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Theorem 21.31 Let \( A \) be an \( \mathrm{{SL}}\left( {n,\mathbb{Z}}\right) \) operator with matrix \( M \) in some integer basis. Suppose that the characteristic polynomial of \( A \) is irreducible over \( \mathbb{Q} \) . Let \( U \) be a fundamental domain of the Klein-Voronoi continued fractions for \( A \) (see D... | Let us prove Theorem 21.31(i). By Proposition 21.28 there exists a nonzero constant \( \alpha \) such that in the system of coordinates \( O{X}_{1}{X}_{2}\ldots {X}_{k}{Y}_{1}{Z}_{1}{Y}_{2}{Z}_{2}\ldots {Y}_{l}{Z}_{l} \) the MD-characteristic \( {\Phi }_{A} \) is expressed as follows:\n\n\[ \alpha \left| {\mathop{\prod... | Yes |
Let us consider the example of the operator \( A \) defined by the matrix \[ \left( \begin{matrix} - 2 & - 4 & - 3 \\ 1 & 2 & 2 \\ - 1 & - 1 & 3 \end{matrix}\right) \] The characteristic polynomial of this operator has three distinct real roots. Therefore, the Klein-Voronoi continued fraction consists of eight sails. T... | The MD-characteristic of these vectors are respectively \( 1,2 \), and 4 . So the minimum of the MD-characteristic (which is 1 in this case) is attained on the vertices of the orbit of the Dirichlet group containing \( \left( {0,0,1}\right) \) . Therefore, there exists a unique \( \varsigma \) -reduced Hessenberg matri... | Yes |
Proposition 21.37 The sequence \( {\alpha }_{1},{\alpha }_{2},{\alpha }_{3},\ldots \) is strictly increasing. | Proof of Proposition 21.37 Consider the coordinate system\n\n\[ O{X}_{1}{X}_{2}\ldots {X}_{k}{Y}_{1}{Z}_{1}{Y}_{2}{Z}_{2}\ldots {Y}_{l}{Z}_{l} \]\n\nintroduced in Sect. 21.3.1.1 above. In these coordinates the form \( {\Delta }_{A} \) is written as follows:\n\n\[ {\Delta }_{A}\left( {{x}_{1},\ldots ,{x}_{n}}\right) = c... | Yes |
Corollary 21.39 For an arbitrary integer \( w,{\alpha }_{w} \geq w \) . | Proof The convexity of a Markov-Davenport form on the intersection of cones \( {C}_{i} \) with the plane \( {\pi }_{ + } \) implies that the minimum \( {\alpha }_{w} \) is attained at an integer vertex of the \( {KVC}{F}_{w}\left( A\right) \) . Therefore, \( {\alpha }_{w} \) is a positive integer for every \( w \geq 0 ... | Yes |
Theorem 21.43 For every positive \( \varepsilon \) there exists \( R > 0 \) such that the following inclusions hold:\n\n\[{\Lambda }_{\varepsilon } \smallsetminus {B}_{R}\left( O\right) \subset {NRS}\left( \Omega \right) \smallsetminus {B}_{R}\left( O\right) \subset {\Lambda }_{-\varepsilon } \smallsetminus {B}_{R}\lef... | We give the proof of this theorem in Sect. 21.6.5 below. | No |
Lemma 21.50 The planar curve \( {\operatorname{Discr}}_{\langle 0,1 \mid 0,0,1\rangle }^{\left( 1,0,0\right) }\left( {m, n}\right) = 0 \) is contained in the domain defined by the inequalities \[ \left\{ \begin{array}{l} \left( {{m}^{2} - {4n} + 3}\right) \left( {{n}^{2} + {4m} + 3}\right) \geq 0, \\ \left( {{m}^{2} - ... | Proof of Lemma 21.50 Note that \[ {\operatorname{Discr}}_{\langle 0,1 \mid 0,0,1\rangle }^{\left( 1,0,0\right) }\left( {m, n}\right) = \left( {{m}^{2} - {4n}}\right) \left( {{n}^{2} + {4m}}\right) - {2mn} - {27}. \] Thus, we have \[ {\operatorname{Discr}}_{\langle 0,1 \mid 0,0,1\rangle }^{\left( 1,0,0\right) }\left( {m... | Yes |
For every \( \Omega = \left\langle {{a}_{11},{a}_{21} \mid {a}_{12},{a}_{22},{a}_{32}}\right\rangle \) there exists an affine (not necessarily integer) transformation of the plane OMN taking the curve \( {\operatorname{Discr}}_{\Omega }^{v}\left( {m, n}\right) = 0 \) to the curve \( {\operatorname{Discr}}_{\langle 0,1 ... | Proof Let \( {H}_{\Omega }^{v}\left( {0,0}\right) = \left( {a}_{i, j}\right) \) . Note that every matrix \( {H}_{\Omega }^{v}\left( {m, n}\right) \) is rational conjugate to the matrix\n\n\[ {H}_{\langle 0,1 \mid 0,0,1\rangle }^{\left( 1,0,0\right) }\left( {{m}^{\prime },{n}^{\prime }}\right) \]\n\nwhere\n\n\[ \left\{ ... | Yes |
Proposition 21.52 Let \( p \) be a nonzero integer point. Then there exists a fundamental domain of the Klein-Voronoi continued fraction for \( A \) with all (integer) orbit-vertices contained in the set \( {\Gamma }_{A}^{0}\left( p\right) \) . | The proof is based on the following lemma. Let\n\n\[ \n{\Gamma }_{A}\left( p\right) = \mathop{\bigcup }\limits_{{k \in \mathbb{Z}}}{\Gamma }_{A}^{k}\left( p\right) \n\] | No |
Lemma 21.53 Let \( p \) be a nonzero integer point. Then one of the Klein-Voronoi sails for \( A \) is contained in the set \( {\Gamma }_{A}\left( p\right) \) . | Proof Notice that the set \( {\Gamma }_{A}\left( p\right) \) is a union of orbits. Let us project \( {\Gamma }_{A}\left( p\right) \) to the half-plane \( {\pi }_{ + } \) . The set \( {\Gamma }_{A}\left( p\right) \) projects to the closure of the complement of the convex hull for the points \( {\Gamma }_{A}\left( {{A}^{... | Yes |
Corollary 21.56 Let \( \Omega = \left\langle {{a}_{11},{a}_{21} \mid {a}_{12},{a}_{22},{a}_{32}}\right\rangle \) and let \( {R}_{1,\Omega, v}^{m, n} \) be an NRS-ray. Then for every \( \varepsilon > 0 \) there exists \( C > 0 \) such that for every \( t > C \) the convex hull of the union of two orbit-vertices\n\n\[ \n... | Proof of Corollary 21.56 Define\n\n\[ \nX = \left( \begin{matrix} {a}_{21}{a}_{32} & - {a}_{32}{a}_{11} & {a}_{11}{a}_{22} - {a}_{21}{a}_{12} \\ 0 & {a}_{32} & - {a}_{11} - {a}_{22} \\ 0 & 0 & 1 \end{matrix}\right) .\n\]\n\nThe operator \( X \) defines two linear functions \( {l}_{1} \) and \( {l}_{2} \) on two variabl... | Yes |
Proposition 21.57 Consider an NRS-ray \( {R}_{2,\Omega, v}^{m, n} \) . Then there exists a constant \( C > 0 \) such that for every \( t > C \) all orbit-vertices of one of the fundamental domains for the Klein-Voronoi continued fraction of the matrix \( {R}_{2,\Omega, v}^{m, n}\left( t\right) \) are contained in the s... | The proof of this proposition is based on a corollary of the following lemma. | No |
Lemma 21.58 Let \( {R}_{2,{\Omega }_{0},{v}_{0}}^{m, n} \) be an NRS-ray. Then for every \( \varepsilon > 0 \) there exists \( C > 0 \) such that for every \( t > C \) the convex hull of the union of two orbit-vertices\n\n\[ \n{T}_{{R}_{2,{\Omega }_{0},{v}_{0}}^{m, n}\left( t\right) }\left( {1,0,0}\right) \;\text{ and ... | Proof First, notice that the Klein-Voronoi continued fractions for the operators \( A \) and \( {A}^{-1} \) coincide.\n\nSecond\n\n\[ \n{H}_{\langle 0,1 \mid 0,0,1\rangle }^{\left( 1,0,0\right) }\left( {m, n + t}\right) = X{H}_{\langle 0,1 \mid 0,0,1\rangle }^{\left( 1,0,0\right) }\left( {-n - t, - m}\right) {X}^{-1}, ... | No |
Problem 31 What is the percentage of \( \varsigma \) -reduced matrices among matrices of a given Hessenberg type \( \Omega \) ? | It is likely that almost all Hessenberg matrices are \( \varsigma \) -reduced (except for some measure-zero subset). | No |
Proposition 22.7 The Markov-Davenport form of an MCRS-group \( \mathfrak{A} \) is defined by \( \mathfrak{A} \) up to a sign. | Proof All regular operators of \( \mathfrak{A} \) have the same set of eigenlines. Hence the forms \( {L}_{i} \) are uniquely defined by the MCRS-group up to a multiplication by a scalar and permutations. Every scalar multiplication is normalized by the determinant in the denominator, while a sign of a permutation is n... | "No" |
