Carleman estimates for one-dimensional degenerate heat equations

In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain. We derive new Carleman estimates for the degenerate parabolic equation $$ w_t + \left( {a\left( x \right)w_x } \right)_x = f,\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where the function a mainly satisfies $$ a \in \mathcal{C}^0 \left( {\left[ {0,1} \right]} \right) \cap \mathcal{C}^1 \left( {\left( {0,1} \right)} \right),a \gt 0 \hbox{on }\left( {0,1} \right) \hbox{and }\frac{1} {{\sqrt a }} \in L^1 \left( {0,1} \right). $$ We are mainly interested in the situation of a degenerate equation at the boundary i.e. in the case where a(0)=0 and / or a(1)=0. A typical example is a(x)=x^α (1 − x)^β with α, β ∈ [0, 2). As a consequence, we deduce null controllability results for the degenerate one dimensional heat equation $$ u_t - (a(x)u_x )_x = h\chi _w ,\quad (t,x) \in (0,T) \times (0,1),\quad \omega \subset \subset (0,1). $$ The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x)=x^α with α ∈[0, 2). Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Blow-up for semilinear parabolic equations with nonlinear memory

In this paper, we consider the semilinear parabolic equation $u_t-\Delta u=u^q\int^{t}_{0}u^p(x,s)ds,\ \ \ x \in \Omega,\ \ t>0$u_t-\Delta u=u^q\int^{t}_{0}u^p(x,s)ds,\ \ \ x \in \Omega,\ \ t>0 with homogeneous Dirichlet boundary conditions, where p, q are nonnegative constants. The blowup criteria and the blowup rate are obtained.     

Entire extensions and exponential decay for semilinear elliptic equations

We consider semilinear partial differential equations in ℝ ⁿ of the form

An estimate of the solutions of a stationary problem for the Navier-Stokes equations

It is shown that the solutions of a nonlinear stationary problem for the Navier-Stokes equations in a bounded domain Ω ? ?³ with boundary conditions $\vec \upsilon \left| {_{\partial \Omega } } \right. = \vec a(x)$ satisfy the inequality $\left. {_{x \in \Omega }^{\sup } } \right|\left. {\vec v(x)} \right| \leqslant c\left( {\left. {_{x \in \partial \Omega }^{\sup } } \right|\left. {\vec a(x)} \right|} \right)$ for arbitrary Reynolds numbers. Bibliography: 9 titles.     

A Blow-up Criterion of Strong Solutions to the Compressible Viscous Heat-Conductive Flows with Zero Heat Conductivity

We study the compressible Navier-Stokes equations of viscous heat-conductive fluids in a periodic domain \mathbbT³\mathbb{T}^{3} with zero heat conductivity k=0. We prove a blow-up criterion for the local strong solutions in terms of the temperature and positive density, similar to the Beale-Kato-Majda criterion for ideal incompressible flows.     

A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations

In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L^\frac21-r( 0,T;[(X)\dot]_r( \mathbbR³) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.     

On the weak solution of the Neumann problem for the 2D Helmholtz equation in a convex cone and Hs regularity

We extend previous results for the Neumann boundary value problem to the case of boundary data from the space $H^{-\frac{1}{2}+\varepsilon}(\Gamma), 0{<}{\varepsilon}{<}\frac{1}{2}We extend previous results for the Neumann boundary value problem to the case of boundary data from the space $H^{-\frac{1}{2}+\varepsilon}(\Gamma), 0{<}{\varepsilon}{<}\frac{1}{2}$, where Γ=?Ω is the boundary of a two‐dimensional cone Ω with angle β<π. We prove that for these boundary conditions the solution of the Helmholtz equation in Ω exists in the Sobolev space H^1+ε(Ω), is unique and depends continuously on the boundary data. Moreover, we give the Sommerfeld representation for these solutions. This can be used to formulate explicit compatibility conditions on the data for regularity properties of the corresponding solution. Copyright © 2010 John Wiley & Sons, Ltd.     

