共查询到20条相似文献,搜索用时 156 毫秒
1.
Shigenori Yanagi 《Mathematical Methods in the Applied Sciences》1993,16(9):609-624
We study the p-system with viscosity given by vt ? ux = 0, ut + p(v)x = (k(v)ux)x + f(∫ vdx, t), with the initial and the boundary conditions (v(x, 0), u(x,0)) = (v0, u0(x)), u(0,t) = u(X,t) = 0. To describe the motion of the fluid more realistically, many equations of state, namely the function p(v) have been proposed. In this paper, we adopt Planck's equation, which is defined only for v > b(> 0) and not a monotonic function of v, and prove the global existence of the smooth solution. The essential point of the proof is to obtain the bound of v of the form b < h(T) ? v(x, t) ? H(T) < ∞ for some constants h(T) and H(T). 相似文献
2.
We consider a family {u? (t, x, ω)}, ? < 0, of solutions to the equation ?u?/?t + ?Δu?/2 + H (t/?, x/?, ?u?, ω) = 0 with the terminal data u?(T, x, ω) = U(x). Assuming that the dependence of the Hamiltonian H(t, x, p, ω) on time and space is realized through shifts in a stationary ergodic random medium, and that H is convex in p and satisfies certain growth and regularity conditions, we show the almost sure locally uniform convergence, in time and space, of u?(t, x, ω) as ? → 0 to the solution u(t, x) of a deterministic averaged equation ?u/?t + H?(?u) = 0, u(T, x) = U(x). The “effective” Hamiltonian H? is given by a variational formula. © 2007 Wiley Periodicals, Inc. 相似文献
3.
This article presents a semigroup approach to the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(ux) in the quasi‐linear parabolic equation ut(x, t)=(k(ux)ux(x, t))x+F(x, t), with Dirichlet boundary conditions u(0, t)=ψ0, u(1, t)=ψ1 and source function F(x, t). The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[·]: ?? → C1[0, T], Ψ[·]: ?? → C1[0, T] via semigroup theory. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
4.
Fred B Weissler 《Journal of Differential Equations》1984,55(2):204-224
The initial value problem on [?R, R] is considered: ut(t, x) = uxx(t, x) + u(t, x)γu(t, ±R) = 0u(0, x) = ?(x), where ? ? 0 and γ is a fixed large number. It is known that for some initial values ? the solution u(t, x) exists only up to some finite time T, and that ∥u(t, ·)∥∞ → ∞ as t → T. For the specific initial value ? = kψ, where ψ ? 0, ψxx + ψγ = 0, ψ(±R) = 0, k is sufficiently large, it is shown that if x ≠ 0, then limt → Tu(t, x) and limt → Tux(t, x) exist and are finite. In other words, blow-up occurs only at the point x = 0. 相似文献
5.
6.
P. V. Tsynaiko 《Ukrainian Mathematical Journal》1998,50(9):1478-1482
We study a periodic boundary-value problem for the quasilinear equation u
tt
−u
xx
=F[u, u
t
, u
x
], u(x, 0)=u(x, π)=0, u(x + ω, t) = u(x, t), x ∈ ℝ t ∈ [0, π], and establish conditions that guarantee the validity of a theorem on unique solvability.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1293–1296, September, 1998. 相似文献
7.
Uğur Yüksel 《Advances in Applied Clifford Algebras》2010,20(1):201-209
This paper deals with the initial value problem of the type
\frac?u(t,x) ?t = Lu(t,x), u(0,x) = u0(x)\frac{\partial u(t,x)} {\partial t} = {\mathcal{L}}u(t,x), \quad u(0,x) = u_{0}(x) 相似文献
8.
J. Bourgain 《Israel Journal of Mathematics》1992,77(1-2):1-16
We study the almost everythere convergence to the initial dataf(x)=u(x, 0) of the solutionu(x, t) of the two-dimensional linear Schrödinger equation Δu=i? t u. The main result is thatu(x, t) →f(x) almost everywhere fort → 0 iff ∈H p (R2), wherep may be chosen <1/2. To get this result (improving on Vega’s work, see [6]), we devise a strategy to capture certain cancellations, which we believe has other applications in related problems. 相似文献
9.
Dinh Nho Ho 《Mathematical Methods in the Applied Sciences》1992,15(8):537-545
The non-characteristic Cauchy problem for the heat equation uxx(x,t) = u1(x,t), 0 ? x ? 1, ? ∞ < t < ∞, u(0,t) = φ(t), ux(0, t) = ψ(t), ? ∞ < t < ∞ is regularizèd when approximate expressions for φ and ψ are given. Properties of the exact solution are used to obtain an explicit stability estimate. 相似文献
10.