Proposition 22.8 All the coefficients of the Markov-Davenport form are either simultaneously real, if there is an even number of minimal invariant planes of the corresponding MCRS-group, or simultaneously complex. | Proof By definition, every MCRS-group contains a real operator with distinct eigenvalues. Linear forms related to real eigenvalues have real coefficients. Pairs of linear forms corresponding to pairs of complex conjugate eigenvalues are complex conjugate up to a multiplicative complex factor. Without loss of generality... | Yes |
Consider the following two operators\n\n\\[ \n\\left( \\begin{matrix} 0 & - 1 \\\\ 1 & 0 \\end{matrix}\\right) \\;\\text{ with eigenvectors }\\left( {I,1}\\right) \\text{ and }\\left( {-I,1}\\right) ,\\]\n\n\\[ \n\\left( \\begin{array}{ll} 1 & 1 \\\\ 4 & 1 \\end{array}\\right) \\;\\text{with eigenvectors}\\left( {1,2}\... | We have\n\n\\[ \n\\left| {{\\Phi }_{{\\mathfrak{A}}_{1}}\\left( v\\right) \\pm {\\Phi }_{{\\mathfrak{A}}_{2}}\\left( v\\right) }\\right| = \\left| {I\\frac{{x}^{2} + {y}^{2}}{2} \\pm \\frac{{y}^{2} - 4{x}^{2}}{4}}\\right|\n\\]\n\nand therefore \\( \\rho \\left( {{\\mathfrak{A}}_{1},{\\mathfrak{A}}_{2}}\\right) = \\frac... | Yes |
Example 23.5 Let us consider an example of a 6-element set \( {S}_{0} \subset {\mathbb{R}}^{3} \) defined as follows\n\n\[ \n{S}_{0} = \left\{ {{s}_{1},{s}_{2},{s}_{3},{s}_{4},{s}_{5},{s}_{6}}\right\} \n\]\n\nwhere\n\n\[ \n{s}_{1} = \left( {3,0,0}\right) ,\;{s}_{2} = \left( {0,3,0}\right) ,\;{s}_{3} = \left( {0,0,3}\ri... | The Minkowski-Voronoi complex contains 5 vertices,6 edges, and 5 faces. Its vertices are\n\n\[ \n{v}_{1} = \left\{ {{s}_{1},{s}_{3},{s}_{4}}\right\} ,\;{v}_{2} = \left\{ {{s}_{3},{s}_{4},{s}_{5}}\right\} ,\;{v}_{3} = \left\{ {{s}_{1},{s}_{4},{s}_{5}}\right\} ,\n\]\n\n\[ \n{v}_{4} = \left\{ {{s}_{2},{s}_{3},{s}_{5}}\rig... | Yes |
Consider the set \( {S}_{0} \) as in Example 23.5. In Fig. 23.1 we show the Minkowski polyhedron (on the left) and the corresponding Minkowski-Voronoi tessellation (on the right). The local minima of the function \( x + y + z \) on the Minkowski polyhedron for \( {S}_{0} \) are the relative minima \( {f}_{1},\ldots ,{f... | The vertices \( {v}_{1},\ldots ,{v}_{5} \) are as follows:\n\n\[ \n{v}_{1} = \left( {3,1,3}\right) ,\;{v}_{2} = \left( {2,2,3}\right) ,\;{v}_{3} = \left( {3,2,2}\right) , \n\] \n\n\[ \n{v}_{4} = \left( {1,3,3}\right) ,\;{v}_{5} = \left( {3,3,1}\right) . \n\] | Yes |
Theorem 23.25 In the above notation the traveler will either exit the pyramid at the rational number\n\n\[ \left\lbrack {0;{a}_{1} : {a}_{2} : \cdots : {a}_{n - 1} : {a}_{n} + 1}\right\rbrack \]\n\nin a finite number of steps or descend to the irrational number\n\n\[ \left\lbrack {0;{a}_{1} : {a}_{2} : \cdots }\right\r... | For instance, in Example 23.24 we exit at point \( \frac{69}{238} = \left\lbrack {0;3 : 2 : 4 : 2 : 3}\right\rbrack \) in finitely many steps. In the example considered in Fig. 23.4 we have the descent sequence \( {LLRR} \) and the exit point \( \frac{3}{7} = \left\lbrack {0;2 : 3}\right\rbrack \) . | No |
Problem 35 Find a natural generalization of the Farey tessellation to higher-dimensional hyperbolic geometry. | In this regard we would like to mention the works [143-147] by V.V. Nikulin on discrete reflection groups in hyperbolic spaces. | No |
Theorem 23.31 The vertices of the triangle \( \bigtriangleup \left( {{a}_{0},\ldots ,{a}_{n}}\right) \) are \( {\widehat{X}}_{n - 1},{\widehat{X}}_{n} \), and \( {X}_{n}\widehat{ + }{X}_{n - 2} \) | As is usual for continued fraction algorithms, the proof is obtained by induction (see the proof in [12]). | No |
Which coordinate bricks are (asymptotically) \( \Pi \) -congruent? | ## Theorem 23.45\n\n(i) Two coordinate bricks of the same dimension are asymptotically \( \Pi \) -congruent if and only if they have the same volume.\n\n(ii) Two relatively rational rectangular coordinate boxes are \( \Pi \) -congruent if and only if they have the same volume. | Yes |
Proposition 23.51 Let \( P = R\left( {{a}_{1},\ldots ,{a}_{n}}\right) \) and \( Q = R\left( {{a}_{1},\ldots ,{a}_{n}}\right) \) be asymptotically \( \Pi \) -congruent coordinate bricks. Then \( P \) and \( Q \) are asymptotically \( \pi \) - congruent, where the hyperplane \( \pi \) is defined by the equation\n\n\[ \n{... | In addition, the decompositions of \( P \) and \( Q \) associated to the bijection \( {\varphi }_{P, Q} \) establish the asymptotic \( \pi \) -congruence. The restriction of \( {\varphi }_{P, Q} \) to parallelepipeds in the decomposition of \( P \) identifies all the translation vectors. | Yes |
We illustrate this with the example of \( a = 9 \) and \( b = {25} \) . The associated decompositions of \( R\left( {9,{25}}\right) \) and of \( R\left( {{25},9}\right) \) consist of two large squares, one average square, three small squares, and two very small squares. This can be read from the regular continued fract... | In Fig. 23.6 we show the layers of equivalent squares as bold rectangles. | No |
Example 23.56 (Jacobi-Perron algorithm ([85, 164, 165])) Historically, the first algorithmic generalization of regular continued fractions is the Jacobi-Perron algorithm (see also [176]). We put\n\n\[ \n\\Delta = \\left\\{ {\\left( {{x}_{0},{x}_{1},\\ldots ,{x}_{n}}\\right) \\mid {x}_{0} \\geq {x}_{i} \\geq 0,1 \\leq i... | In Substep \( {k.3} \) we remember the \( n \) -dimensional vector\n\n\[ \n{a}_{k - 1} = \\left( {\\left\\lfloor \\frac{{x}_{0, k - 1}}{{x}_{1, k - 1}}\\right\\rfloor ,\\left\\lfloor \\frac{{x}_{2, k - 1}}{{x}_{1, k - 1}}\\right\\rfloor ,\\ldots ,\\left\\lfloor \\frac{{x}_{n, k - 1}}{{x}_{1, k - 1}}\\right\\rfloor }\\r... | Yes |
Proposition 23.65 Suppose that a Hessenberg continued fraction for a real number \( x \) is defined by a sequence \( \left( {{a}_{1},{a}_{2},{a}_{3},0,0,\ldots }\right) \) . Then we have | \[ \begin{array}{l} {a}_{2} - \frac{{a}_{3}}{{a}_{1} - \frac{{a}_{2} - \frac{{a}_{3}}{{a}_{1} - \cdots }}{{a}_{1} - \frac{{a}_{2} - \cdots }}{{a}_{1} - \cdots }}}\text{. } \\ x = - {a}_{1} + \frac{{a}_{2} - \frac{{a}_{3}}{{a}_{1} - \cdots }}{{a}_{1} - \frac{{a}_{2} - \cdots }}{{a}_{1} - \cdots }} \\ \end{array} \] | No |
Let \( {u}_{0} \) be a bounded continuous function on \( {\mathbb{R}}^{3} \) . Then the function\n\n\[ u\left( {t, x}\right) = \int {u}_{0}\left( {x - {ty}}\right) W\left( y\right) {dy} \]\n\n(4.3)\n\nis continuous on \( \lbrack 0, + \infty ) \times {\mathbb{R}}^{3} \) and \( {\mathcal{C}}^{\infty } \) on \( \left( {0,... | From the formula (4.3), we see that \( u \) is continuous on \( \lbrack 0, + \infty ) \times {\mathbb{R}}^{3} \) and that \( u\left( {0,\text{.}}\right) = {u}_{0} \) . From the equality, for \( t > 0 \) ,\n\n\[ u\left( {t, x}\right) = {W}_{t} * {u}_{0} = \int {W}_{t}\left( {x - y}\right) u\left( y\right) {dy} \]\n\nwe ... | Yes |
Theorem 4.4\n\nLet \( {\overrightarrow{F}}_{0} \) be a \( {\mathcal{C}}^{2} \) vector field on \( {\mathbb{R}}^{3} \) such that\n\n\[ \mathop{\sup }\limits_{{\left| \alpha \right| \leq 2}}\mathop{\sup }\limits_{{x \in {\mathbb{R}}^{3}}}{\left( 1 + \left| x\right| \right) }^{\beta }\left| {{\partial }^{\alpha }{\overrig... | ## Proof:\n\nWe begin by proving the uniqueness. Let us assume that we have two solutions \( \left( {{\overrightarrow{F}}_{1},{\overrightarrow{H}}_{1}}\right) \) and \( \left( {{\overrightarrow{F}}_{2},{\overrightarrow{H}}_{2}}\right) \) . Since \( {\overrightarrow{H}}_{2} - {\overrightarrow{H}}_{1} \) is irrotational ... | Yes |
