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1.
Let u, p be a weak solution of the stationary Navier-Stokes equations in a bounded domain N, 5N . If u, p satisfy the additional conditions
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2.
Singular perturbations of two-point boundary problems for systems of ordinary differential equations
W. A. Harris Jr. 《Archive for Rational Mechanics and Analysis》1960,5(1):212-225
Asymptotic solutions of linear systems of ordinary differential equations are employed to discuss the relationship of the solution of a certain “complete” boundary problem.
$$\begin{gathered} \left\{ \begin{gathered} {\text{ }}\frac{{d{\text{ }}x_1 }}{{d{\text{ }}t}} = A_{11} (t,\varepsilon ){\text{ }}x_1 (t,\varepsilon ){\text{ }} + \cdots + A_{1p} (t,\varepsilon ){\text{ }}x_p (t,\varepsilon ) \hfill \\ \varepsilon ^{h_2 } \frac{{d{\text{ }}x_2 }}{{d{\text{ }}t}} = A_{21} (t,\varepsilon ){\text{ }}x_1 (t,\varepsilon ){\text{ }} + \cdots + A_{2p} (t,\varepsilon ){\text{ }}x_p (t,\varepsilon ) \hfill \\ {\text{ }} \vdots {\text{ }} \vdots {\text{ }} \vdots \hfill \\ \varepsilon ^{h_p } \frac{{d{\text{ }}x_2 }}{{d{\text{ }}t}} = A_{p1} (t,\varepsilon ){\text{ }}x_1 (t,\varepsilon ){\text{ }} + \cdots + A_{pp} (t,\varepsilon ){\text{ }}x_p (t,\varepsilon ) \hfill \\ \end{gathered} \right\} \hfill \\ {\text{ }}R(\varepsilon ){\text{ }}x(a,{\text{ }}\varepsilon ){\text{ }} + {\text{ }}S(\varepsilon ){\text{ }}x(b,{\text{ }}\varepsilon ) = c(\varepsilon ){\text{ }} \hfill \\ \end{gathered}$$ 相似文献
3.
In this paper, exact solutions are constructed for stationary election beams that are degenerate in the Cartesian (x,y,z), axisymmetric (r,θ,z), and spiral (in the planes y=const (u,y,v)) coordinate systems. The degeneracy is determined by the fact that at least two coordinates in such a solution are cyclic or are integrals of motion. Mainly, rotational beams are considered. Invariant solutions for beams in which the presence of vorticity resulted in a linear dependence of the electric-field potential ? on the above coordinates were considered in [1], In degenerate solutions, the presence of vorticity results in a quadratic or more complex dependence of the potential on the coordinates that are integrals of motion. In [2] and in a number of papers referred to in [2], the degenerate states of irrotational beams are described. The known degenerate solutions for rotational beams apply to an axisymmetric one-dimensional (r) beam with an azimuthal velocity component [3] and to relativistic conical flow [1]. The equations used below follow from the system of electron hydrodynamic equations for a stationary relativistic beam $$\begin{array}{*{20}c} {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}\left[ {\sqrt \gamma g^{\beta \beta } g^{\alpha \alpha } \left( {\frac{{\partial A_\alpha }}{{\partial q^\beta }} - \frac{{\partial A_\beta }}{{\partial q^\alpha }}} \right)} \right]} = 4\pi \rho \sqrt \gamma g^{\alpha \alpha } u_\alpha ,} \\ {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}\left( {\sqrt \gamma g^{\beta \beta } \frac{{\partial \varphi }}{{\partial q^\beta }}} \right)} = 4\pi \rho \sqrt {\gamma u} ,\sum\limits_{\beta = 1}^3 {g^{\beta \beta } u_\beta ^2 + 1 = u^2 } } \\ \begin{gathered} \frac{\eta }{c}u\frac{{\partial \mathcal{E}}}{{\partial q^\alpha }} = \sum\limits_{\beta = 1}^3 {g^{\beta \beta } u_\beta } \left( {\frac{{\partial p_\beta }}{{\partial q^\alpha }} - \frac{{\partial p_\alpha }}{{\partial q^\beta }}} \right), \hfill \\ \begin{array}{*{20}c} {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}(\sqrt \gamma g^{\beta \beta } \rho u_\beta ) = 0,u \equiv \frac{\eta }{{c^2 }}(\varphi + \mathcal{E}) + 1,} } \\ {cu_\alpha \equiv \frac{\eta }{c}A_\alpha + p_\alpha ,\alpha ,\beta = 1,2,3,\gamma \equiv g_{11} g_{22} g_{33} } \\ \end{array} \hfill \\ \end{gathered} \\ \end{array} $$ where qβ denotes orthogonal coordinates with the metric tensor gββ (β=1,2,3); Aα is the magnetic potential; Aα = (uα/u)c is the electron velocity; ρ is the scalar space-charge density (ρ > 0); is the energy in eV; pα is the generalized momentum of an electron per unit mass; η is the electron charge-mass ratio. 相似文献
4.
