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Corollary 1.34 If \( u, v \in {L}^{2}\left( {0, T;{H}_{0}^{1}}\right) \) and \( {\partial }_{t}u,{\partial }_{t}v \in {L}^{2}\left( {0, T;{H}^{-1}}\right) \) then the mapping \( t \mapsto \langle u, v\rangle \) is absolutely continuous and\n\n\[ \langle u\left( t\right), v\left( t\right) \rangle = \langle u\left( s\rig... | Proof We have \( \left( {u + v}\right) \in {L}^{2}\left( {0, T;{H}_{0}^{1}}\right) \) and \( {\partial }_{t}\left( {u + v}\right) \in {L}^{2}\left( {0, T;{H}^{-1}}\right) \), and so \( t \mapsto \parallel u + v{\parallel }^{2} \) is absolutely continuous. Hence\n\n\[ 2\langle u, v\rangle = \parallel u + v{\parallel }^{... | Yes |
Lemma 2.3 Each \( u \in H\left( {\mathbb{T}}^{3}\right) \) is weakly divergence free and moreover\n\n\[ \langle u,\nabla \varphi \rangle = 0\;\text{ for all }\;\varphi \in {H}^{1}\left( {\mathbb{T}}^{3}\right) . \] | Proof If \( u \in H \) is given by\n\n\[ u\left( x\right) = \mathop{\sum }\limits_{{l \in {\dot{\mathbb{Z}}}^{3}}}{\widehat{u}}_{l}{\mathrm{e}}^{\mathrm{i}l \cdot x} \]\n\nand \( {\varphi }_{k}\left( x\right) = {\mathrm{e}}^{-\mathrm{i}k \cdot x} \) then\n\n\[ {\int }_{{\mathbb{T}}^{3}}u\left( x\right) \cdot \nabla {\v... | Yes |
Theorem 2.7 The space \( {\mathbb{L}}^{2}\left( {\mathbb{R}}^{3}\right) \) can be decomposed as\n\n\[ {\mathbb{L}}^{2}\left( {\mathbb{R}}^{3}\right) = H\left( {\mathbb{R}}^{3}\right) \oplus G\left( {\mathbb{R}}^{3}\right) \]\n\nwhere\n\n\[ H\left( {\mathbb{R}}^{3}\right) \mathrel{\text{:=}} \left\{ {u : u \in {\mathbb{... | Proof See Exercise 2.4. | No |
Lemma 2.9 The Leray projector \( \mathbb{P} \) on the torus and on the whole space commutes with any derivative:\n\n\[ \mathbb{P}\left( {{\partial }_{j}u}\right) = {\partial }_{j}\left( {\mathbb{P}u}\right) \;j = 1,2,3 \] \n\nfor all \( u \in {\dot{\mathbb{H}}}^{1} \) . | Proof We consider the case of a torus; for the proof in case of the whole space see Exercise 2.6. Let \( u \) be given by \( u\left( x\right) = {\widehat{u}}_{k}{\mathrm{e}}^{\mathrm{i}k \cdot x} \) . For any such \( u \) we have \( {}^{1} \n\n\[ \mathbb{P}{\partial }_{j}\left( u\right) = \mathbb{P}\left( {\mathrm{i}{k... | No |
Lemma 2.11 The functions in \( H\left( \Omega \right) \) are weakly divergence free. Moreover if \( u \in H \) then\n\n\[{\int }_{\Omega }u\left( x\right) \cdot \nabla \psi \left( x\right) \mathrm{d}x = 0\;\text{ for all }\;\psi \in {H}^{1}\left( \Omega \right) . | Proof Let \( u \in H,\psi \in {H}^{1}\left( \Omega \right) ,{\varphi }_{n} \in {C}_{c,\sigma }^{\infty }\left( \Omega \right) \) with \( {\varphi }_{n} \rightarrow u \) in \( H \) . Then\n\n\[{\int }_{\Omega }{\varphi }_{n}\left( x\right) \cdot \nabla \psi \left( x\right) \mathrm{d}x = - {\int }_{\Omega }\left( {\opera... | Yes |
Lemma 2.12 Let \( \Omega \) be a bounded open set with smooth boundary. \( {}^{2} \) If \( u \in E\left( \Omega \right) \) then the normal component of \( u \) is well defined on the boundary as a bounded linear functional on the space of traces of functions in \( {H}^{1}\left( \Omega \right) \) and the Gauss formula h... | Proof We sketch the argument; for more details see Proposition 1.4 in Constantin & Foias (1988) or Theorem 1.2 in Temam (1977).\n\nSuppose that \( {\varphi }_{n} \in {C}^{1}\left( \bar{\Omega }\right) \) with\n\n\[ {\varphi }_{n} \rightarrow u\text{ in }{\mathbb{L}}^{2}\text{ and }\operatorname{div}{\varphi }_{n} \righ... | Yes |
Corollary 2.13 Let \( \Omega \) be a smooth bounded domain. If \( u \in H\left( \Omega \right) \) then we have \( u \cdot n = 0 \) on \( \partial \Omega \) in the sense of Lemma 2.12. | Proof Since \( \operatorname{div}u = 0 \) for \( u \in H \), the corollary follows immediately from (2.4) and (2.5). | No |