If the longitudinal line method is applied to the Cauchy problem ut = uxx, u(0, x) = u0(x) with a bounded function u0, one is led to a linear initial value problem v¢(t)=A v(t), v(0)=wv'(t)=A v(t),\, v(0)=w in l¥ (\Bbb Z)l^\infty (\Bbb Z). Using Banach limit techniques we study the asymptotic behaviour of the solutions of these problems as t tends to infinity. 相似文献
11.
Patrick S. Hagan 《Studies in Applied Mathematics》1981,64(1):57-88
We consider the class of equations ut=f(uxx, ux, u) under the restriction that for all a,b,c. We first consider this equation over the unbounded domain ? ∞ < x < + ∞, and we show that very nearly every bounded nonmonotonic solution of the form u(t, x)=?(x?ct) is unstable to all nonnegative and all nonpositive perturbations. We then extend these results to nonmonotonic plane wave solutions u(t, x, y)=?(x?ct) of ut = F(uxx, uxy, ux, uy, u). Finally, we consider the class of equations ut=f(uxx, ux, u) over the bounded domain 0 < x < 1 with the boundary conditions u(t, x)=A at x=0 and u(t, x)=B at x=1, and we find the stability of all steady solutions u(t, x)=?(x). 相似文献
12.
Peter Lesky 《Mathematical Methods in the Applied Sciences》1992,15(7):453-468
We study the initial-boundary value problem for ?t2u(t,x)+A(t)u(t,x)+B(t)?tu(t,x)=f(t,x) on [0,T]×Ω(Ω??n) with a homogeneous Dirichlet boundary condition; here A(t) denotes a family of uniformly strongly elliptic operators of order 2m, B(t) denotes a family of spatial differential operators of order less than or equal to m, and u is a scalar function. We prove the existence of a unique strong solution u. Furthermore, an energy estimate for u is given. 相似文献
13.
Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solution approach 总被引:1,自引:0,他引:1
The problem of determining the pair w:={F(x,t);T0(t)} of source terms in the parabolic equation ut=(k(x)ux)x+F(x,t) and Robin boundary condition −k(l)ux(l,t)=v[u(l,t)−T0(t)] from the measured final data μT(x)=u(x,T) is formulated. It is proved that both components of the Fréchet gradient of the cost functional can be found via the same solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is derived. The obtained results permit one to prove existence of a quasi-solution of the considered inverse problem, as well as to construct a monotone iteration scheme based on a gradient method. 相似文献
14.
We consider the periodic boundary-value problem u
tt
− u
xx
= g(x, t), u(0, t) = u(π, t) = 0, u(x, t + ω) = u(x, t). By representing a solution of this problem in the form u(x, t) = u
0(x, t) + ũ(x, t), where u
0(x, t) is a solution of the corresponding homogeneous problem and ũ(x, t) is the exact solution of the inhomogeneous equation such that ũ(x, t + ω) u x = ũ(x, t), we obtain conditions for the solvability of the inhomogeneous periodic boundary-value problem for certain values of the
period ω. We show that the relation obtained for a solution includes known results established earlier.
__________
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 912–921, July, 2005. 相似文献
15.
John Lagnese 《Journal of Functional Analysis》1973,13(3):302-316
Let V?, W?, W and X be Hilbert spaces (0 < ? ? 1) with V? ? W? ? W ? X algebraically and topologically, each space being dense in the one that follows it. For each t? [0, T] let a?(t; u, v), b?(t; u, v) and b(t; u, v) be continuous sesqui-linear forms on V?, W? and W, respectively, which satisfy certain ellipticity conditions. Consider the two equations a?(t; u?, v) + b?(t; u?, v) = 〈f?, v〉 (v?V?) and (u′, v)x + b(t; u, v) = 〈f, v〉 (v?W). Estimates are obtained on the rate of convergence of u? to u, assuming a?(t; u, v) → (u, v)x and b?(t; u, v) → b(t; u, v) in an appropriate sense. These results are then applied to singular perturbation of a class of parabolic boundary value problems. 相似文献
16.
Ebru Ozbilge 《Mathematical Methods in the Applied Sciences》2008,31(11):1333-1344
In this article, a semigroup approach is presented for the mathematical analysis of the inverse coefficient problems of identifying the unknown diffusion coefficient k(u(x, t)) in the quasi‐linear parabolic equation ut(x, t)=(k(u(x, t))ux(x, t))x, with Dirichlet boundary conditions ux(0, t)=ψ0, u(1, t)=ψ1. The main purpose of this work is to analyze the distinguishability of the input–output mappings Φ[·] : ??→C1[0, T], Ψ[·] : ??→C1[0, T] using semigroup theory. In this article, it is shown that if the null space of semigroups T(t) and S(t) consists of only a zero function, then the input–output mappings Φ[·] and Ψ[·] have the distinguishability property. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
17.