K. I. Kim 《Journal of Applied Mechanics and Technical Physics》1968,9(1):20-23
In a lightly ionized plasma, charged-particle drift due to collisions with neutral atoms occurs at different velocities: $$\begin{array}{*{20}c} {v_{Ea} = \mp \frac{{b_a E}}{{1 + (\omega _a \tau _a )^2 }},v_{ \bot a} = \frac{{b_a E(\omega _a \tau _a )}}{{1 + (\omega _a \tau _a )^2 }}} \\ {\left( {b_a = \frac{{|e|\tau _a }}{{m_a }},\omega _a = \frac{{|e|\tau _a }}{{m_a }}} \right),} \\ \end{array} $$ where ba is the mobility of particles of the type a;ωa is the Larmor frequency; the upper sign refers to electrons and the lower sign to ions. A difference in the charged-particle drift velocities can cause instability of an inhomogeneous lightly ionized plasma. Let us consider the following example. Assume that in the initial state of the plasma there is a concentration gradient along the x-axis, that the external electric field is directed along the x-axis, and that the magnetic field coincides with the z-axis. In this system, under the influence of a Lorentz force the charged particles will move in a direction opposite to the y-axis. Since electrons have a higher velocity than ions, an electric field is induced in this direction. This electric field, together with the magnetic field, causes particle drift in the negative direction of the x-axis. Consequently, if the concentration gradient in the initial state is directed opposite to the x-axis this state cannot be stable. Instability of this kind has been examined by Simon [1]. On the basis of studies by Kadomtsev and Nedospasov [2], as well as by Rosenbluth and Longmire [3], Simon developed a theory of instability of a lightly ionized plasma in crossed fields with an inhomogeneous density distribution in the direction of the external electric field. Somewhat later, Simon's theory was developed [4]. In devices with inhomogeneous plasma flow in which the plasma (conducting) layers alternate with nonconducting layers, the external electric field and concentration are normal to one another. We shall bear this case in mind below and shall examine the instability of a lightly ionized plasma in crossed fields when the concentration inhomogeneity is in a direction perpendicular to the external electric field. 相似文献
5.
At the clamped edge of a thin plate, the interior transverse deflection ω(x 1, x2) of the mid-plane x 3=0 is required to satisfy the boundary conditions ω=?ω/?n=0. But suppose that the plate is not held fixed at the edge but is supported by being bonded to another elastic body; what now are the boundary conditions which should be applied to the interior solution in the plate? For the case in which the plate and its support are in two-dimensional plane strain, we show that the correct boundary conditions for ω must always have the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakqaabeqaaiaabEhacaqGTaWa% aSaaaeaacaGG0aGaae4vamaaCaaaleqabaGaamOqaaaaaOqaaiaaco% dadaqadaqaaiaacgdacqGHsislcaqG2baacaGLOaGaayzkaaaaaiaa% bIgadaahaaWcbeqaaiaackdaaaGcdaWcaaqaaiaabsgadaahaaWcbe% qaaiaackdaaaGccaqG3baabaGaaeizaiaabIhafaqabeGabaaajaaq% baqcLbkacaGGYaaajaaybaqcLbkacaGGXaaaaaaakiabgUcaRmaala% aabaGaaiinaiaabEfadaahaaWcbeqaaiaadAeaaaaakeaacaGGZaWa% aeWaaeaacaGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGOb% WaaWbaaSqabeaacaGGZaaaaOWaaSaaaeaacaqGKbWaaWbaaSqabeaa% caGGZaaaaOGaae4DaaqaaiaabsgacaqG4bqcaaubaeqabiqaaaqcaa% saaiaacodaaKaaafaajugGaiaacgdaaaaaaOGaeyypa0Jaaiimaiaa% cYcaaeaadaWcaaqaaiaabsgacaqG3baabaGaaeizaiaabIhaliaacg% daaaGccqGHsisldaWcaaqaaiaacsdacqqHyoqudaahaaWcbeqaaiaa% bkeaaaaakeaacaGGZaWaaeWaaeaacaGGXaGaeyOeI0IaaeODaaGaay% jkaiaawMcaaaaacaqGObWaaSaaaeaacaqGKbWaaWbaaSqabeaacaGG% YaaaaOGaae4DaaqaaiaabsgacaqG4bqbaeqabiqaaaqcaauaaKqzGc% GaaiOmaaqcaawaaKqzGcGaaiymaaaaaaGccqGHRaWkdaWcaaqaaiaa% csdacqqHyoqudaahaaWcbeqaaiaabAeaaaaakeaacaGGZaWaaeWaae% aacaGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGObWaaWba% aSqabeaacaGGYaaaaOWaaSaaaeaacaqGKbWaaWbaaSqabeaacaGGZa% aaaOGaae4DaaqaaiaabsgacaqG4bqcaaubaeqabiqaaaqcaasaaiaa% codaaKaaafaajugGaiaacgdaaaaaaOGaeyypa0JaaiimaiaacYcaaa% aa!993A!