Let \( {\overrightarrow{u}}_{0} \) be a \( {\mathcal{C}}^{2} \) vector field on \( {\mathbb{R}}^{3} \) and let \( \overrightarrow{f} \) be a time-dependent vector field such that:\n\n- \( \mathop{\sup }\limits_{{\left| \alpha \right| \leq 2}}\mathop{\sup }\limits_{{x \in {\mathbb{R}}^{3}}}\left| {{\partial }^{\alpha }{... | In Theorem 4.5, the solution \( \left( {\overrightarrow{u}, p}\right) \) is given by\n\n\[ \np\left( {t, x}\right) = - \frac{1}{{4\pi }\left| x\right| } * \operatorname{div}\overrightarrow{f}\left( {t, x}\right) = - \mathop{\sum }\limits_{{j = 1}}^{3}{f}_{j}\left( {t, x}\right) * {\partial }_{j}G \n\]\n\nand\n\n\[ \n\o... | Yes |
Theorem 4.7\n\nLet \( {\overrightarrow{u}}_{0} \) be a \( {\mathcal{C}}^{2} \) vector field on \( {\mathbb{R}}^{3} \) and let \( \overrightarrow{f} \) be a time-dependent vector field such that:\n\n\[ \text{-}\mathop{\sup }\limits_{{\left| \alpha \right| \leq 2}}\mathop{\sup }\limits_{{x \in {\mathbb{R}}^{3}}}\left( {1... | ## Proof:\n\nWe are going to estimate the size of the vector fields \( {\overrightarrow{v}}_{n} \) given by the equations (4.20) and (4.21). First of all, we rewrite the \ | No |
Theorem 4.11\n\nLet \( \\left( {\\overrightarrow{u}, p}\\right) \) be the classical solution of the Navier-Stokes problem on a strip \( \\left\\lbrack {0, T}\\right\\rbrack \\times {\\mathbb{R}}^{3} \), associated to the regular data \( \\left( {{\\overrightarrow{u}}_{0},\\overrightarrow{f}}\\right) \). Assume moreover... | Proof:\n\nWe first write \( \\overrightarrow{u} \) as a solution of a Stokes system\n\n\[ \n\\overrightarrow{u} = {W}_{\\nu t} * {\\overrightarrow{u}}_{0} + {\\int }_{0}^{t}\\mathcal{O}\\left( {\\nu \\left( {t - s}\\right) }\\right) : : \\overrightarrow{g}\\left( s\\right) {ds}\n\]\n\nwith forcing term\n\n\[ \n\\overri... | Yes |
Let \( B \) be a bounded bilinear operator on a Banach space \( E \): \[ \parallel B\left( {u, v}\right) {\parallel }_{E} \leq {C}_{0}\parallel u{\parallel }_{E}\parallel v{\parallel }_{E} \] Then, when \( {\begin{Vmatrix}{u}_{0}\end{Vmatrix}}_{E} \leq \frac{1}{4{C}_{0}} \), the equation \[ u = {u}_{0} + B\left( {u, u}... | The method used by Oseen, which we developed in Chapter 4, Section 4.6, is very efficient. We introduce a development of the solution \( {u}_{\epsilon } \) of the equation \( {u}_{\epsilon } = {u}_{0} + {\epsilon B}\left( {{u}_{\epsilon },{u}_{\epsilon }}\right) \) as a power series in \( \epsilon \) . We find (at leas... | Yes |
Let \( {f}_{0} \) be non-negative and measurable and let \( {f}_{n} \) be inductively defined as\n\n\[ \n{f}_{n + 1}\left( x\right) = {f}_{0}\left( x\right) + {\int }_{X}K\left( {x, y}\right) {f}_{n}^{2}\left( y\right) {d\mu }\left( y\right) \n\]\n\n(5.8)\n\nLet \( f = \mathop{\sup }\limits_{{n \in \mathbb{N}}}{f}_{n}\... | Proof:\n\nDue to the inequalities \( {f}_{0} \geq 0 \) and \( K \geq 0 \), we find by induction that \( 0 \leq \)\n\n--- \n\n\( {f}_{n} \), so that \( {f}_{n + 1} \) is well defined (with values in \( \left\lbrack {0, + \infty }\right\rbrack \) ); we get moreover (by induction, as well) that \( {f}_{n} \leq {f}_{n + 1}... | Yes |
Theorem 5.3\n\nLet \( \\left( {X,\\delta ,\\mu }\\right) \) be a space of homogeneous type, with homogeneous dimension Q. Let\n\n\[ \n{K}_{\\alpha }\\left( {x, y}\\right) = \\frac{1}{\\delta {\\left( x, y\\right) }^{Q - \\alpha }}\n\]\n\n(5.16)\n\n(where \( 0 < \\alpha < Q/2 \) ) and \( {E}_{{K}_{\\alpha }} \) the asso... | Proof:\n\nIt is enough to see that \( A{t}^{\\frac{Q}{Q - \\alpha }} \\leq {\\int }_{\\rho \\left( {x, y}\\right) < t}{d\\mu }\\left( y\\right) \\leq B{t}^{\\frac{Q}{Q - \\alpha }} \) (with \( \\rho \\left( {x, y}\\right) = \) \( \\left. \\frac{1}{K\\left( {x, y}\\right) }\\right) \) and that \( 1 < \\frac{Q}{Q - \\alp... | Yes |
If \( f \in {\dot{M}}^{p, q}\left( X\right) \) and if \( 0 < \alpha < \frac{Q}{q} \), then\n\n\[ \left| {{\int }_{X}\frac{1}{\delta {\left( x, y\right) }^{Q - \alpha }}f\left( y\right) {d\mu }\left( y\right) }\right| \leq {C}_{p, q,\alpha }{\left( {\mathcal{M}}_{f}\left( x\right) \right) }^{1 - \frac{\alpha q}{Q}}\para... | Proof:\n\nLet \( R > 0 \) . We have\n\n\[ \left| {{\int }_{\rho \left( {x, y}\right) < R}\frac{f\left( y\right) }{\delta {\left( x, y\right) }^{Q - \alpha }}{d\mu }\left( y\right) }\right| \leq \mathop{\sum }\limits_{{j = 0}}^{{+\infty }}{\int }_{\frac{R}{{2}^{j + 1}} \leq \rho \left( {x, y}\right) < \frac{R}{{2}^{j}}}... | Yes |
Let \( 0 < \alpha < Q/2 \) and \( 2 < p \leq \frac{Q}{\alpha } \). Then we have:\n\n\[ \n{\dot{M}}^{p,\frac{Q}{\alpha }}\left( X\right) \subset {\mathcal{V}}^{\alpha } = \mathcal{M}\left( {{W}^{\alpha } \mapsto {L}^{2}}\right) \subset {\dot{M}}^{2,\frac{Q}{\alpha }}\left( X\right) \n\] | Proof:\n\nFor \( f \in {\dot{M}}^{p,\frac{Q}{\alpha }}\left( X\right) \) and \( g \in {\dot{M}}^{p,\frac{Q}{\alpha }}\left( X\right) \), we have \( {fg} \in {\dot{M}}^{\frac{p}{2},\frac{Q}{2\alpha }}\left( X\right) \). We have \( p/2 > 1 \) and \( \alpha < Q/q \) with \( q = \frac{Q}{2\alpha } \), hence, since \( \lamb... | Yes |
If, for all \( 0 < t < T \) and \( x \in {\mathbb{R}}^{3},\left| {\overrightarrow{U}\left( {t, x}\right) }\right| \leq \Omega \left( {t, x}\right) \) with \( \Omega \in {\Gamma }_{\nu, T} \), then the equation\n\n\[ \overrightarrow{u} = \overrightarrow{U} - {\int }_{0}^{t}\mathop{\sum }\limits_{{j = 1}}^{3}{\partial }_... | We define \( {\Omega }_{0}\left( {t, x}\right) \) as \( {\Omega }_{0}\left( {t, x}\right) = \left| {\overrightarrow{U}\left( {t, x}\right) }\right| \) for \( 0 < t < T \), and\n\n\[ {\Omega }_{n + 1}\left( {t, x}\right) = {\Omega }_{0}\left( {t, x}\right) + {C}_{0}{\int }_{0}^{t}{\int }_{{\mathbb{R}}^{3}}\frac{1}{{\nu ... | Yes |
There exists a constant \( {\epsilon }_{0} \) such that, for all \( t \in \mathbb{R} \), all \( x \in {\mathbb{R}}^{3} \) and all \( \nu > 0 \) , we have  | Proof:\n\nLet \( I = {\iint }_{\mathbb{R} \times {\mathbb{R}}^{3}}\frac{1}{{\nu }^{2}{\left( t - s\right) }^{2} + {\left| x - y\right| }^{4}}\frac{1}{{\left( \sqrt{\nu \left| s\right| } + \left| y\right| \right) }^{2}}{dyds} \) . We have\n\n\[ I \leq {\int }_{{\mathbb{R}}^{3}}{\begin{Vmatrix}\frac{1}{{\nu }^{2}{\left( ... | Yes |
There exists a constant \( {\epsilon }_{2} \) such that, for all \( t \in \mathbb{R} \), all \( x \in {\mathbb{R}}^{3} \) and all \( \nu > 0 \) , we have \[ {\int }_{{\mathbb{R}}^{3}}\frac{\sqrt{\nu \left| t\right| }}{\nu {t}^{2} + {\left| x - y\right| }^{4}}\frac{1}{{\left| y\right| }^{2}}{dy} \leq {\epsilon }_{1}\fra... | Let \( I = {\int }_{{\mathbb{R}}^{3}}\frac{\sqrt{\nu \left| t\right| }}{\nu {t}^{2} + {\left| x - y\right| }^{4}}\frac{1}{{\left| y\right| }^{2}}{dy} \). We have, using the duality between the Lorentz spaces \( {L}^{3/2,\infty } \) and \( {L}^{3,1} \), \[ I \leq {\begin{Vmatrix}\frac{\sqrt{\nu \left| t\right| }}{\nu {t... | Yes |
If \( {u}_{0} \in {L}^{2} \), then \( {W}_{\nu t} * {u}_{0} \in \mathcal{C}\left( {\lbrack 0, + \infty ),{L}^{2}}\right) \) with\n\n\[ \mathop{\sup }\limits_{{t > 0}}{\begin{Vmatrix}{W}_{\nu t} * {u}_{0}\end{Vmatrix}}_{2} = {\begin{Vmatrix}{u}_{0}\end{Vmatrix}}_{2} \]\n\n(7.8)\n\nMoreover, \( {W}_{\nu t} * {u}_{0} \in ... | Proof:\n\nTo check that \( {W}_{\nu t} * {u}_{0} \in \mathcal{C}\left( {\lbrack 0, + \infty }\right) ,{L}^{2}) \), just use the spatial Fourier transform\n\n\[ {\mathcal{F}}_{x}\left( {{W}_{\nu t} * {u}_{0}}\right) \left( \xi \right) = {e}^{-{\nu t}{\left| \xi \right| }^{2}}{\widehat{u}}_{0}\left( \xi \right) \]\n\nTo ... | Yes |