Theorem 2.16 (Helmholtz-Weyl decomposition on \( \Omega \subset {\mathbb{R}}^{3} \) ) Let \( \Omega \) be an open bounded set in \( {\mathbb{R}}^{3} \) with Lipschitz boundary. Then\n\n\[ \n{\mathbb{L}}^{2}\left( \Omega \right) = H\left( \Omega \right) \oplus G\left( \Omega \right) \n\]\n\nwhere \( H \) and \( G \) are... | Proof Given \( u \in {\mathbb{L}}^{2}\left( \Omega \right) \), let \( {\phi }_{1} \in {H}^{1} \) be the weak solution of\n\n\[ \n\Delta {\phi }_{1} = \operatorname{div}u\;\text{ in }\Omega ,\;{\left. {\phi }_{1}\right| }_{\partial \Omega } = 0.\n\]\n\n(Notice that the problem is well posed since div \( u \in {H}^{-1} \... | Yes |
Theorem 2.18 On every domain\n\n\[ \langle \mathbb{P}u, v\rangle = \langle u,\mathbb{P}v\rangle \text{ for all }u, v \in {\mathbb{L}}^{2}. \] | Proof If \( u = {h}_{1} + \nabla {g}_{1} \) and \( v = {h}_{2} + \nabla {g}_{2} \) then\n\n\[ \langle \mathbb{P}u, v\rangle = \left\langle {{h}_{1},{h}_{2} + \nabla {g}_{2}}\right\rangle = \left\langle {{h}_{1},{h}_{2}}\right\rangle = \left\langle {{h}_{1} + \nabla {g}_{1},{h}_{2}}\right\rangle = \langle u,\mathbb{P}v\... | Yes |
Example 2.19 Consider \( u\left( {{x}_{1},{x}_{2},{x}_{3}}\right) = \left( {{x}_{2}, - {x}_{1},0}\right) \) on the cylinder\n\n\[ Q = \left\{ {\left( {{x}_{1},{x}_{2},{x}_{3}}\right) \in {\mathbb{R}}^{3} : {x}_{1}^{2} + {x}_{2}^{2} < 1, - 1 < {x}_{3} < 1}\right\} . \]\n\nThen \( u \) is divergence free and its normal c... | It follows that\n\n\[ {\partial }_{1}\mathbb{P}u = {\partial }_{1}u = \left( {0, - 1,0}\right) \]\n\nwhile\n\n\[ \mathbb{P}{\partial }_{1}u = \mathbb{P}\left( {0, - 1,0}\right) = \left( {0,0,0}\right) ,\]\n\nsince \( \left( {0, - 1,0}\right) = \nabla \left( {-{x}_{2}}\right) \) . | No |
Theorem 2.22 On the whole space \( {\mathbb{R}}^{3} \) and on the torus \( {\mathbb{T}}^{3} \) we have\n\n\[ - \mathbb{P}{\Delta u} = - \Delta \mathbb{P}u \]\n\nfor all \( u \in {\mathbb{H}}^{2} \) . Hence\n\n\[ {Au} = - {\Delta u} \]\n\nfor all \( u \in D\left( A\right) \) on \( {\mathbb{T}}^{3} \) and \( {\mathbb{R}}... | From Theorem 2.22 it follows that on the torus the Stokes operator is given by the explicit formula\n\n\[ {Au}\left( x\right) = \mathop{\sum }\limits_{{k \in {\dot{\mathbb{Z}}}^{3}}}{\left| k\right| }^{2}{\widehat{u}}_{k}{\mathrm{e}}^{\mathrm{i}k \cdot x},\;u \in D\left( A\right) ,\]\n\nso \( {Au} = {\Lambda }^{2}u \),... | No |
Theorem 2.23 On the torus \( {\mathbb{T}}^{3} \) and on a smooth bounded domain we have\n\n\[ \parallel u{\parallel }_{{\mathbb{H}}^{m + 2}} \leq c\parallel {Au}{\parallel }_{{\mathbb{H}}^{m}} \]\n\nfor all \( u \in V \) such that \( {Au} \in {\mathbb{H}}^{m} \) and all \( m \in \mathbb{N} \) . On the whole space we ha... | Proof On a smooth bounded domain the result follows, as we mentioned before, from the regularity results for the Stokes problem (for more details see Galdi, 2011).\n\nOn the torus we have\n\n\[ \parallel {Au}{\parallel }_{{\dot{\mathbb{H}}}^{m}}^{2} = {\begin{Vmatrix}\mathop{\sum }\limits_{{k \in {\dot{\mathbb{Z}}}^{3}... | Yes |
Lemma 2.26 Let \( \Omega \) be a smooth bounded domain or the torus. If \( u, v \in D\left( {A}^{s}\right) \) then\n\n\[ \left\langle {{A}^{{s}_{1}}u,{A}^{{s}_{2}}v}\right\rangle = \left\langle {u,{A}^{s}v}\right\rangle ,\;{s}_{1} + {s}_{2} = s,\;{s}_{1} \geq 0,\;{s}_{2} \geq 0. \] | Proof See Exercise 2.10. | No |
Theorem 2.27 In the case of the torus \( D\left( {A}^{s/2}\right) = {\dot{H}}^{s}\left( {\mathbb{T}}^{3}\right) \) and\n\n\[ \parallel u{\parallel }_{{\dot{H}}^{s}} = \begin{Vmatrix}{{A}^{s/2}u}\end{Vmatrix}. \] | Proof Simply note that \( {A}^{s/2}u = {\Lambda }^{s}u \), which follows from (1.18) and the argument used in the proof of Theorem 2.24. | Yes |
Lemma 3.2 Let \( u \in V \) and \( v, w \in {H}^{1} \). Then\n\n\[ \langle \left( {u \cdot \nabla }\right) v, w\rangle = - \langle \left( {u \cdot \nabla }\right) w, v\rangle . \]\n\nIn particular\n\n\[ \langle \left( {u \cdot \nabla }\right) v, v\rangle = 0. \] | Proof First, observe that for \( u \in {C}_{c,\sigma }^{\infty }\left( \Omega \right) \) and \( v, w \in {C}^{1}\left( \Omega \right) \) we have from an integration by parts (using Einstein's summation convention)\n\n\[ \langle \left( {u \cdot \nabla }\right) v, w\rangle = \int {u}_{j}\left( {{\partial }_{j}{v}_{i}}\ri... | No |