\[\begin{gathered}{\text{w - }}\frac{{4{\text{W}}^B }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^2 \frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} + \frac{{4{\text{W}}^F }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^3 \frac{{{\text{d}}^3 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}3 \\1 \\\end{array} }} = 0, \hfill \\\frac{{{\text{dw}}}}{{{\text{dx}}1}} - \frac{{4\Theta ^{\text{B}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}\frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} + \frac{{4\Theta ^{\text{F}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^2 \frac{{{\text{d}}^3 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}3 \\1 \\\end{array} }} = 0, \hfill \\\end{gathered}\]with exponentially small error as L/h→∞, where 2h is the plate thickness and L is the length scale of ω in the x 1-direction. The four coefficients W B, WF, Θ B , Θ F are computable constants which depend upon the geometry of the support and the elastic properties of the support and the plate, but are independent of the length of the plate and the loading applied to it. The leading terms in these boundary conditions as L/h→∞ (with all elastic moduli remaining fixed) are the same as those for a thin plate with a clamped edge. However by obtaining asymptotic formulae and general inequalities for Θ B , W F, we prove that these constants take large values when the support is ‘soft’ and so may still have a strong influence even when h/L is small. The coefficient W F is also shown to become large as the size of the support becomes large but this effect is unlikely to be significant except for very thick plates. When h/L is small, the first order corrected boundary conditions are w=0,% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakeaadaWcaaqaaiaabsgacaqG% 3baabaGaaeizaiaabIhaliaacgdaaaGccqGHsisldaWcaaqaaiaacs% dacqqHyoqudaahaaWcbeqaaiaabkeaaaaakeaacaGGZaWaaeWaaeaa% caGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGObWaaSaaae% aacaqGKbWaaWbaaSqabeaacaGGYaaaaOGaae4DaaqaaiaabsgacaqG% 4bqbaeqabiqaaaqcaauaaKqzGcGaaiOmaaqcaawaaKqzGcGaaiymaa% aaaaGccqGH9aqpcaGGWaGaaiilaaaa!5DD4!\[\frac{{{\text{dw}}}}{{{\text{dx}}1}} - \frac{{4\Theta ^{\text{B}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}\frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} = 0,\]which correspond to a hinged edge with a restoring couple proportional to the angular deflection of the plate at the edge. 相似文献
6.
Ref. [1] discussed the existence of positive solutions of quasilinear two-point boundary problems: but it restricts O相似文献
7.
Philippe G. Ciarlet Véronique Lods Bernadette Miara 《Archive for Rational Mechanics and Analysis》1996,136(2):163-190
We consider as in Part I a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R 3, whereω?R 2 is a bounded and connected open set with a Lipschitz-continuous boundary, and?∈l 3 (?;R 3). The shells are clamped on a portion of their lateral face, whose middle line is?(γ 0), whereγ 0 is any portion of?ω withlength γ 0>0. We make an essential geometrical assumption on the middle surfaceS and on the setγ 0, which states that the space of inextensional displacements $$\begin{gathered} V_F (\omega ) = \{ \eta = (\eta _i ) \in H^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \hfill \\ \eta _i = \partial _v \eta _3 = 0 on \gamma _0 ,\gamma _{\alpha \beta } (\eta ) = 0 in \omega \} , \hfill \\ \end{gathered}$$ where $\gamma _{\alpha \beta }$ (η) are the components of the linearized change is metric tensor ofS, contains non-zero functions. This assumption is satisfied in particular ifS is a portion of cylinder and?(γ 0) is contained in a generatrix ofS. We show that, if the applied body force density isO(? 2) with respect to?, the fieldu(?)=(u i (?)), whereu i (?) denote the three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity, once “scaled” so as to be defined over the fixed domain Ω=ω×]?1, 1[, converges as?→0 inH 1(Ω) to a limitu, which is independent of the transverse variable. Furthermore, the averageζ=1/2ts ?1 1 u dx 3, which belongs to the spaceV F (ω), satisfies the (scaled) two-dimensional equations of a “flexural shell”, viz., $$\frac{1}{3}\mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta )\rho _{\alpha \beta } (\eta )\sqrt {a } dy = \mathop \smallint \limits_\omega \left\{ {\mathop \smallint \limits_{ - 1}^1 f^i dx_3 } \right\} \eta _i \sqrt {a } dy$$ for allη=(η i ) ∈V F (ω), where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor of the surfaceS, $$\begin{gathered} \rho _{\alpha \beta } (\eta ) = \partial _{\alpha \beta } \eta _3 - \Gamma _{\alpha \beta }^\sigma \partial _\sigma \eta _3 + b_\beta ^\sigma \left( {\partial _\alpha \eta _\sigma - \Gamma _{\alpha \sigma }^\tau \eta _\tau } \right) \hfill \\ + b_\alpha ^\sigma \left( {\partial _\beta \eta _\sigma - \Gamma _{\beta \sigma }^\tau \eta _\tau } \right) + b_\alpha ^\sigma {\text{|}}_\beta \eta _\sigma - c_{\alpha \beta } \eta _3 \hfill \\ \end{gathered} $$ are the components of the linearized change of curvature tensor ofS, $\Gamma _{\alpha \beta }^\sigma$ are the Christoffel symbols ofS, $b_\alpha ^\beta$ are the mixed components of the curvature tensor ofS, andf i are the scaled components of the applied body force. Under the above assumptions, the two-dimensional equations of a “flexural shell” are therefore justified. 相似文献
8.