If \( \left. {f \in {L}^{2}\left( {0, + \infty }\right) ,{L}^{2}}\right) \) and \( F\left( {t, x}\right) = {\int }_{0}^{t}{W}_{\nu \left( {t - s}\right) } * f\left( {s,\text{.}}\right) {ds},{thenF} \) belongs to \( \mathcal{C}\left( {\lbrack 0, + \infty ),{\dot{H}}^{1}}\right) \) and we have\n\n\[ \n\parallel \overrigh... | Proof: Just write:\n\n\[ \n\parallel \overrightarrow{\nabla }F\left( {t,.}\right) {\parallel }_{2} = \parallel \sqrt{-\Delta }F\left( {t,.}\right) {\parallel }_{2} \n\]\n\n\[ \n= \mathop{\sup }\limits_{{{\begin{Vmatrix}{u}_{0}\end{Vmatrix}}_{2} = 1}}\left| {\int \left( {{\int }_{0}^{t}\sqrt{-\Delta }\left( {{W}_{\nu \l... | Yes |
Proposition 7.2\n\nLet \( s > 1/2 \) . Then we have\n\n\[ \n{\begin{Vmatrix}{\left( -\Delta \right) }^{s/2}\left( uv\right) - u{\left( -\Delta \right) }^{s/2}v\end{Vmatrix}}_{2} \leq C\left( {\parallel u{\parallel }_{{\dot{H}}^{3/2}}\parallel v{\parallel }_{{\dot{H}}^{s}} + \parallel u{\parallel }_{{\dot{H}}^{s + 1}}\p... | Proof:\n\nWe compute the norm of the Fourier transform; let\n\n\[ \nI = \int {\left| \int \widehat{u}\left( \xi - \eta \right) \widehat{v}\left( \eta \right) \left( {\left| \xi \right| }^{s} - {\left| \eta \right| }^{s}\right) d\eta \right| }^{2}{d\xi } \leq 2\left( {{I}_{1} + {I}_{2}}\right) \n\]\n\nwhere\n\n\[ \n{I}_... | Yes |
If \( \omega \) is a radially decreasing function on \( {\mathbb{R}}^{3} \) and \( f \) a locally integrable function, then\n\n\[ \left| {{\int }_{{\mathbb{R}}^{3}}\omega \left( {x - y}\right) f\left( y\right) {dy}}\right| \leq \parallel \omega {\parallel }_{1}\mathop{\sup }\limits_{{r > 0}}\frac{1}{\left| B\left( 0, r... | or equivalently\n\n\[ \left| {\omega * f}\right| \leq \parallel \omega {\parallel }_{1}{M}_{f} \]\n\nwhere \( {M}_{f} \) is the Hardy-Littlewood maximal function of \( f \) . | Yes |
Let \( 1 \leq {p}_{0} < {p}_{1} \leq + \infty \) and \( 0 < \theta < 1 \) . Then \( {\left\lbrack {L}^{{p}_{0}},{L}^{{p}_{1}}\right\rbrack }_{\theta ,\infty } = {L}^{p, * } \) with \( \frac{1}{p} = \frac{1 - \theta }{{p}_{0}} + \frac{\theta }{{p}_{1}} \). | Proof:\n\nIf \( f \in {\left\lbrack {L}^{{p}_{0}},{L}^{{p}_{1}}\right\rbrack }_{\theta ,\infty } \) with norm \( M \) and \( \lambda > 0 \), we write \( f = {f}_{\lambda } + {g}_{\lambda } \) with\n\n\( {\begin{Vmatrix}{f}_{\lambda }\end{Vmatrix}}_{{p}_{0}} \leq M{\mu }^{-\theta },{\begin{Vmatrix}{g}_{\lambda }\end{Vma... | Yes |
If \( \mathcal{S} \) is dense in \( E \) (so that \( {E}^{\prime } \subset {\mathcal{S}}^{\prime } \) ), we have \( {\left( {\dot{B}}_{E,1}^{s}\right) }^{\prime } = {\dot{B}}_{{E}^{\prime },\infty }^{-s} \) for \( - {\beta }_{{E}^{\prime }} < s < {\beta }_{E} \) | Let \( \omega \in \mathcal{S} \) be such that its Fourier transform \( \widehat{\omega } \) is compactly supported and is identically equal to 1 on a neighborhood of 0 . Let \( {\Delta }_{j} \) be the convolution operator with \( {2}^{3\left( {j + 1}\right) }\omega \left( {{2}^{j + 1}x}\right) - {2}^{3j}\omega \left( {... | Yes |
Theorem 8.1\n\nLet \( E \) be a Banach space such that:\n\n- \( E \subset {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{3}\right) \)\n\n- \( E \) is stable under convolution with \( {L}^{1} : \parallel f * g{\parallel }_{E} \leq \parallel f{\parallel }_{1}\parallel g{\parallel }_{E} \)\n\n- \( E \) is stable under bounde... | ## Proof:\n\nPointwise product maps \( E \times {L}^{\infty } \) and \( {L}^{\infty } \times E \) to \( E \), hence maps \( {\left\lbrack E,{L}^{\infty }\right\rbrack }_{1/2,1} \times \) \( {\left\lbrack E,{L}^{\infty }\right\rbrack }_{1/2,1} \) to \( E \) . Moreover convolution with \( {W}_{t} \) maps \( E \) to \( E ... | Yes |
Theorem 8.5 Let \( E \subset {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{3}\right) \) be a Banach space such that:\n\n- \( E \) is stable under convolution with \( {L}^{1} : \parallel f * g{\parallel }_{E} \leq \parallel f{\parallel }_{1}\parallel g{\parallel }_{E} \)\n\n- \( E \) is stable under bounded pointwise mult... | The proof is based on the inequality, for \( 0 < t < T \) ,\n\n\[ {\begin{Vmatrix}{W}_{\nu t} * f\end{Vmatrix}}_{\infty } \leq {C}_{\nu, T}{t}^{\left( {\alpha - 1}\right) /2}\parallel f{\parallel }_{E} \]\n\n(which is valid as well for \( T = + \infty \) when \( E \subset {\dot{B}}_{\infty ,\infty }^{-1 + \alpha } \) )... | Yes |
Theorem 9.1\n\nThe bilinear operator\n\n\[ B\left( {\overrightarrow{F},\overrightarrow{G}}\right) = {\int }_{0}^{t}\mathop{\sum }\limits_{{j = 1}}^{3}{\partial }_{j}\mathcal{O}\left( {\nu \left( {t - s}\right) }\right) : : \left( {{F}_{j}\overrightarrow{G}}\right) {ds} = {\int }_{0}^{t}{W}_{\nu \left( {t - s}\right) } ... | In order to prove Theorem 9.1, we follow the strategy of Auscher and Frey [8]. We begin with the following lemma:\n\nLemma 9.1\n\nIf \( \overr | No |
Let, for \( T \in (0, + \infty \rbrack \) , \[ \parallel h{\parallel }_{{E}_{T}} = \mathop{\sup }\limits_{{0 < t < T}}\sqrt{t}\parallel h\left( {t,.}\right) {\parallel }_{\infty } + \mathop{\sup }\limits_{{0 < t < T}}\mathop{\sup }\limits_{{{x}_{0} \in {\mathbb{R}}^{3}}}\frac{1}{{t}^{3/4}}\sqrt{{\int }_{0}^{t}{\int }_{... | As usual, by Picard's iterative scheme, using the estimate given by Theorem 9.1. The Koch and Tataru theorem gives criteria for local or global existence: - if \( {\overrightarrow{u}}_{0} \in {bm}{o}^{-1} \) and \( {W}_{\nu t} * {\overrightarrow{u}}_{0} \in {E}_{\infty } \), then \( {\overrightarrow{u}}_{0} \) belongs ... | Yes |
Let \( {\overrightarrow{u}}_{0} \in {BM}{O}^{-1} \) with \( \operatorname{div}{\overrightarrow{u}}_{0} = 0 \) . Let \( B \) be the bilinear operator\n\n\[ B\left( {\overrightarrow{V},\overrightarrow{W}}\right) = {\int }_{0}^{t}{W}_{\nu \left( {t - s}\right) } * \mathbb{P}\operatorname{div}\left( {\overrightarrow{V}\lef... | Case \( - 1 < \sigma < 0 \) :\n\nWe introduce the path space\n\n\[ X = \left\{ {\overrightarrow{U}/{t}^{-\sigma /2}\parallel \overrightarrow{U}\left( {t,.}\right) {\parallel }_{p} \in {L}^{q}\left( \frac{dt}{t}\right) }\right\} \]\n\nWe easily check that, if \( \overrightarrow{V} \in X \) and \( \mathop{\sup }\limits_{... | Yes |
Let \( {\overrightarrow{u}}_{0} \in {BM}{O}^{-1} \) with \( \operatorname{div}{\overrightarrow{u}}_{0} = 0 \) . Let \( B \) be the bilinear operator \[ B\left( {\overrightarrow{V},\overrightarrow{W}}\right) = {\int }_{0}^{t}{W}_{\nu \left( {t - s}\right) } * \mathbb{P}\operatorname{div}\left( {\overrightarrow{V}\left( ... | Proof: We would like to control a product \( {fg} \) when \( f \in {\dot{B}}_{p, q}^{\sigma } \cap {\dot{B}}_{\infty ,\infty }^{-1} \) and \( g \in {\dot{B}}_{p, q}^{\sigma } \cap {L}^{\infty } \) . Using the Littlewood-Paley decomposition, we write \[ {fg} = \pi \left( {f, g}\right) + \rho \left( {f, g}\right) \] wher... | Yes |
If \( \left( {\overrightarrow{{v}_{0}},{w}_{0}}\right) \in {\left( {L}^{2}\left( {\mathbb{R}}^{2}\right) \right) }^{3} \) with \( \operatorname{div}{\overrightarrow{v}}_{0} = 0 \) and \( \left( {\overrightarrow{g}, h}\right) \in {\left( {L}^{2}\left( \left( 0, T\right) ,{H}^{-1}\left( {\mathbb{R}}^{2}\right) \right) \r... | The existence of \( \overrightarrow{v} \) has been proved in Theorem 10.1. For the existence of \( w \) , we write \( w \) as a fixed point of the transform\n\n\[ \omega \mapsto {W}_{\nu t}^{\left( 2\right) } * {w}_{0} + {\int }_{0}^{t}{W}_{\nu \left( {t - s}\right) }^{\left( 2\right) } * \left( {h - \operatorname{div}... | Yes |