Lemma 3.4 If \( u \in {L}^{\infty }\left( {0, T;H}\right) \cap {L}^{2}\left( {0, T;V}\right) \) then\n\n\[ \left( {u \cdot \nabla }\right) u \in {L}^{4/3}\left( {0, T;{L}^{6/5}}\right) . \] | Proof From the Hölder inequality with exponents \( 5/3 \) and \( 5/2 \) we have\n\n\[ \parallel \left( {u \cdot \nabla }\right) u{\parallel }_{{L}^{6/5}} \leq {\left( \int {\left| \nabla u\right| }^{6/5}{\left| u\right| }^{6/5}\right) }^{5/6} \]\n\n\[ \leq {\left( \int {\left| \nabla u\right| }^{2}\right) }^{1/2}{\left... | Yes |
Lemma 3.5 If \( u \in {L}^{\infty }\left( {0, T;H}\right) \cap {L}^{2}\left( {0, T;V}\right) \) then\n\n\[ u \in {L}^{r}\left( {0, T;{L}^{s}\left( \Omega \right) }\right) \]\n\nwhere\n\n\[ \frac{2}{r} + \frac{3}{s} = \frac{3}{2},\;2 \leq s \leq 6. \]\n\nIn particular\n\n\[ u \in {L}^{{10}/3}\left( {\left( {0, T}\right)... | Proof We interpolate the \( {L}^{s} \) norm between \( {L}^{2} \) and \( {L}^{6} \)\n\n\[ \parallel u{\parallel }_{{L}^{s}} \leq \parallel u{\parallel }_{{L}^{2}}^{\left( {6 - s}\right) /{2s}}\parallel u{\parallel }_{{L}^{6}}^{\left( {{3s} - 6}\right) /{2s}}. \]\n\nIt follows that\n\n\[ {\int }_{0}^{T}\parallel u{\para... | Yes |
Lemma 3.6 If \( u \) is a weak solution of the Navier-Stokes equations then\n\n\[ \n{\int }_{{t}_{1}}^{{t}_{2}}\langle \nabla u,\nabla \phi \rangle + {\int }_{{t}_{1}}^{{t}_{2}}\langle \left( {u \cdot \nabla }\right) u,\phi \rangle = \left\langle {u\left( {t}_{1}\right) ,\phi }\right\rangle - \left\langle {u\left( {t}_... | Proof Take any \( \phi \in {C}_{c,\sigma }^{\infty }\left( \Omega \right) \) and choose \( \alpha \in {C}_{c}^{\infty }\lbrack 0,\infty ) \) with \( \alpha \left( t\right) = 1 \) for all \( t \in \left\lbrack {{t}_{1},{t}_{2}}\right\rbrack \) . Then \( \widetilde{\varphi } \) given by\n\n\[ \n\widetilde{\varphi }\left(... | Yes |
Lemma 3.7 If \( u \) is a weak solution of the Navier-Stokes equations then\n\n\[ \n{\partial }_{t}u \in {L}^{4/3}\left( {0, T;{V}^{ * }}\right) .\n\]\n\nIn particular, one can modify \( u \) on the set of measure zero in such a way that\n\n\[ \nu \in C\left( {\left\lbrack {0, T}\right\rbrack ;{V}^{ * }}\right) \n\]\n\... | Proof From Lemma 3.6 we have, for almost all \( t > 0 \) and for every choice of \( \varphi \in {C}_{c,\sigma }^{\infty }\left( \Omega \right) \)\n\n\[ \n\langle u\left( t\right) ,\varphi \rangle - \left\langle {{u}_{0},\varphi }\right\rangle = {\int }_{0}^{t}\langle g,\varphi {\rangle }_{{V}^{ * } \times V}\n\]\n\nwhe... | Yes |
Theorem 3.8 Weak solutions of the Navier-Stokes equations are \( {L}^{2} \) -weakly continuous in time, i.e.\n\n\[ \mathop{\lim }\limits_{{t \rightarrow {t}_{0}}}\langle u\left( t\right), v\rangle = \left\langle {u\left( {t}_{0}\right), v}\right\rangle \;\text{ for all }\;v \in {L}^{2}\left( \Omega \right) . | Proof We take \( v \in {L}^{2} \) if \( \Omega \) is the whole space or a bounded subset of \( {\mathbb{R}}^{3} \) ; and we take \( v \in {\dot{L}}^{2} \) for the torus, since on \( {\mathbb{T}}^{3} \) we can assume that \( v \in {\dot{L}}^{2} \), because \( u \) then has zero average and in consequence \( \langle u\le... | Yes |
Corollary 3.9 Let \( u \) be a weak solution of the Navier-Stokes equations. Then for every \( T > 0 \) the value of \( u\left( T\right) \) is uniquely determined as the weak limit in the space \( {L}^{2} \) : | \[ u\left( t\right) \rightharpoonup u\left( T\right) \;\text{ weakly in }{L}^{2}\text{ when }t \rightarrow T. \] In particular, for any given weak solution \( u \) for every \( T > 0 \) the value of \( u\left( T\right) \) is uniquely determined by the values of \( u\left( t\right) \) on the time interval \( \left( {0, ... | Yes |