《Applied Mathematics Letters》2002,15(6):727-734
This paper is concerned with the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws ut + f(u)x = uxx on [0, 1], with the boundary condition u(0, t) = u−(t) → u−, u(1, t) = u+(t) → u+, as t → +∞ and the initial data u(x,0) = u0(x) satisfying u0(0) = u−(0), u0(1) = u+(1), where u± are given constants, u− ≠ u+ and f is a given function satisfying f″(u) > 0 for u under consideration. By means of an elementary energy estimates method, both the global existence and the asymptotic behavior are obtained. When u− 〈 u+, which corresponds to rarefaction waves in inviscid conservation laws, no smallness conditions are needed. While for u− > u+, which corresponds to shock waves in inviscid conservation laws, it is established for weak shock waves, that is, |u−−u+| is small. Moreover, when u±(t) ≡ u±, t ≥ 0, exponential decay rates are both obtained. 相似文献
18.
As shown by an example, the integral function f :
n
, defined by f(x) = a
b[B(x, t)]+
g(t) dt, may not be a strongly semismooth function, even if g(t) 1 and B is a quadratic polynomial with respect to t and infinitely many times smooth with respect to x. We show that f is a strongly semismooth function if g is continuous and B is affine with respect to t and strongly semismooth with respect to x, i.e., B(x, t) = u(x)t + v(x), where u and v are two strongly semismooth functions in
n
. We also show that f is not a piecewise smooth function if u and v are two linearly independent linear functions, g is continuous and g 0 in [a, b], and n 2. We apply the first result to the edge convex minimum norm network interpolation problem, which is a two-dimensional interpolation problem. 相似文献
19.
Consider the Cauchy problem ∂u(x, t)/∂t = ℋu(x, t) (x∈ℤd, t≥ 0) with initial condition u(x, 0) ≡ 1 and with ℋ the Anderson Hamiltonian ℋ = κΔ + ξ. Here Δ is the discrete Laplacian, κ∈ (0, ∞) is a diffusion constant,
and ξ = {ξ(x): x∈ℤ
d
} is an i.i.d.random field taking values in ℝ. G?rtner and Molchanov (1990) have shown that if the law of ξ(0) is nondegenerate,
then the solution u is asymptotically intermittent.
In the present paper we study the structure of the intermittent peaks for the special case where the law of ξ(0) is (in the
vicinity of) the double exponential Prob(ξ(0) > s) = exp[−e
s
/θ] (s∈ℝ). Here θ∈ (0, ∞) is a parameter that can be thought of as measuring the degree of disorder in the ξ-field. Our main result
is that, for fixed x, y∈ℤ
d
and t→∈, the correlation coefficient of u(x, t) and u(y, t) converges to ∥w
ρ∥−2
ℓ2Σz ∈ℤd
w
ρ(x+z)w
ρ(y+z). In this expression, ρ = θ/κ while w
ρ:ℤd→ℝ+ is given by w
ρ = (v
ρ)⊗
d
with v
ρ: ℤ→ℝ+ the unique centered ground state (i.e., the solution in ℓ2(ℤ) with minimal l
2-norm) of the 1-dimensional nonlinear equation Δv + 2ρv log v = 0. The uniqueness of the ground state is actually proved only for large ρ, but is conjectured to hold for any ρ∈ (0, ∞).
empty
It turns out that if the right tail of the law of ξ(0) is thicker (or thinner) than the double exponential, then the correlation
coefficient of u(x, t) and u(y, t) converges to δ
x, y
(resp.the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation
structure.
Received: 5 March 1997 / Revised version: 21 September 1998 相似文献
20.
The system of equations (f (u))t − (a(u)v + b(u))x = 0 and ut − (c(u)v + d(u))x = 0, where the unknowns u and v are functions depending on
, arises within the study of some physical model of the flow of miscible fluids in a porous medium. We give a definition for
a weak entropy solution (u, v), inspired by the Liu condition for admissible shocks and by Krushkov entropy pairs. We then prove, in the case of a natural
generalization of the Riemann problem, the existence of a weak entropy solution only depending on x/t. This property results from the proof of the existence, by passing to the limit on some approximations, of a function g such that u is the classical entropy solution of ut − ((cg + d)(u))x = 0 and simultaneously w = f (u) is the entropy solution of wt − ((ag + b)(f(−1)(w)))x
= 0. We then take v = g(u), and the proof that (u, v) is a weak entropy solution of the coupled problem follows from a linear combination of the weak entropy inequalities satisfied
by u and f (u). We then show the existence of an entropy weak solution for a general class of data, thanks to the convergence proof of
a coupled finite volume scheme. The principle of this scheme is to compute the Godunov numerical flux with some interface
functions ensuring the symmetry of the finite volume scheme with respect to both conservation equations. 相似文献
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