Let \( w \) be a weight on \( {\mathcal{R}}^{3} \) such that \( w \in {\mathcal{A}}_{2}\left( {{\mathcal{R}}^{3},{dx}}\right) \). Then:\n\n\[ \text{-}\left| {{W}_{\nu t} * f\left( x\right) }\right| \leq {\mathcal{M}}_{f}\left( x\right) \text{for}f \in {L}^{2}\left( {wdx}\right) \] | From the inequality \( \left| {{W}_{\nu t} * f\left( x\right) }\right| \leq {\mathcal{M}}_{f}\left( x\right) \) (Lemma 7.4), we get\n\n\[ \mathop{\sup }\limits_{{t > 0}}{\begin{Vmatrix}{W}_{\nu t} * f\left( t,.\right) \end{Vmatrix}}_{{L}^{2}\left( {wdx}\right) } \leq {\begin{Vmatrix}{\mathcal{M}}_{f}\end{Vmatrix}}_{{L}... | Yes |
The operator \( f \mapsto {\int }_{-\infty }^{t}{W}_{\nu \left( {t - s}\right) } * {fds} \) is bounded from \( {L}^{1}\left( {\left( {-\infty , + \infty }\right) ,{\dot{B}}_{\infty ,\infty }^{-1}\left( {\mathbb{R}}^{3}\right) }\right) \), \( {L}^{\infty }{B}_{\infty ,\infty }^{-3} \) or \( {L}^{p,\infty }{\dot{B}}_{\in... | Let \( u = {\int }_{-\infty }^{t}{W}_{\nu \left( {t - s}\right) } * {fds} \). For \( 1 \leq p \leq + \infty \), and \( \tau > 0 \), we have \[ {\begin{Vmatrix}{W}_{\nu \tau \Delta } * u\end{Vmatrix}}_{\infty } \leq {C}_{p}{\int }_{-\infty }^{t}\frac{1}{{\left( \nu \left( \tau + t - s\right) \right) }^{\frac{3}{2} - \fr... | Yes |
Let \( \overrightarrow{u} \) be a solution to\n\n\[ \left\{ \begin{array}{l} {\partial }_{t}\overrightarrow{u} = {\nu \Delta }\overrightarrow{u} \\ \overrightarrow{\nabla } \land \overrightarrow{u} = \lambda \overrightarrow{u} \end{array}\right. \]\n\nwhere \( \lambda \neq 0 \) . Then\n\n\[ \text{-}\Delta \overrightarr... | From \( \overrightarrow{\nabla } \land \overrightarrow{u} = \lambda \overrightarrow{u} \), we get that \( \operatorname{div}\overrightarrow{u} = 0 \) . Then, we have \( \Delta \overrightarrow{u} = - \overrightarrow{\nabla } \land \left( {\overrightarrow{\nabla } \land \overrightarrow{u}}\right) = \) \( - {\lambda }^{2}... | Yes |
Let \( {\overrightarrow{u}}_{0} \in {H}^{1} \) with \( \operatorname{div}{\overrightarrow{u}}_{0} = 0 \) and \( \overrightarrow{f} \in {L}^{2}\left( {\left( {0, + \infty }\right) ,{L}^{2}}\right) \) . Let \( \overrightarrow{u} \) be a solution of \[ {\partial }_{t}\overrightarrow{u} = {\nu \Delta }\overrightarrow{u} + ... | From (11.8), we get \[ \frac{d}{dt}\parallel \overrightarrow{u}{\parallel }_{{\dot{H}}^{1}}^{2} \leq - {2\nu }\parallel \Delta \overrightarrow{u}{\parallel }_{2}^{2} + 2\parallel \Delta \overrightarrow{u}{\parallel }_{2}\parallel \overrightarrow{u}.\overrightarrow{\nabla }\overrightarrow{u}{\parallel }_{2} + 2\parallel... | Yes |
Lemma 11.1 (Grönwall’s lemma)\n\nIf \( u \geq 0 \) is defined on \( \lbrack 0, T) \) and satisfies\n\n\[ u\left( t\right) \leq {a}_{0} + {\int }_{0}^{t}\Phi \left( {u\left( s\right) }\right) \omega \left( s\right) {ds} \]\n\nwith \( \Phi \geq 0 \) a non-decreasing function such that\n\n\[ {\int }_{1}^{+\infty }\frac{dt... | ## Proof:\n\nLet \( {a}_{1} = \max \left( {1,{a}_{0}}\right) \) and define \( A\left( t\right) = {a}_{1} + {\int }_{0}^{t}\Phi \left( {u\left( s\right) }\right) \omega \left( s\right) {ds} \). We have \( \frac{d}{dt}A = \) \( {\omega \Phi }\left( u\right) \leq {\omega \Phi }\left( A\right) \), so that \( {\int }_{0}^{t... | Yes |
Let \( {\dot{\overrightarrow{u}}}_{0} \in {H}^{1} \) with \( \operatorname{div}{\overrightarrow{u}}_{0} = 0 \) and \( \overrightarrow{f} \in {L}^{2}\left( {\left( {0, + \infty }\right) ,{H}^{1}}\right) \) . Let \( \overrightarrow{u} \) be a solution of \[ {\partial }_{t}\overrightarrow{u} = {\nu \Delta }\overrightarrow... | The first problem is of course to include the pressure \( \varpi \) in the estimates, and to exclude the term \( \overrightarrow{u}.\overrightarrow{\nabla }\overrightarrow{u} \) as we have no good control on this term (except for its divergence, as \( \operatorname{div}\left( {\overrightarrow{u} \cdot \overrightarrow{\... | Yes |
Let \( {\overrightarrow{u}}_{0} \in {H}^{3} \) with \( \operatorname{div}{\overrightarrow{u}}_{0} = 0 \) and \( \overrightarrow{f} \in {L}^{2}\left( {\left( {0, + \infty }\right) ,{H}^{2}}\right) \) . Let \( \overrightarrow{u} \) be a solution of \[ {\partial }_{t}\overrightarrow{u} = {\nu \Delta }\overrightarrow{u} + ... | \[ \parallel \overrightarrow{u}\left( {T,.}\right) {\parallel }_{{\dot{H}}^{3}}^{2} \leq {C}_{0}{e}^{\left( {{\begin{Vmatrix}{\overrightarrow{u}}_{0}\end{Vmatrix}}_{{H}^{3}}^{2} + \frac{1}{\nu }\parallel \overrightarrow{f}{\parallel }_{{L}^{2}{H}^{2}}^{2}}\right) {e}^{{C}_{0}{\int }_{0}^{T}\parallel \overrightarrow{\om... | Yes |
Let \( {\overrightarrow{u}}_{0} \in {H}^{3} \) with \( \operatorname{div}{\overrightarrow{u}}_{0} = 0 \) and \( \overrightarrow{f} \in {L}^{2}\left( {\left( {0, + \infty }\right) ,{H}^{2}}\right) \) . Let \( \overrightarrow{u} \) be a solution of \[ {\partial }_{t}\overrightarrow{u} = {\nu \Delta }\overrightarrow{u} + ... | Let \( {M}_{R}\left( t\right) = \mathop{\sup }\limits_{{\left| {\overrightarrow{\omega }\left( {t, x}\right) }\right| > R,\left| {\overrightarrow{\omega }\left( {t, y}\right) }\right| > R}}\frac{\left| \overrightarrow{\xi }\left( t, x\right) \land \overrightarrow{\xi }\left( t, y\right) \right| }{{\left| x - y\right| }... | Yes |
Theorem 12.1 (Rellich-Lions theorem)\n\nLet \( I \) be an open interval of \( \mathbb{R} \) and \( \Omega \) an open subset of \( {\mathbb{R}}^{d} \) . Let \( {\left( {u}_{n}\right) }_{n \in \mathbb{N}} \) be a sequence of measurable functions on \( I \times \Omega \) such that, for every \( \varphi \in \mathcal{D}\lef... | ## Proof:\n\nFirst, we consider a fixed \( \varphi \) . Let \( {v}_{n} = \varphi {u}_{n} \) . We have \( \mathop{\sup }\limits_{{n \in \mathbb{N}}}{\begin{Vmatrix}{v}_{n}\end{Vmatrix}}_{{L}^{2}\left( {I,{H}^{\alpha }\left( {\mathbb{R}}^{d}\right) }\right) } < \) \( + \infty \) and \( \mathop{\sup }\limits_{{n \in \math... | Yes |
Let \( {\overrightarrow{u}}_{0} \in {L}^{2} \), with \( \operatorname{div}{\overrightarrow{u}}_{0} = 0 \), and \( \overrightarrow{f} \in {L}_{t}^{2}{H}_{x}^{-1} \) on \( \left( {0, T}\right) \times {\mathbb{R}}^{3} \) . Then\n\n- for \( \epsilon > 0 \), the problem associated to the mollifier \( {\theta }_{\epsilon } \... | - First step: Local existence of \( {\overrightarrow{u}}_{\epsilon } \) .\n\nWe start from the obvious inequality\n\n\[ \n{\begin{Vmatrix}\overrightarrow{u} * {\theta }_{\epsilon }\end{Vmatrix}}_{\infty } \leq {\epsilon }^{-3/2}\parallel \overrightarrow{u}{\parallel }_{2}\parallel \theta {\parallel }_{2} \n\]\n\nThus, ... | Yes |
Let \( {\overrightarrow{u}}_{0} \in {L}^{2} \) and \( \overrightarrow{f} \in {L}^{2}\left( {\left( {0, + \infty }\right) ,{L}^{2}}\right) \cap {L}^{2}\left( {\left( {0, + \infty }\right) ,{\dot{H}}^{-1}}\right) \) . Then the Navier-Stokes problem\n\n\[ \n{\partial }_{t}\overrightarrow{u} + \mathbb{P}\left( {\overrighta... | We construct \( \overrightarrow{u} \) by Leray’s mollification (see Theorem 12.2). The global control of \( \overrightarrow{u} \) in \( {L}_{t}^{\infty }{L}^{2} \cap {L}_{t}^{2}{\dot{H}}^{1} \) is provided by the inequality\n\n\[ \n\parallel \overrightarrow{u}\left( {t,.}\right) {\parallel }_{2}^{2} + \nu {\int }_{0}^{... | Yes |
Let \( {\overrightarrow{u}}_{0} \in {L}^{2} \), with \( \operatorname{div}{\overrightarrow{u}}_{0} = 0 \), and \( \overrightarrow{f} \in {L}_{t}^{2}{H}_{x}^{-1} \) on \( \left( {0, T}\right) \times {\mathbb{R}}^{3} \). Assume that the Navier-Stokes problem \[ {\partial }_{t}\overrightarrow{u} + \mathbb{P}\left( {\overr... | Proof: If \( {\overrightarrow{u}}_{1} \in {L}_{t}^{\infty }{L}^{2} \cap {L}_{t}^{2}{H}^{1} \cap {\mathbb{X}}_{T} \) with \( \operatorname{div}{\overrightarrow{u}}_{1} = 0 \) and \( \overrightarrow{v} \in {L}^{2}{H}^{1} \), we write \[ {\int }_{0}^{t}\left\langle {\left( {{\overrightarrow{u}}_{1} \cdot \overrightarrow{\... | Yes |