Lemma 3.12 Let \( u \) be a weak solution of the Navier-Stokes equations, take \( \alpha \in {H}^{1}\left( {0, T}\right) \) and \( a \in \mathcal{N} \) . Then (3.3) holds for\n\n\[ \varphi \left( {x, t}\right) = \alpha \left( t\right) a\left( x\right) \]\n\nfor almost all \( s \in \left\lbrack {0, T}\right\rbrack \) . | Proof From the assumption we know that there is \( g = \dot{\alpha } \in {L}^{2}\left( {0, T}\right) \) (by \( \dot{\alpha } \) we denote the time derivative) and\n\n\[ \alpha \left( t\right) = {c}_{0} + {\int }_{0}^{t}g\left( s\right) \mathrm{d}s. \]\n\nNow let\n\n\[ {\alpha }_{\varepsilon }\left( t\right) \mathrel{\t... | Yes |
Proposition 3.13 If \( u \) is a weak solution of the Navier-Stokes equations then\n\n\[{\int }_{0}^{\infty } - \left\langle {u,{\partial }_{t}\varphi }\right\rangle + {\int }_{0}^{\infty }\langle \nabla u,\nabla \varphi \rangle + {\int }_{0}^{\infty }\langle \left( {u \cdot \nabla }\right) u,\varphi \rangle = \left\la... | Proof Take any \( \varphi \in {\widetilde{\mathcal{D}}}_{\sigma } \) . From Lemma 3.11 it follows that (3.3) is valid for \( \varphi \) for almost all \( s > 0 \) . Since \( \varphi \left( s\right) = 0 \) for large \( s \) it follows that the equation above is satisfied. | No |
Lemma 4.1 Let \( \Omega \) be the torus \( {\mathbb{T}}^{3} \) or a smooth bounded domain in \( {\mathbb{R}}^{3} \) . If \( u \in D\left( {A}^{s/2}\right), s > 0 \), then \( {P}_{n}u \rightarrow u \) in \( {H}^{s}\left( \Omega \right) \) and\n\n\[ \n{\begin{Vmatrix}{P}_{n}u\end{Vmatrix}}_{{H}^{s}} \leq c\parallel u{\pa... | Proof See Exercise 4.1 for the case of the torus (the case of a bounded domain is similar). | No |
Theorem 4.3 (Aubin-Lions Lemma; simple version) Let \( H \) be \( H\left( {\mathbb{T}}^{3}\right) \) or \( H\left( \Omega \right) \), where \( \Omega \) is a smooth bounded subset of \( {\mathbb{R}}^{3} \). Assume that \( p, q > 1 \) and \[ {\begin{Vmatrix}{u}_{n}\end{Vmatrix}}_{{L}^{q}\left( {0, T;V}\right) } + {\begi... | This additional strong convergence in \( {L}^{2} \) is enough to pass to the limit in (4.2) for any fixed \( \varphi \in {\widetilde{\mathcal{D}}}_{\sigma } \) and thereby conclude that the limit function \( u \) satisfies the weak formulation of the Navier-Stokes equations for all \( \varphi \in {\widetilde{\mathcal{D... | No |
Lemma 4.5 If \( {u}_{0} \in H\left( {\mathbb{T}}^{3}\right) \) then there exists a weak solution of the Navier-Stokes equations corresponding to \( {u}_{0} \) and such that\n\n\[{\partial }_{t}u \in {L}^{4/3}\left( {0, T;{H}^{-1}}\right) . | Proof The proof is exactly the same as before but differs in Step 3 where we can prove a stronger uniform bound on \( {\partial }_{t}{u}_{n} \) :\n\n\[{\begin{Vmatrix}{\partial }_{t}{u}_{n}\end{Vmatrix}}_{{L}^{4/3}\left( {0, T;{H}^{-1}}\right) } \leq C\text{ for all }n \in \mathbb{N}.\n\nIndeed, in the case of the toru... | Yes |
Theorem 4.6 Let \( \Omega \) be a bounded domain in \( {\mathbb{R}}^{3} \) (or the torus \( {\mathbb{T}}^{3} \) ). The weak solution u of the Navier-Stokes equations constructed in Theorem 4.4 satisfies the strong energy inequality: | Proof It follows from (4.13) that for almost all \( s, t \in \lbrack 0,\infty ) \) we have \[ {u}_{n}\left( s\right) \rightarrow u\left( s\right) \text{ and }{u}_{n}\left( t\right) \rightarrow u\left( t\right) \] strongly in \( {L}^{2}\left( \Omega \right) \), so in particular \[ {\begin{Vmatrix}{u}_{n}\left( s\right) ... | Yes |
For the weak solutions constructed in Theorem 4.4 we have\n\n\[ u\\left( t\\right) \\rightarrow u\\left( 0\\right) \\;\\text{ strongly in }\\;{L}^{2}\\left( \\Omega \\right) \\;\\text{ as }\\;t \\rightarrow {0}^{ + }.\n\] | Proof From the energy inequality in (4.17),\n\n\[ \\parallel u\\left( t\\right) {\\parallel }^{2} + 2{\\int }_{0}^{t}\\parallel \\nabla u\\left( s\\right) {\\parallel }^{2}\\mathrm{\\;d}s \\leq \\parallel u\\left( 0\\right) {\\parallel }^{2},\n\]\n\nwe obtain\n\n\[ \\mathop{\\limsup }\\limits_{{t \\rightarrow {0}^{ + }... | Yes |
