems related to nonlinear monotone potential operators

This article deals with boundary-value problems (BVPs) for the second-order nonlinear differential equations with monotone potential operators of type Au := ??(k(|?u|²)?u(x)) + q(u ²)u(x), x ∈ Ω ? R ⁿ . An analysis of nonlinear problems shows that the potential of the operator A as well as the potential of related BVP plays an important role not only for solvability of these problems and linearization of the nonlinear operator, but also for the strong convergence of solutions of corresponding linearized problems. A monotone iterative scheme for the considered BVP is proposed.     

On Principal Eigenvalues for Indefinite Problems in Euclidean Space

For the eigenvalue problem—λΔu = q(x)u in IR^d, with the weight function q changing sign, conditions are discussed for existence of eigenvalues with positive decaying eigenfunctions.     

Kernel regression estimation for continuous spatial processes

We investigate here a kernel estimate of the spatial regression function r(x) = E(Y _u | X _u = x), x ∈ ℝ^d, of a stationary multidimensional spatial process { Z _u = (X _u, Y _u), u ∈ ℝ ^N }. The weak and strong consistency of the estimate is shown under sufficient conditions on the mixing coefficients and the bandwidth, when the process is observed over a rectangular domain of ℝ^N. Special attention is paid to achieve optimal and suroptimal strong rates of convergence. It is also shown that this suroptimal rate is preserved by using a suitable spatial sampling scheme.      

Remarks on the total variation of the Jacobian

The total variation TV(u) of the Jacobian determinant of nonsmooth vector fields u has recently been studied in [2] [3]. We focus on the subclass u(x) = φ(x/|x|) of homogeneous extensions of smooth functions In the case n = 2, we explicitely compute TV(u) for some relevant examples exhibiting a gap with respect to the total variation |Det Du| of the distributional determinant. We then provide examples of functions with |DetDu| = 0 and TV(u) = + ∞. We finally show that this gap phenomenon doesn’t occur if n ≥ 3.     

Regularity of the obstacle problem for a fractional power of the laplace operator

Given a function φ and s ∈ (0, 1), we will study the solutions of the following obstacle problem:

• u ≥ φ in ?ⁿ,
• (??)^su ≥ 0 in ?ⁿ,
• (??)^su(x) = 0 for those x such that u(x) > φ(x),
• lim_{|x| → + ∞} u(x) = 0.

We show that when φ is C^{1, s} or smoother, the solution u is in the space C^{1, α} for every α < s. In the case where the contact set {u = φ} is convex, we prove the optimal regularity result u ∈ C^{1, s}. When φ is only C^{1, β} for a β < s, we prove that our solution u is C^{1, α} for every α < β. © 2006 Wiley Periodicals, Inc.     

Finite‐difference approximation for the u(k)‐derivative with O(hM−k+1) accuracy: An analytical expression

An approximation of function u(x) as a Taylor series expansion about a point x₀ at M points x_i, ～ i = 1,2,…,M is used where x_i are arbitrary‐spaced. This approximation is a linear system for the derivatives u^(k) with an arbitrary accuracy. An analytical expression for the inverse matrix A ^?1 where A = [A_ik] = (x_i ? x₀)^k is found. A finite‐difference approximation of derivatives u^(k) of a given function u(x) at point x₀ is derived in terms of the values u(x_i). © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Principal Eigenvalues and Anti – Maximum Principle for Some Quasilinear Elliptic Equations on ℝN

We improve some previous existence and nonexistence results for positive principal eigenvalues of the problem —Δ_pu = λg(x)ψ^p(u), x ∈ ℝ^N, lim_{‖x‖⇒+∞}u(x) = 0. Also we discuss existence, nonexistence and antimaximum principle questions concerning the perturbed problem —Δ_pu = λg(x)ψ^p(u) + f(x), x∈ ℝ^N.     

The conformal plate buckling equation

The linear equation Δ²u = 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over ?² has the particularly interesting nonlinear generalization Δ_g²u = 1, where Δ_g = e^?2u Δ is the Laplace‐Beltrami operator for the metric g = e^2ug₀, with g₀ the standard Euclidean metric on ?². This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order Δu(x)+K_g(x) exp(2u(x)) = 0 and Δ K_g(x) + exp(2u(x)) = 0, with x ∈ ?², describing a conformally flat surface with a Gauss curvature function K_g that is generated self‐consistently through the metric's conformal factor. We study this conformal plate buckling equation under the hypotheses of finite integral curvature ∫ K_g exp(2u)dx = κ, finite area ∫ exp(2u)dx = α, and the mild compactness condition K₊ ∈ L¹(B₁(y)), uniformly w.r.t. y ∈ ?². We show that asymptotically for |x|→∞ all solutions behave like u(x) = ?(κ/2π)ln |x| + C + o(1) and K(x) = ?(α/2π) ln|x| + C + o(1), with κ ∈ (2π, 4π) and . We also show that for each κ ∈ (2π, 4π) there exists a K^* and a radially symmetric solution pair u, K, satisfying K(u) = κ and maxK = K^*, which is unique modulo translation of the origin, and scaling of x coupled with a translation of u. © 2001 John Wiley & Sons, Inc.     

Nonlinear equations with mixed derivatives

The nonlinear hyperbolic equation ∂²u(x, y)/∂x ∂y + g(x, y)f(u(x, y)) = 0 with u(x, 0) = φ(x) and u(0, y) = Ψ(y), considered by [1.], 31–45) under appropriate smoothness conditions, is solvable by the author's decomposition method (“Stochastic Systems,” Academic Press, 1983 and “Nonlinear Stochastic Operator Equations,” Academic Press, 1986).     

Problem with periodicity conditions for a degenerating equation of mixed type

For the equation K(t)u _xx + u _tt − b ² K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t| ^m , m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability of the boundary value problem u(0, t) = u(1, t), u _x (0, t) = u _x (1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1.     

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation u_t = (uⁿ)_xx + lf(u)/(ò_-1¹ f(u)dx)²{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s ^n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}.     

The cone of positive harmonic functions for scale-invariant diffusions

Consider a scale invariant diffusion whose state space is a closed cone in R ^d , minus the vertex. Then the process is either recurrent, transient to ∞ or transient to the vertex of the cone. In the latter case, the diffusion has finite lifetime (a.s.) and converges to the vertex at the lifetime. The Martin boundary consists of two points, and the corresponding minimal harmonic functions are of the form 1 and |x| ^α ψ(x/|x|).     

The Two-Dimensional Exterior Stokes Problem,Existence, Regularity and Decay Properties

The Stokes problem −Δu+∇p = f, div u = g in Ω, u∣_∂Ω = h is investigated for two-dimensional exterior domains Ω. By means of potential theory, existence, uniqueness and regularity results for weak solutions are proved in weighted Sobolev spaces with weights proportional to ∣x∣^δ as ∣x∣→∞. For f = 0,g = 0, explicit decay formulas are obtained for the solutions u and p. Finally, the results are compared with the theory of r-generalized solutions, i.e. ∇u∈L^r.