If \( \overrightarrow{u} \in {\dot{B}}_{\infty ,\infty }^{r},\overrightarrow{v} \in {L}^{2} \) with \( \operatorname{div}\overrightarrow{v} = 0 \) and \( \overrightarrow{w} \in {\dot{B}}_{2,1}^{1 - r} \), then\n\n\[ \left| {\int \overrightarrow{u} \cdot \operatorname{div}\left( {\overrightarrow{v} \otimes \overrightarr... | Let \( T \) be the operator \( \left( {\overrightarrow{v},\overrightarrow{w}}\right) \mapsto T\left( {\overrightarrow{v},\overrightarrow{w}}\right) = \operatorname{div}\left( {\left( {\mathbb{P}\overrightarrow{v}}\right) \otimes \overrightarrow{w}}\right) \) . If \( \overrightarrow{v} \in {L}^{2} \) and \( \overrightar... | Yes |
If \( I \) is an interval of \( \mathbb{R} \), then the pointwise multiplication by \( {1}_{I} \) is bounded on \( {\dot{H}}^{r}\left( \mathbb{R}\right) \) with \( 0 < r < 1/2 \) : | It is enough to prove the theorem for \( I = \left( {0, + \infty }\right) \) as\n\n\[ {1}_{\left( a, b\right) }\left( t\right) = {1}_{\left( 0, + \infty \right) }\left( {t - a}\right) \left( {1 - {1}_{(}0,\infty }\right) \left( {t - b}\right) \]\n\n(for \( t \neq b \) ). But for \( I = \left( {0, + \infty }\right) \), ... | Yes |
Let \( {\overrightarrow{u}}_{0} \in {L}^{2} \) with \( \operatorname{div}{\overrightarrow{u}}_{0} = 0 \) and \( \overrightarrow{f} \in {L}^{2}\left( {\left( {0, + \infty }\right) ,{H}^{1}}\right) \) . Assume that \( \overrightarrow{u} \) and \( \overrightarrow{v} \) are two solutions of the Navier-Stokes equations on \... | We may of course assume that \( {r}_{2} \leq {r}_{1} \) .\n\nCase \( {r}_{1} > 0 \) :\n\nIn that case, we know that we have weak-strong uniqueness (Proposition 12.4). As \( \overrightarrow{v} \) satisfies the Leray energy (in)equality and \( \overrightarrow{u} \in {L}^{\frac{2}{{r}_{1} + 1}}\left( {\left( {0, T}\right)... | Yes |
Let \( {\mathbb{X}}_{T}^{r} \) be defined, for \( - 1 \leq r \leq 1 \) as\n\n\[ \text{-}{\mathbb{X}}_{T}^{1} = {L}^{1}\left( {\left( {0, T}\right) ,\operatorname{Lip}}\right) \]\n\n\[ \text{- for} - 1 < r < 1,{\mathbb{X}}_{T}^{r} = {L}^{\frac{2}{1 + r}}\left( {\left( {0, T}\right) ,{\dot{B}}_{\infty ,\infty }^{r}}\righ... | ## Proof:\n\nStep 1: almost strong solutions.\n\nThe first step is to check that a solution \( \overrightarrow{u} \in {L}^{\infty }{L}^{2} \cap {L}^{2}{H}^{1} \cap {\mathbb{X}}_{T}^{r} \) (for some \( r \in \left\lbrack {-1,1}\right\rbrack ) \) is indeed an almost strong solution. This has already been proved for \( \l... | No |
Let \( \mathbb{X} \) satisfy the assumptions of Theorem 12.10. Assume that, for some \( {\epsilon }_{1} > 0 \) (depending only on \( \mathbb{X} \) ), we have the following assumptions on \( \overrightarrow{f} \) and \( {\overrightarrow{u}}_{0} \) :\n\n- in the steady case: \( \overrightarrow{f} = \Delta \overrightarrow... | Due to Theorems 10.8 and 10.12, we know that the existence result (Theorem 12.10) holds, as well as the stability result (Theorem 12.11) in the case \( {\overrightarrow{v}}_{0} = 0 \) .\n\nWhen \( {\overrightarrow{v}}_{0} \neq 0 \), we begin by solving the Navier-Stokes problem with initial\n\nvalue \( {\overrightarrow... | Yes |
If \( u \in \mathbb{X} \) and \( v \in {L}^{2}{L}^{2} \), then \( {uv} \in {L}_{t}^{1}{L}^{2} + {L}_{t}^{2}{H}^{-1} \) . | The dual of \( {L}_{t}^{1}{L}^{2} + {L}_{t}^{2}{H}^{-1} \) is \( {L}_{t}^{\infty }{L}^{2} \cap {L}_{t}^{2}{H}^{1} \) . Thus, for \( w \in {L}_{t}^{1}{L}^{2} + {L}_{t}^{2}{H}^{-1} \) ,\n\n\[ \parallel w{\parallel }_{{L}_{t}^{1}{L}^{2} + {L}_{t}^{2}{H}^{-1}} \approx \sup \{ \left| {\langle w}\right| z\rangle |/\parallel ... | Yes |
Let \( Q = I \times \Omega \), where \( I = \left( {a, b}\right) \) and \( \Omega = B\left( {{x}_{0}, r}\right) \). Let \( \overrightarrow{u} \in {L}^{\infty }\left( {I,{L}^{2}\left( \Omega \right) }\right) \cap {L}^{2}\left( {I,{H}^{1}\left( \Omega \right) }\right) ,\overrightarrow{f} \in {L}^{2}\left( {I,{L}^{2}\left... | ## Proof:\n\nFirst step: a linear heat equation.\n\nLet \( \overrightarrow{\omega } = \operatorname{curl}\overrightarrow{u} \). We consider a function \( \phi \in \mathcal{D}\left( {\mathbb{R} \times {\mathbb{R}}^{3}}\right) \) which is equal to 1 on \( \left\lbrack {c, b}\right\rbrack \times B\left( {{x}_{0},\rho }\ri... | Yes |
Let \( Q = I \times \Omega \), where \( I = \left( {a, b}\right) \) and \( \Omega = B\left( {{x}_{0}, r}\right) \). Let \( \overrightarrow{u} \in {L}^{\infty }\left( {I,{L}^{2}\left( \Omega \right) }\right) \cap {L}^{2}\left( {I,{H}^{1}\left( \Omega \right) }\right) ,\overrightarrow{f} \in {L}^{2}\left( {I,{H}^{1}\left... | With no loss of generality, we may assume, as \( Q \) is bounded, that \( 5 < q \leq 6 \). We write \( \lambda = 1 - \frac{q - 5}{5q} \in \left( {0,1}\right) \). We are going to show that, for \( a < c < b \) and \( 0 < \rho < r \) and \( {Q}_{0} = \left( {c, b}\right) \times B\left( {{x}_{0},\rho }\right) \), we have ... | Yes |
Theorem 13.9 (Dimension of the singular set)\n\nLet \( \overrightarrow{u} \) be a weak solution for the Navier-Stokes equations on \( \left( {0, T}\right) \times {\mathbb{R}}^{3} \) , which is a suitable solution on the cylinder \( {Q}_{0} = \left( {a, b}\right) \times B\left( {{x}_{0},{r}_{0}}\right) \) (with pressure... | Proof:\n\nLet \( \delta > 0 \) . Let \( \left( {t, x}\right) \in \sum \) . According to Theorem 13.8, we know that\n\n\[ \mathop{\limsup }\limits_{{r \rightarrow 0}}\frac{1}{r}{\iint }_{Q\left( {x, r}\right) }{\left| \overrightarrow{\nabla } \otimes \overrightarrow{u}\right| }^{2}{dyds} \geq {\epsilon }^{ * } > 0. \]\n... | Yes |
Theorem 13.10\n\nLet \( \Omega \) be a bounded domain of \( \mathbb{R} \times {\mathbb{R}}^{3} \) . Let \( \left( {\overrightarrow{u}, p}\right) \) a weak solution on \( \Omega \) of the Navier-Stokes equations\n\n\[ \n{\partial }_{t}\overrightarrow{u} = {\nu \Delta }\overrightarrow{u} - \overrightarrow{u} \cdot \overr... | ## Proof:\n\nWe are now going to prove Proposition 13.5 and Theorem 13.10. With no loss of generality (due to the local character of the properties studied in Proposition 13.5 and Theorem 13.10), we may assume that \( \Omega = \left( {a, b}\right) \times B \) where \( B \) is an open ball in \( {\mathbb{R}}^{3} \) .\n\... | Yes |
Let \( \overrightarrow{u} \) be a local Leray solution on \( \left( {0, T}\right) \times {\mathbb{R}}^{3} \) to the problem \[ \left\{ \begin{matrix} {\partial }_{t}\overrightarrow{u} = {\nu \Delta }\overrightarrow{u} + \mathbb{P}\operatorname{div}\left( {\mathbb{F} - \overrightarrow{u} \otimes \overrightarrow{u}}\righ... | Proof: We use the suitability of \( \overrightarrow{u} \) and apply the local energy inequality to the test function \( \phi \left( {s, x}\right) = {\gamma }_{{t}_{0},\eta }\left( s\right) \varphi \left( x\right) \) (with \( \varphi \in \mathcal{B} : \varphi \left( x\right) = {\varphi }_{{x}_{0}}\left( x\right) = {\var... | Yes |
Theorem 15.3\n\nLet \( \\overrightarrow{\\omega } \) be a vector field on \( {Q}_{ + } = \\left( {-1,0}\\right) \\times {\\mathbb{R}}_{ + }^{3} \) (where \( {\\mathbb{R}}_{ + }^{3} = {\\mathbb{R}}^{2} \\times \\left( {0, + \\infty }\\right) \) ) such that\n\n- for every bounded subdomain \( \\Omega \) of \( {Q}_{ + },\... | The reader will find the proof of Theorem 15.3 in the papers of Escauriaza, Seregin and Šverák [132, 133] or in Seregin’s book [369]. | No |