Corollary 4.8 Ifs is a time such that (4.16) holds then \( \parallel u\left( \cdot \right) \parallel \) is right continuous at \( s \) . | Proof Passing to the limit as \( t \rightarrow {s}^{ + } \) in (4.16) gives\n\n\[ \mathop{\limsup }\limits_{{t \rightarrow {s}^{ + }}}\parallel u\left( t\right) \parallel \leq \parallel u\left( s\right) \parallel \]\n\nSince \( u\left( s\right) \) is a weak limit of \( u\left( t\right) \) when \( t \rightarrow {s}^{ + ... | Yes |
On the domain \( {\mathbb{T}}^{3} \) or \( {\mathbb{R}}^{3} \) suppose that\n\n\[ - {\Delta p} = {\partial }_{i}{\partial }_{j}\left( {{u}_{i}{u}_{j}}\right) \]\n\nto ensure uniqueness we impose an additional condition: on the torus we require \( {\int }_{{\mathbb{T}}^{3}}p = 0 \) and on the whole space we require \( p... | Proof We concentrate on the case of the whole space, which is the simplest situation, given that we require the Calderón-Zygmund Theorem. As we have already observed, it follows from (5.6) that\n\n\[ p = {\left( -\Delta \right) }^{-1}{\partial }_{i}{\partial }_{j}\left( {{u}_{i}{u}_{j}}\right) \]\n\nwhere \( {\left( -\... | Yes |
Corollary 5.2 If \( u \) is a weak solution in the absence of boundaries then the corresponding pressure \( p \) satisfies \( {}^{1} \n\n\[ \np \in {L}^{r}\left( {0, T;{L}^{s}}\right) \;\text{ for }\;2/r + 3/s = 3,\;s > 1 \n\] | Proof It is easy to see that (5.13) follows from (5.11) together with the bound \( \parallel p{\parallel }_{{L}^{s}} \leq {C}_{s}\parallel u{\parallel }_{{L}^{2s}}^{2} \) (equation (5.7)). (Note that we must exclude \( s = 1 \) since the Calderón-Zygmund estimate in (5.10) is not valid in \( {L}^{1} \) .)\n\nTo bound \... | Yes |
Theorem 5.6 For every initial condition \( {u}_{0} \in H\left( {\mathbb{T}}^{3}\right) \) there exists a weak solution \( u \) that satisfies the Navier-Stokes equations in the distributional sense: there exists a \( p \in {L}_{\text{loc }}^{4/3}\left( {0,\infty ;{W}^{1,6/5}\left( {\mathbb{T}}^{3}\right) }\right) \) su... | Proof Let \( u \) be a weak solution constructed in the proof of Theorem 5.5.\n\nChoose \( T > 0 \) and take \( \varphi \in {\mathcal{D}}_{\sigma } \) with compact support in \( {\mathbb{T}}^{3} \times \left( {0, T}\right) \) . Then\n\n\[- {\int }_{0}^{T}\left\langle {u,{\partial }_{t}\varphi }\right\rangle + {\int }_{... | Yes |
Proposition 5.8 On a bounded domain \( \Omega \) every weak solutions satisfies the Navier-Stokes equations in the sense of distributions, i.e.\n\n\[{\int }_{0}^{\infty }\left\langle {u,{\partial }_{t}\varphi }\right\rangle + {\int }_{0}^{\infty }\langle u,{\Delta \varphi }\rangle + {\int }_{0}^{\infty }\langle u \otim... | Proof We have shown that (5.32) holds almost everywhere in \( \Omega \times \left( {0, T}\right) \) . Multiplying this equation by \( \varphi \in {C}_{c}^{\infty }\left( {\Omega \times \left( {0, T}\right) }\right) \), integrating over the spacetime domain \( \Omega \times \left( {0, T}\right) \), and finally integrati... | Yes |
Lemma 6.3 Suppose that \( u \) is a strong solution of the Navier-Stokes equations on \( \left\lbrack {0, T}\right\rbrack \) . Then\n\n\[{\int }_{0}^{T}\left\langle {{\partial }_{t}u - {\Delta u} + \left( {u \cdot \nabla }\right) u, w}\right\rangle = 0\]\n\n(6.1)\n\nfor all \( w \in {L}^{2}\left( {0, T;H}\right) \) . | Proof First we use Lemma 1.28 to find a sequence \( {w}_{n} \in {\widetilde{\mathcal{D}}}_{\sigma } \) (see Definition 3.10) such that\n\n\[{w}_{n} \rightarrow w\text{ in }{L}^{2}\left( {0, T;H}\right)\]\n\nSince \( {w}_{n} \in {\widetilde{\mathcal{D}}}_{\sigma } \) we conclude from Lemma 3.11 that\n\n\[{\int }_{0}^{T}... | Yes |
Theorem 6.4 Let \( \Omega \) be the torus \( {\mathbb{T}}^{3} \), a smooth bounded domain in \( {\mathbb{R}}^{3} \) or the whole space \( {\mathbb{R}}^{3} \) . Suppose that \( u \) is a strong solution of the Navier-Stokes equations on \( \left\lbrack {0, T}\right\rbrack \) . Then there exists a function \( p \in {L}^{... | Proof Let \( \mathbb{P} \) denote the Leray projector from Chapter 2. For almost all times \( s \in \left\lbrack {0, T}\right\rbrack \) we have\n\n\[{\partial }_{t}u\left( s\right) - {\Delta u}\left( s\right) + \left( {u\left( s\right) \cdot \nabla }\right) u\left( s\right) \in {L}^{2}\left( \Omega \right)\]\n\nand we ... | Yes |