Carleman estimates for degenerate parabolic operators with applications to null controllability

We prove an estimate of Carleman type for the one dimensional heat equation $$ u_t - \left( {a\left( x \right)u_x } \right)_x + c\left( {t,x} \right)u = h\left( {t,x} \right),\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where a(·) is degenerate at 0. Such an estimate is derived for a special pseudo-convex weight function related to the degeneracy rate of a(·). Then, we study the null controllability on [0, 1] of the semilinear degenerate parabolic equation $$ u_t - \left( {a\left( x \right)u_x } \right)_x + f\left( {t,x,u} \right) = h\left( {t,x} \right)\chi _\omega \left( x \right), $$ where (t, x) ∈(0, T) × (0, 1), ω=(α, β) ⊂⊂ [0, 1], and f is locally Lipschitz with respect to u. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Explicit relation between the solutions of the heat and the Hermite heat equation

There are lots of results on the solutions of the heat equation \frac?u?t = \mathop?ⁿ_i=1\frac?²?x²_iu,\frac{\partial u}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\frac{\partial^2}{\partial x^{2}_{i}}u, but much less on those of the Hermite heat equation \frac?U?t = \mathop?ⁿ_i=1(\frac?²?x²_i - x²_i) U\frac{\partial U}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\left(\frac{\partial^2}{\partial x^{2}_{i}} - x^{2}_{i}\right) U due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005).     

Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation u_t + Lu + a(x) |u|^q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb R^N{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò₀¹ s^-1 (meas\nolimits {x ? W: |a(x)| ￡ s })^q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.     

On the rate of convergence of some difference schemes for second order elliptic equations

The purpose of this paper is to estimate the rate of convergence for some natural difference analogues of Dirichlet's problem for uniformly elliptic differential equations, $$\begin{gathered} \sum\limits_{j,k = 1}^N {\frac{\partial }{{\partial x_j }}} \left( {a_{jk} \frac{{\partial u}}{{\partial x_k }}} \right) = F in R, \hfill \\ u = f on B, \hfill \\ \end{gathered}$$ in aN-dimensional domainR with boundaryB. These schemes will in general not be of positive type, and the analysis will therefore be carried out in discreteL ²-norms rather than in the maximum norm. Since our approximation of the boundary condition is rather crude, we will only arrive at a rate of convergence of first order for smoothF andf. Special emphasis will be put on appraising the dependence of the rate of convergence on the regularity ofF andf.     

On the convergence rate of a recursively defined sequence

Consider the following recursively defined sequence: $\tau _1 = 1,\sum\limits_{j = 1}^n {\frac{1} {{\sum\nolimits_{s = j}^n {\tau _s } }}} = 1forn \geqslant 2, $ , which originates from a heat conduction problem first studied by Myshkis (1997). Chang, Chow, and Wang (2003) proved that $\tau _n = \log n + O(1) for large n.$ . In this note, we refine this result to $\tau _n = \log n + \gamma + O\left( {\frac{1} {{\log n}}} \right). $ . where γ is the Euler constant.     

Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem

We consider the space A(\mathbbT)A(\mathbb{T}) of all continuous functions f on the circle \mathbbT\mathbb{T} such that the sequence of Fourier coefficients [^(f)] = { [^(f)]( k ), k ? \mathbbZ }\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\} belongs to l ¹(ℤ). The norm on A(\mathbbT)A(\mathbb{T}) is defined by || f ||_A(\mathbbT) = || [^(f)] ||_{l¹ (\mathbbZ)}\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}. According to the well-known Beurling-Helson theorem, if f:\mathbbT ? \mathbbT\phi :\mathbb{T} \to \mathbb{T} is a continuous mapping such that || e^inf ||_A(\mathbbT) = O(1)\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1), n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that || e^inf ||_A(\mathbbT) = o( log| n | )\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right). We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right), then φ is linear.     

Study of the classical solution to the one-dimensional mixed problem for a class of semilinear long-wave equations

Many problems in mathematical physics are reduced to one- or multidimensional initial and initial-boundary value problems for, generally speaking, strongly nonlinear Sobolev-type equations. In this work, local and global classical solvability is studied for the one-dimensional mixed problem with homogeneous Riquier-type boundary conditions for a class of semilinear long-wave equations

Convergence rates of the LIL for random fields in Hilbert spaces

Considering the positive d-dimensional lattice point Z ₊ ^d (d ≥ 2) with partial ordering ≤, let {X _k: k ∈ Z ₊ ^d } be i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with mean zero and covariance operator Σ, and set $ S_n = \sum\limits_{k \leqslant n} {X_k } $ S_n = \sum\limits_{k \leqslant n} {X_k } , n ∈ Z ₊ ^d . Let σ _i ², i ≥ 1, be the eigenvalues of Σ arranged in the non-increasing order and taking into account the multiplicities. Let l be the dimension of the corresponding eigenspace, and denote the largest eigenvalue of Σ by σ ². Let logx = ln(x ∨ e), x ≥ 0. This paper studies the convergence rates for $ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) $ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) . We show that when l ≥ 2 and b > −l/2, E[‖X‖²(log ‖X‖) ^d−2(log log ‖X‖) ^b+4] < ∞ implies $ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} $ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} , where Γ(·) is the Gamma function and $ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } $ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } .     