If the forcing term \( \overrightarrow{f} \) satisfies\n\n\[ \operatorname{div}\overrightarrow{f} = 0 \]\n\nand, for every \( T < + \infty \) ,\n\n\[ \overrightarrow{f} \in {L}_{t}^{p}\left( {\left( {0, T}\right) ,{L}^{q}\left( {\mathbb{R}}^{3}\right) }\right) \text{with}\frac{2}{p} + \frac{3}{q} = 3\text{and}3/2 < q <... | Proof:\n\nIf we look for a solution \( \overrightarrow{u} \) of equations (15.4) such that \( \overrightarrow{u} \in {L}^{\infty }\left( {\left\lbrack {0,{T}_{0}}\right\rbrack ,{L}_{x}^{3}}\right) \) , then we may remark that \( \overrightarrow{u} \) belongs to the closure of \( \mathcal{D}\left( {\left( {0,{T}_{0}}\ri... | Yes |
Let \( {T}_{1} < {T}_{2} \) be two real numbers and \( p, q > 1 \) with \( \frac{3}{2} < q < 3 \) and \( \frac{2}{p} + \frac{3}{q} < 2 \) . Let \( \overrightarrow{f} \in {L}_{t}^{p}{L}_{x}^{q}\left( {\left( {{T}_{1},{T}_{2}}\right) \times {\mathbb{R}}^{3}}\right) \) with \( \operatorname{div}\overrightarrow{f} = 0 \) a... | From Theorem 15.9, it is enough to show that for \( r < \frac{1}{2}{r}_{1} \), we have\n\n\[ \n\frac{1}{r}{\iint }_{{Q}_{r}\left( {{t}_{0},{x}_{0}}\right) \cap \Omega }{\left| \overrightarrow{\nabla } \otimes \overrightarrow{u}\right| }^{2}{dsdx} \leq \frac{1}{2}{\epsilon }^{ * }\n\]\n\nAs in the proof of Theorem 13.8,... | Yes |
Let \( {T}_{1} < {T}_{2} \) be two real numbers and \( p, q > 1 \) with \( \frac{3}{2} < q < 3 \) and \( \frac{2}{p} + \frac{3}{q} < 2 \) . Let \( \overrightarrow{f} \in {L}_{t}^{p}{L}_{x}^{q}\left( {\left( {{T}_{1},{T}_{2}}\right) \times {\mathbb{R}}^{3}}\right) \) with \( \operatorname{div}\overrightarrow{f} = 0 \) a... | If there exists some \( r \in \left( {0\sqrt{{t}_{0} - {T}_{1}}}\right) \) such that\n\n\[ \n\frac{1}{{r}^{2}}{\iint }_{{Q}_{r}\left( {{t}_{0},{x}_{0}}\right) \cap \Omega }{\left| \overrightarrow{u}\left( t, x\right) \right| }^{3}{dtdx} \geq {\epsilon }_{0}\min \left( {\frac{1}{{M}_{0}},\frac{1}{{M}_{0}^{3}}}\right) ,\... | No |
Theorem 15.11\n\nLet \( {T}_{1} < {T}_{2} \) be two real numbers and \( p, q > 1 \) with \( \frac{3}{2} < q < 3 \) and \( \frac{2}{p} + \frac{3}{q} < 2 \) . Let \( {\overrightarrow{f}}_{n} \) be a sequence of vector fields such that \( {\overrightarrow{f}}_{n} \in {L}_{t}^{p}{L}_{x}^{q}\left( {\left( {{T}_{1},{T}_{2}}\... | Proof:\n\nLet\n\n\[ \n{M}_{0} = \mathop{\sup }\limits_{{n \in \mathbb{N}}}{\begin{Vmatrix}{\overrightarrow{u}}_{n}\end{Vmatrix}}_{{L}^{\infty }{L}^{3}} + {\begin{Vmatrix}{\overrightarrow{u}}_{n}\end{Vmatrix}}_{{L}^{\infty }{L}^{3}}^{2} + {\sqrt{{T}_{2} - {T}_{1}}}^{5 - \frac{2}{{\tau }_{0}}}{\begin{Vmatrix}{\overrighta... | Yes |
Lemma 16.1\n\nLet \( u \) be a function on \( {\mathbb{R}}^{3} \) such that:\n\n- \( u \) is locally square integrable on \( {\mathbb{R}}^{3} \smallsetminus \{ 0\} \)\n\n- \( u \) is homogeneous: for \( \lambda > 0, u\left( {\lambda x}\right) = {\lambda }^{-1}u\left( x\right) \) .\n\nThen \( u \) belongs to \( {L}_{\te... | Proof:\n\nWe have\n\n\[ \parallel \gamma {\parallel }_{{L}^{2}\left( {S}^{2}\right) }^{2} = {\int }_{1 < \left| x\right| < 2}{\left| u\left( x\right) \right| }^{2}{dx} \]\n\nand\n\n\[ {\int }_{B\left( {{x}_{0},1}\right) }{\left| u\left( x\right) \right| }^{2}{dx} \leq 2\parallel \gamma {\parallel }_{{L}^{2}\left( {S}^{... | Yes |
Let \( u \) be a function on \( {\mathbb{R}}^{3} \) of the form\n\n\[ u\left( x\right) = \frac{1}{\left| x\right| }\gamma \left( \frac{x}{\left| x\right| }\right) \]\n\nwhere \( \gamma \in {L}^{2}\left( {S}^{2}\right) \) . Then the following assertions are equivalent:\n\n- \( u \) is locally (essentially) bounded on \(... | Proof:\n\nWe have of course, for \( R > 1 \) ,\n\n\[ \parallel \gamma {\parallel }_{{L}^{\infty }\left( {S}^{2}\right) } = \text{ (ess.)sup. }{}_{1/R < \left| x\right| < R}\left| x\right| \left| {u\left( x\right) }\right| \]\n\nIf \( W \) is the heat kernel, so that \( {e}^{t\Delta }u = \frac{1}{{t}^{3/2}}W\left( \frac... | Yes |
Let \( {\overrightarrow{u}}_{0} \in {L}^{2} \), with \( \operatorname{div}{\overrightarrow{u}}_{0} = 0 \), and \( \overrightarrow{f} \in {L}_{t}^{2}{H}_{x}^{-1} \) on \( \left( {0, T}\right) \times {\mathbb{R}}^{3}\left( {T < + \infty }\right) \) . Let \( K \in {L}^{1} \cap {L}^{2} \) with \( \int {Kdx} = 1 \) . Then\n... | ## Proof:\n\n- First step: Local existence of \( {\overrightarrow{u}}_{\left( \alpha \right) } \) .\n\nWe have\n\n\[ \n{\begin{Vmatrix}{\overrightarrow{u}}_{\alpha }\end{Vmatrix}}_{\infty } \leq {\alpha }^{-3/2}\parallel \overrightarrow{u}{\parallel }_{2}\parallel K{\parallel }_{2} \n\]\n\nThus, for \( \overrightarrow{... | Yes |
Let \( {\overrightarrow{u}}_{0} \in {L}^{2} \), with \( \operatorname{div}{\overrightarrow{u}}_{0} = 0 \), and \( \overrightarrow{f} \in {L}_{t}^{2}{H}_{x}^{-1} \) on \( \left( {0, T}\right) \times {\mathbb{R}}^{3}\left( {T < + \infty }\right) \). Then for \( \alpha > 0 \), the problem \[ {\partial }_{t}\overrightarrow... | - First step: Local existence of \( {\overrightarrow{u}}_{\left( \alpha \right) } \). This step is very close to the same step in the proof for the Leray- \( \alpha \) model (Theorem 17.1). We just have to estimate, for \( \overrightarrow{u} \) and \( \overrightarrow{v} \) in \( {L}^{\infty }{L}^{2} \cap {L}^{2}{H}^{1}... | No |
Let \( {\overrightarrow{u}}_{0} \in {L}^{2} \), with \( \operatorname{div}{\overrightarrow{u}}_{0} = 0 \), and \( \overrightarrow{f} \in {L}_{t}^{2}{H}_{x}^{-1} \) on \( \left( {0, T}\right) \times {\mathbb{R}}^{3}\left( {T < + \infty }\right) \). - for \( \alpha > 0 \), the problem \[ {\partial }_{t}\overrightarrow{u}... | Proof: - First step: Local existence of \( {\overrightarrow{u}}_{\left( \alpha \right) } \). We just have to estimate, for \( \overrightarrow{u} \) and \( \overrightarrow{v} \) in \( {L}^{\infty }{L}^{2} \cap {L}^{2}{H}^{1} \) with \( \operatorname{div}\overrightarrow{u} = 0 \) and for \( 0 < {T}_{0} < T \), the norms ... | Yes |
Theorem 17.4\n\nLet \( {\overrightarrow{u}}_{0} \in {L}^{2} \), with \( \operatorname{div}{\overrightarrow{u}}_{0} = 0 \), and \( \overrightarrow{f} \in {L}_{t}^{2}{H}_{x}^{-1} \) on \( \left( {0, T}\right) \times {\mathbb{R}}^{3}\left( {T < + \infty }\right) \) .\n\n- for \( \alpha > 0 \), the problem\n\n\[ \n{\partia... | ## Proof:\n\n- First step: Local existence of \( {\overrightarrow{u}}_{\left( \alpha \right) } \) .\n\nWe just have to estimate, for \( \overrightarrow{u} \) and \( \overrightarrow{v} \) in \( {L}^{\infty }{L}^{2} \cap {L}^{2}{H}^{1} \) with \( \operatorname{div}\overrightarrow{u} = 0 \) and for \( 0 < {T}_{0} < T \), ... | No |
Let \( {\overrightarrow{u}}_{0} \in {L}^{2} \), with \( \operatorname{div}{\overrightarrow{u}}_{0} = 0 \), and \( \overrightarrow{f} \in {L}_{t}^{2}{H}_{x}^{-1} \) on \( \left( {0, T}\right) \times {\mathbb{R}}^{3}\left( {T < + \infty }\right) \). Then, for \( N \in \mathbb{N} \), the problem \[ {\partial }_{t}\overrig... | Proof: ## - First step: Global existence of \( {\overrightarrow{u}}_{\left( N\right) } \) . Let \( {X}_{N} \) be the linear space generated by \( {\overrightarrow{w}}_{0},\ldots ,{\overrightarrow{w}}_{N} \), endowed with the norm \( \parallel \overrightarrow{w}{\parallel }_{{X}_{N}} = \parallel \overrightarrow{w}{\para... | No |