Theorem 6.5 If \( u \) is a strong solution of the Navier-Stokes equations on \( \left\lbrack {0, T}\right\rbrack \) then \( u \) satisfies the energy equality\n\n\[ \frac{1}{2}{\begin{Vmatrix}u\left( {t}_{1}\right) \end{Vmatrix}}^{2} + {\int }_{{t}_{0}}^{{t}_{1}}\parallel \nabla u\left( s\right) {\parallel }^{2}\mathr... | Proof Let \( u \) be a strong solution and let \( {\chi }_{\left\lbrack {t}_{0},{t}_{1}\right\rbrack } \) be the characteristic function of the interval \( \left\lbrack {{t}_{0},{t}_{1}}\right\rbrack \), where \( 0 \leq {t}_{0} < {t}_{1} \leq T \).\n\nThen \( u{\chi }_{\left\lbrack {t}_{0},{t}_{1}\right\rbrack } \in {L... | Yes |
Lemma 6.6 Let \( u \) be a weak solution of the Navier-Stokes equations. If\n\n\[ v \in {L}^{2}\left( {0, T;{H}^{2} \cap V}\right) \;\text{ and }\;{\partial }_{t}v \in {L}^{2}\left( {0, T;{L}^{2}}\right) \]\n\nthen \( {}^{2} \) for all \( t \in \left\lbrack {0, T}\right\rbrack \)\n\n\[ - {\int }_{0}^{t}\left\langle {u,... | Proof We will give a proof valid for the torus and a smooth bounded domain, but a similar argument works also for the whole space. \( {}^{3} \) Define \( {v}_{n} \mathrel{\text{:=}} {P}_{n}\left( v\right) \) , where \( {P}_{n} \) is the projection onto the first \( n \) eigenfunctions of the Stokes operator. Since \( {... | No |
Lemma 6.7 Let \( u \in {L}^{2}\left( {0, T;D\left( A\right) }\right) \) with \( {\partial }_{t}u \in {L}^{2}\left( {0, T;H}\right) \) . Then the function \( t \mapsto \parallel \nabla u{\parallel }^{2} \) is absolutely continuous on \( \left\lbrack {0, T}\right\rbrack \) and\n\n\[ \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d... | Proof The result is valid on all three domains considered in this book; we give a proof in the case of the torus or a smooth bounded domain. Take a sequence \( {u}_{n} \in {C}^{1}\left( {\left\lbrack {0, T}\right\rbrack ;{P}_{n}H}\right) \) converging to a strong solution \( u \) in the following way:\n\n\[ {u}_{n} \ri... | No |
Corollary 6.9 Let \( \Omega \) be a smooth bounded domain in \( {\mathbb{R}}^{3},\lambda \geq 1 \) and\n\n\[ \n{\Omega }_{\lambda } \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{3} : x = {\lambda y}, y \in \Omega }\right\} .\n\]\n\nThere exists a constant \( c > 0 \), independent of \( \lambda \), with the following... | Proof Our task is to show that \( c \) does not depend on \( \lambda \) . So take any choice of \( {v}_{0} \in V\left( {\Omega }_{\lambda }\right) \), and let \( {\begin{Vmatrix}\nabla {v}_{0}\end{Vmatrix}}_{{L}^{2}\left( {\Omega }_{\lambda }\right) }^{2} = M \) . Define\n\n\[ \n{u}_{0}\left( x\right) \mathrel{\text{:=... | Yes |
Lemma 6.11 There exists a constant \( {c}^{\prime } > 0 \) (depending only on the domain) such that if \( u \) is a Leray-Hopf weak solution of the Navier-Stokes equations for some initial data \( {u}_{0} \in V \) that blows up at time \( {T}^{ * } > 0 \) then\n\n\[ \parallel \nabla u\left( t\right) {\parallel }^{2} \g... | Proof We know that \( u\left( t\right) \in V \) and the strong energy inequality is satisfied from \( s = t \) for all \( t < {T}^{ * } \), see Theorem 6.5. For all such \( t \), using Theorem 6.8, we can construct a strong solution \( v \) with initial condition \( v\left( 0\right) = u\left( t\right) \), whose maximal... | Yes |
Theorem 6.12 Let \( \Omega \) be the torus \( {\mathbb{T}}^{3} \) or a smooth bounded domain in \( {\mathbb{R}}^{3} \). There exists a constant \( C > 0 \) (depending on \( \Omega \) ) such that if\n\n\[ \begin{Vmatrix}{\nabla {u}_{0}}\end{Vmatrix} < C \]\n\nthen the strong solution corresponding to \( {u}_{0} \in V \)... | Proof Instead of working with the Galerkin approximations we will make estimates directly for \( u \) . We know that \( {u}_{0} \) gives rise to a strong solution that exists at least on a certain time interval \( \lbrack 0, T) \) . On this time interval for each time \( t \in \left( {0, T}\right) \) we take the \( {L}... | Yes |