We consider as in Parts I and II a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R 3, whereω?R 2 is a bounded and connected open set with a Lipschitz-continuous boundary, and? ∈ ?3 (?;R 3). The shells are clamped on a portion of their lateral face, whose middle line is?(γ 0), whereγ 0 is a portion of?ω withlength γ 0>0. For all?>0, let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $u_i^\varepsilon g^{i,\varepsilon }$ of the points of the shell, obtained by solving the three-dimensional problem; let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $\zeta _i^\varepsilon$ a i of the points of the middle surfaceS, obtained by solving the two-dimensional model ofW.T. Koiter, which consists in finding $$\zeta ^\varepsilon = \left( {\zeta _i^\varepsilon } \right) \in V_K (\omega ) = \left\{ {\eta = (\eta _\iota ) \in {\rm H}^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \eta _i = \partial _v \eta _3 = 0 on \gamma _0 } \right\}$$ such that $$\begin{gathered} \varepsilon \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta ^\varepsilon )\gamma _{\alpha \beta } (\eta )\sqrt a dy + \frac{{\varepsilon ^3 }}{3} \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta ^\varepsilon )\rho _{\alpha \beta } (\eta )\sqrt a dy \hfill \\ = \mathop \smallint \limits_\omega p^{i,\varepsilon } \eta _i \sqrt a dy for all \eta = (\eta _i ) \in V_K (\omega ), \hfill \\ \end{gathered}$$ where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor ofS, $\gamma _{\alpha \beta }$ (η) and $\rho _{\alpha \beta }$ (η) are the components of the linearized change of metric and change of curvature tensors ofS, and $p^{i,\varepsilon }$ are the components of the resultant of the applied forces. Under the same assumptions as in Part I, we show that the fields $\frac{1}{{2_\varepsilon }}\smallint _{ - \varepsilon }^\varepsilon u_i^\varepsilon g^{i,\varepsilon } dx_3^\varepsilon$ and $\zeta _i^\varepsilon$ a i , both defined on the surfaceS, have the same principal part as? → 0, inH 1 (ω) for the tangential components, and inL 2(ω) for the normal component; under the same assumptions as in Part II, we show that the same fields again have the same principal part as? → 0, inH 1 (ω) for all their components. For “membrane” and “flexural” shells, the two-dimensional model ofW.T. Koiter is therefore justified. 相似文献
9.
F. S. Churikov 《Fluid Dynamics》1966,1(3):70-71
It is known that the nonlinear system of equations of plane steady isentropic potential gas flow can be linearized and transformed to a single equivalent linear differential equation of second order. For the case of a perfect gas this equation has the form [1]
$$\begin{gathered} \frac{{1 - \tau ^2 }}{{\tau ^2 (1 - \alpha \tau ^2 )}} \frac{{\partial ^2 \Phi }}{{\partial \theta ^2 }} + \frac{{\partial ^2 \Phi }}{{\partial \tau ^2 }} + \frac{{\tau (1 - \tau ^2 )}}{{\tau ^2 (1 - \alpha \tau ^2 )}} \frac{{\partial \Phi }}{{\partial \tau }} = 0, \hfill \\ (\tau = w/c_k , w = \sqrt {u^2 + \upsilon ^2 } , \alpha = (\gamma - 1)/(\gamma + 1); \gamma = c_p /c_\upsilon ). (0.1) \hfill \\ \end{gathered} $$ 相似文献
10.
W. Czernous 《Nonlinear Oscillations》2011,13(4):595-612
We consider the initial boundary-value problem for a system of quasilinear partial functional differential equations of the
first order
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