Let \( {\overrightarrow{u}}_{0} \in {L}^{2} \), with \( \operatorname{div}{\overrightarrow{u}}_{0} = 0 \), and \( \overrightarrow{f} \in {L}_{t}^{2}{H}_{x}^{-1} \) on \( \left( {0, T}\right) \times {\mathbb{R}}^{3}\left( {T < + \infty }\right) \). Let \( \left( {X}_{N}\right) \) be a sequence of closed subspaces of \( ... | First step: Global existence of \( {\overrightarrow{u}}_{\left( N\right) } \). We have \( {P}_{N}\overrightarrow{f} \in {L}^{2}{X}_{N} \). Moreover the linear operator \( \overrightarrow{w} \mapsto \nu {P}_{N}\left( {\Delta \overrightarrow{w}}\right) \) and the bilinear operator \( \left( {\overrightarrow{v},\overright... | Yes |
Let \( {\overrightarrow{u}}_{0} \in {L}^{2} \), with \( \operatorname{div}{\overrightarrow{u}}_{0} = 0 \), and \( \overrightarrow{f} \in {L}_{t}^{2}{H}_{x}^{-1} \) on \( \left( {0, T}\right) \times {\mathbb{R}}^{3}\left( {T < + \infty }\right) \). Then\n\n- for \( \alpha > 0 \), the problem\n\n\[ \n{\partial }_{t}\over... | ## Proof:\n\n- First step: Local existence of \( {\overrightarrow{u}}_{\left( \alpha \right) } \).\n\nLet \( {e}^{-t{\Delta }^{2}} \) be the operator defined by\n\n\[ \n\mathcal{F}\left( {{e}^{-t{\Delta }^{2}}g}\right) \left( \xi \right) = {e}^{-t{\left| \xi \right| }^{4}}\widehat{g}\left( \xi \right)\n\]\n\nWe have th... | Yes |
Theorem 1 The steady-state availability of the system is\n\n\[ A = \frac{{g}^{ * }\left( {\lambda p}\right) {g}^{ * }\left( {{2\lambda p} + \lambda {p}^{2}}\right) \left( {\left( {{p}^{2} + p + 4}\right) {\bar{H}}^{ * }\left( {\lambda p}\right) - \left( {3 - p}\right) {\bar{H}}^{ * }\left( {{2\lambda p} + \lambda {p}^{... | Proof The availability of the system is\n\n\[ \begin{aligned} A\left( t\right) = & {\int }_{0}^{\infty }{P}_{0}\left( {t, x}\right) \mathrm{d}x + {\int }_{0}^{\infty }{P}_{1}\left( {t, x}\right) \mathrm{d}x \\ & + {\int }_{0}^{\infty }{P}_{2}\left( {t, x}\right) \mathrm{d}x + {\int }_{0}^{\infty }{P}_{3}\left( {t, y}\r... | Yes |
Corollary 1 The steady state probability that the repairman is on vacation is\n\n\[ \n{P}_{V} = \frac{{\left( 1 + p\right) }^{2}{g}^{ * }\left( {\lambda p}\right) {g}^{ * }\left( {{2\lambda p} + \lambda {p}^{2}}\right) }{\alpha K}. \n\] | Proof As the state set of the repairman on vacation is \( U = \{ 0,1,2,5\} \), the probability that the repairman is on vacation at time \( t \) is\n\n\[ \n{P}_{V}\left( t\right) = P\left( {S\left( t\right) \in V}\right) = {\int }_{0}^{\infty }{P}_{0}\left( {t, x}\right) \mathrm{d}x + {\int }_{0}^{\infty }{P}_{1}\left(... | Yes |
Corollary 2 The steady state probability that the system is waiting for repair is\n\n\[ \begin{matrix} {P}_{w} & = & \left( {{\left( 1 + p\right) }^{2}{\alpha }^{-1} - \left( {{p}^{2} + p + 4}\right) {\overline{H}}^{ * }\left( {\lambda p}\right) + \left( {3 - p}\right) {\overline{H}}^{ * }\left( {{2\lambda p} + \lambda... | Proof The probability that the system is waiting for repair, namely the system is in failure but the repairman is on vacation is \( {P}_{w}\left( t\right) \). From assumptions of the system, we have\n\n\[ {P}_{w}\left( t\right) = {\int }_{0}^{\infty }{P}_{5}\left( {t, x}\right) \mathrm{d}x \]\n\nTaking the Laplace tran... | Yes |
Theorem 2 According to the result in [19], the steady-state failure frequency of the system is\n\n\[ M = \frac{\left( {{p}^{2} + p + 4}\right) \left( {1 - {h}^{ * }\left( {\lambda p}\right) }\right) {g}^{ * }\left( {\lambda p}\right) {g}^{ * }\left( {{2\lambda p} + \lambda {p}^{2}}\right) }{K} \]\n\n\[ - \frac{\left( {... | Proof According to the result in [19], we have that\n\n\[ W\left( t\right) = \lambda {p}^{2}{\int }_{0}^{\infty }{P}_{1}\left( {t, x}\right) \mathrm{d}x + {\lambda p}{\int }_{0}^{\infty }{P}_{2}\left( {t, x}\right) \mathrm{d}x + \lambda {p}^{2}{\int }_{0}^{\infty }{P}_{3}\left( {t, y}\right) \mathrm{d}y + {\lambda p}{\... | Yes |
Theorem 3 The Laplace transform formula of the reliability of the system is\n\n\[ \n\begin{matrix} {R}^{ * }\left( s\right) & = & \frac{\left( {\left( {{p}^{2} + p + 4}\right) {\overline{H}}^{ * }\left( {s + {\lambda p}}\right) - \left( {3 - p}\right) {\overline{H}}^{ * }\left( {s + {2\lambda p} + \lambda {p}^{2}}\righ... | Proof Directly by calculating\n\n\[ \n\operatorname{MTTFF} = {\int }_{0}^{\infty }R\left( t\right) \mathrm{d}t = \mathop{\lim }\limits_{{s \rightarrow {0}^{ + }}}{R}^{ * }\left( s\right) \]\n\nimplies the result. | Yes |
Example 1.2 Let \( m \) be any positive integer, and \( p \) a prime, and let \( {\left\{ {C}_{n}^{m}\right\} }_{n \geq m} \) be the sequence of binomial coefficients. Then, by the Kummer theorem, | \[ {\operatorname{ord}}_{p}\left( {C}_{n{p}^{t}}^{m}\right) = t + \nu ,\text{ for all }t \geq T, \] where \( T = \max \left\{ {0,\left\lbrack {{\log }_{p}m}\right\rbrack - {\operatorname{ord}}_{p}\left( n\right) + 1}\right\} \) and \( \nu = {\operatorname{ord}}_{p}\left( n\right) - {\operatorname{ord}}_{p}\left( m\righ... | No |
Lemma 2.1 \( {}^{\left\lbrack 4\right\rbrack } \) Let \( m \geq n \) be two positive integers. By Euclidean division, we the obtain the integral quotient and remainder polynomials \( q\left( x\right) \) and \( r\left( x\right) \) satisfying\n\n\[ \n{T}_{m}\left( x\right) = q\left( x\right) {T}_{n}\left( x\right) + r\le... | (1) Assume that there exists an integer \( l \) such that \( m = \left( {{2l} - 1}\right) n \) . Then\n\n\[ \nq\left( x\right) = 2\mathop{\sum }\limits_{{k = 1}}^{{l - 1}}{\left( -1\right) }^{k}{T}_{m - \left( {{2k} - 1}\right) n}\left( x\right) + {\left( -1\right) }^{l - 1},\;r\left( x\right) = 0.\n\]\n\n(2) Assume th... | Yes |
Proposition 2.3 Let \( {n}_{1},{n}_{2} \) be two positive integers. For all \( m \in \mathbb{N} \), we have\n\n\[ \gcd \left( {{T}_{{n}_{1}}\left( m\right) ,{T}_{{n}_{2}}\left( m\right) }\right) = \left\{ \begin{array}{ll} {T}_{\gcd \left( {{n}_{1},{n}_{2}}\right) }\left( m\right) , & \text{ if }{\operatorname{ord}}_{2... | Proof By Lemmas 2.1,2.2, there exist polynomials \( {u}_{{n}_{1}}\left( x\right) ,{v}_{{n}_{2}}\left( x\right) \in \mathbb{Z}\left\lbrack x\right\rbrack \) such that\n\n\[ {u}_{{n}_{1}}\left( x\right) {T}_{{n}_{1}}\left( x\right) + {v}_{{n}_{2}}\left( x\right) {T}_{{n}_{2}}\left( x\right) = \left\{ \begin{array}{ll} {T... | Yes |
Proposition 3.1 Let \( n \in {\mathbb{N}}^{ * } \), and \( p \) an odd prime factor of \( {T}_{n}\left( m\right) \) . Then, for any odd positive integer \( t \) satisfying \( p \mid t \), we have \( p \mid \frac{{T}_{nt}\left( m\right) }{{T}_{n}\left( m\right) } \) . | Proof Proposition 2.3 implies \( {T}_{n}\left( m\right) \mid {T}_{nt}\left( m\right) \), since \( {\operatorname{ord}}_{2}\left( n\right) = {\operatorname{ord}}_{2}\left( {nt}\right) \) . By (2) of Proposition 1.5, the condition \( p \mid {T}_{n}\left( m\right) \) implies \( {\alpha }^{n} \equiv - {\widetilde{\alpha }}... | Yes |
Corollary 3.2 For all \( t \in \mathbb{N},{p}^{e + t} \mid {T}_{n{p}^{t}}\left( m\right) \), where \( e = {\operatorname{ord}}_{p}\left( {{T}_{n}\left( m\right) }\right) \geq 1 \) . | Proof It is obvious by induction on \( t \) . | No |
Proposition 3.3 Let \( n, t \in {\mathbb{N}}^{ * } \) . Suppose \( p \) is a prime such that \( p \nmid {T}_{n}\left( m\right) \) and \( \left( {t,{p}^{2} - 1}\right) = 1 \) . Then (i) \( p \nmid {T}_{nt}\left( m\right) \) ; (ii) \( p \nmid {T}_{n{p}^{x}}\left( m\right) \) for any \( x \in \mathbb{N} \) . | Proof It is clear that (ii) follows (i). Hence it suffices to prove (i). Let \( p = 2 \) . If \( 2 \mid {T}_{nt}\left( m\right) \), then, by (3) of Proposition 1.5, \( 2 \mid m \) and \( {nt} \) is odd. Hence \( n \) is odd and \( 2 \mid {T}_{n}\left( m\right) \) . This contradicts the assumption \( 2 \nmid {T}_{n}\lef... | Yes |
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