Theorem 6.14 (Ladyzhenskaya) Let \( {u}_{0} \in V\left( \Omega \right) \), where \( \Omega \) is a smooth domain in \( {\mathbb{R}}^{2} \) . Then \( {u}_{0} \) gives rise to a strong global-in-time solution of the Navier-Stokes equations. | Proof (sketch) Taking the \( {L}^{2} \) inner product of the 2D Navier-Stokes equations with \( {Au} \), we then integrate by parts and obtain, just as in the \( 3\mathrm{D} \) case, the inequality\n\n\[ \n\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\parallel \nabla u{\parallel }^{2} + \parallel {Au}{\parallel }^{2} \leq ... | No |
Theorem 6.15 There exists a constant \( c > 0 \) such that every initial condition \( {u}_{0} \in {H}^{1}\left( {\mathbb{R}}^{3}\right) \) gives rise to a strong solution of the Navier-Stokes equations on the time interval \( \left\lbrack {0, c{\begin{Vmatrix}\nabla {u}_{0}\end{Vmatrix}}^{-4}}\right\rbrack \) . | Proof We will give a proof corresponding to the construction of weak solutions in Section 4.4.\n\nLet \( {B}_{n} = B\left( {0,{R}_{n}}\right) \) denote the ball centred at zero and of radius \( {R}_{n} \), where \( {R}_{n} \rightarrow \infty \), as \( n \rightarrow \infty \) . For a given \( {u}_{0} \in V\left( {\mathb... | Yes |
Corollary 7.2 For every \( {u}_{0} \in V \cap {H}^{k}, k \geq 2 \) there exists a local-in-time strong solution \( u \in {L}^{\infty }\left( {0, T;{H}^{k}}\right) \cap {L}^{2}\left( {0, T;{H}^{k + 1}}\right) \) . | We can now use Theorem 7.1 to prove the smoothness of strong solutions in the space variables on the time interval \( \left( {0, T}\right) \) . We require no more regularity of the initial condition than \( {u}_{0} \in V \) . (The proof is based on that of Theorem 10.6 in Constantin & Foias, 1988.) | No |
Theorem 7.3 Let \( u \) be a strong solution of the Navier-Stokes equations \( {}^{2} \) on the time interval \( \left\lbrack {0, T}\right\rbrack \), with initial condition \( {u}_{0} \in V \) . Then for all \( 0 < \varepsilon < T \) we have\n\n\[ u \in C\left( {\left\lbrack {\varepsilon, T}\right\rbrack ;{H}^{m}}\righ... | Proof Take any \( \varepsilon > 0 \) . From Theorem 7.1 it follows that for almost all \( s \in \left( {0,\varepsilon }\right) \) we have \( u\left( s\right) \in {H}^{2} \), so we can choose an \( {s}_{1} \in \left( {0,\varepsilon }\right) \) such that\n\n\( u\left( {s}_{1}\right) \in V \cap {H}^{2} \) . From Theorem 7... | Yes |
Lemma 7.4 Let \( u \) be a strong solution of the Navier-Stokes equations on the time interval \( \left\lbrack {0, T}\right\rbrack \) . Then for every \( \varepsilon > 0 \), and all \( j, k \in \mathbb{N} \) we have\n\n\[{\partial }_{t}^{j}u \in {L}^{\infty }\left( {\varepsilon, T;{H}^{k}}\right) . | Proof We have shown that if \( {u}_{0} \in V \) then for every \( \varepsilon \in \left( {0, T}\right) \) the Galerkin approximations \( {u}_{n} \) are uniformly bounded in \( {L}^{\infty }\left( {\varepsilon, T;{H}^{k}}\right) \) for all \( k \in \mathbb{N} \) . From the equality\n\n\[{\partial }_{t}{u}_{n} = - A{u}_{... | Yes |
Theorem 7.6 Let \( u \) be a Leray-Hopf weak solution of the Navier-Stokes equations corresponding to an initial condition \( {u}_{0} \in V \) . If\n\n\[ \n{\int }_{0}^{T}\parallel \nabla u\left( s\right) {\parallel }_{\infty }\mathrm{d}s < \infty \n\]\n\nthen \( u \) is strong on \( \left\lbrack {0, T}\right\rbrack \)... | Proof See Exercise 7.4. | No |
Theorem 8.1 Any global-in-time Leray-Hopf weak solution \( u \) is eventually strong: there exists a \( {T}^{ * } > 0 \) such that \( u \in {C}^{\infty }\left( {\bar{\Omega } \times \left( {{T}^{ * },\infty }\right) }\right) \) . | Proof From Theorem 6.13 we know that there is a \( C > 0 \) such that\n\n\[ \begin{Vmatrix}{u}_{0}\end{Vmatrix}\begin{Vmatrix}{\nabla {u}_{0}}\end{Vmatrix} \leq C\; \Rightarrow \;u\text{ is strong on }\left( {0,\infty }\right) .\n\]\n\nFrom the energy inequality (4.17) we deduce that\n\n\[ \parallel u\left( s\right) \p... | Yes |