On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data

We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations

l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      17. Heat content and Brownian motion for some regions with a fractal boundary      M. van den Berg 《Probability Theory and Related Fields》1994,100(4):439-456 LetD be an open, bounded set in euclidean space m (m=2, 3, ...) with boundary D. SupposeD has temperature 0 at timet=0, while D is kept at temperature 1 for allt>0. We use brownian motion to obtain estimates for the solution of corresponding heat equation and to obtain results for the asymptotic behaviour ofE D (t), the amount of heat inD at timet, ast0+. For the triadic von Koch snowflakeK our results imply that       18. Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle      Martin Costabel  Frédérique Le Lou?r 《Integral Equations and Operator Theory》2012,73(1):17-48 We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a penetrable bounded obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The latter are typically bounded on the space of tangential vector fields of mixed regularity T H-\frac12(divG,G){\mathsf T \mathsf H^{-\frac{1}{2}}({\rm div}_{\Gamma},\Gamma)}. Using Helmholtz decomposition, we can base their analysis on the study of pseudo-differential integral operators in standard Sobolev spaces, but we then have to study the Gateaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem.      19. On the rates of the other law of the logarithm      Li-Xin Zhang  You-You Chen 《数学学报(英文版)》2012,28(4):781-792 Let X, X1 , X2 , . . . be i.i.d. random variables, and set Sn = X1 +···+Xn , Mn = maxk≤n |Sk|, n ≥1. Let an = o( (n)(1/2)/logn). By using the strong approximation, we prove that, if EX = 0, VarX = σ2 0 and E|X| 2+ε ∞ for some ε 0, then for any r 1, lim ε1/(r-1)(1/2) [ε-2-(r-1)]∞∑n=1 nr-2 P{Mn ≤εσ (π2n/(8log n))(1/2) + an } = 4/π . We also show that the widest a n is o( n(1/2)/logn).      20. Blow-up for semilinear parabolic equations with nonlinear memory   总被引：4，自引：0，他引：4   Li  Yuxiang  Xie  Chunhong 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2004,55(1):15-27 In this paper, we consider the semilinear parabolic equation with homogeneous Dirichlet boundary conditions, where p, q are nonnegative constants. The blowup criteria and the blowup rate are obtained.

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Heat content and Brownian motion for some regions with a fractal boundary

LetD be an open, bounded set in euclidean space ^m (m=2, 3, ...) with boundary D. SupposeD has temperature 0 at timet=0, while D is kept at temperature 1 for allt>0. We use brownian motion to obtain estimates for the solution of corresponding heat equation and to obtain results for the asymptotic behaviour ofE _D (t), the amount of heat inD at timet, ast0⁺. For the triadic von Koch snowflakeK our results imply that

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a penetrable bounded obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The latter are typically bounded on the space of tangential vector fields of mixed regularity T H^-\frac12(div_G,G){\mathsf T \mathsf H^{-\frac{1}{2}}({\rm div}_{\Gamma},\Gamma)}. Using Helmholtz decomposition, we can base their analysis on the study of pseudo-differential integral operators in standard Sobolev spaces, but we then have to study the Gateaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem.     

On the rates of the other law of the logarithm

Let X, X1 , X2 , . . . be i.i.d. random variables, and set Sn = X1 +···+Xn , Mn = maxk≤n |Sk|, n ≥1. Let an = o( (n)(1/2)/logn). By using the strong approximation, we prove that, if EX = 0, VarX = σ2 0 and E|X| 2+ε ∞ for some ε 0, then for any r 1, lim ε1/(r-1)(1/2) [ε-2-(r-1)]∞∑n=1 nr-2 P{Mn ≤εσ (π2n/(8log n))(1/2) + an } = 4/π . We also show that the widest a n is o( n(1/2)/logn).     

Blow-up for semilinear parabolic equations with nonlinear memory

In this paper, we consider the semilinear parabolic equation with homogeneous Dirichlet boundary conditions, where p, q are nonnegative constants. The blowup criteria and the blowup rate are obtained.