Lemma 8.3 The set \( \mathcal{T} \) of singular times of a Leray-Hopf weak solution is compact. | Proof From Theorem 8.1 we know that \( \mathcal{T} \) must be bounded. It is also easy to see that \( \mathcal{T} \) is closed, because it follows from the definition of singular points that every accumulation point of \( \mathcal{T} \) must be singular. | Yes |
Lemma 8.4 A Leray-Hopf weak solution is smooth on the open set \( \Omega \times \mathcal{R} \) . | Proof By definition, for any \( {t}_{0} \in \mathcal{R} \) there is some \( \delta > 0 \) and \( M > 0 \) such that\n\n\[ \parallel u\left( s\right) {\parallel }_{{H}^{1}} \leq M\;\text{ for almost all }\;s \in \left( {{t}_{0} - \delta ,{t}_{0} + \delta }\right) . \]\n\nIt follows from Theorems 6.8 and 7.5 that there e... | Yes |
For a compact set \( X \subset {\mathbb{R}}^{n} \) we can also define the box-counting dimension as\n\n\[ \n{\dim }_{\mathrm{B}}\left( X\right) = \mathop{\limsup }\limits_{{\varepsilon \rightarrow {0}^{ + }}}\frac{\log M\left( {X,\varepsilon }\right) }{-\log \varepsilon } \n\]\n\nwhere \( M\left( {X,\varepsilon }\right... | Proof First we show that\n\n\[ \nM\left( {X,\varepsilon }\right) \leq N\left( {X,\varepsilon }\right) . \n\]\n\nTo prove (8.3) suppose that \( {c}_{i} \in X \) are the centres of disjoint balls of radius \( \varepsilon > 0, i = 1,2,\ldots, M\left( {X,\varepsilon }\right) \) . Since no ball of radius \( \varepsilon \) c... | Yes |
Lemma 8.7 Suppose that \( K\left( \varepsilon \right) = N\left( {X,\varepsilon }\right) \) or \( M\left( {X,\varepsilon }\right) \) and define\n\n\[ \nd = \mathop{\limsup }\limits_{{\varepsilon \rightarrow {0}^{ + }}}\frac{\log K\left( \varepsilon \right) }{-\log \varepsilon }.\n\]\n\nThen\n\n(i) for every \( 0 < {d}^{... | Proof (i) It follows from (8.5) that there exists a sequence \( {\varepsilon }_{j} \rightarrow {0}^{ + } \) such that\n\n\[ \n{\log }_{{\varepsilon }_{j}}K\left( {\varepsilon }_{j}\right) \rightarrow - d\n\]\n\nso for \( \delta = d - {d}^{\prime } > 0 \) we can find \( N \) such that for all \( j > N \) we must have\n\... | No |
Lemma 8.11 For any compact set \( X \subset {\mathbb{R}}^{n} \) we have\n\n\[{\dim }_{\mathrm{H}}\left( X\right) \leq {\dim }_{\mathrm{B}}\left( X\right)\] | Proof Suppose that \( {\dim }_{\mathrm{B}}\left( X\right) = d \) and take any \( s > d \) . We want to prove that for every \( \delta > 0 \) it is possible to cover the set \( X \) with a countable family of balls of radii \( {r}_{1},{r}_{2},{r}_{3},\ldots \) such that \( \mathop{\sum }\limits_{{i = 1}}^{\infty }{r}_{i... | Yes |
Corollary 8.12 If \( X \subset {\mathbb{R}}^{n} \) is compact and \( {\dim }_{\mathrm{B}}\left( X\right) < n \) then \( X \) has zero Lebesgue measure. | Proof By Lemma 8.11 it follows that \( {\dim }_{\mathrm{H}}\left( X\right) \leq {\dim }_{\mathrm{B}}\left( X\right) < n \), and hence \( {\mathcal{H}}^{n}\left( X\right) = 0 \) . Since \( {\mathcal{H}}^{n} \) is proportional to the \( n \) -dimensional Lebesgue measure the result follows. | Yes |
Theorem 8.13 If \( u \) is a Leray-Hopf weak solution of the Navier-Stokes equations then the box-counting dimension of the set \( \mathcal{T} \) of singular times corresponding to \( u \) is no greater than \( 1/2 \) . | Proof Fix some small \( \varepsilon > 0 \) . Let \( F \) be a maximal family of disjoint intervals of radius \( \varepsilon \) centred at points of the compact (see Lemma 8.3) set \( \mathcal{T} \) . Let \( {t}_{0},{t}_{1},{t}_{2},\ldots ,{t}_{M\left( {\mathcal{T},\varepsilon }\right) } \) be the centres of the interva... | Yes |
Theorem 8.14 (Epochs of regularity) If \( u \) is a Leray-Hopf weak solution of the Navier-Stokes equations then \( u \) is smooth on an open set of times\n\n\[ \mathcal{R} = \mathop{\bigcup }\limits_{{i = 1}}^{\infty }\left( {{a}_{i},{b}_{i}}\right) \]\n\nthat is of full measure. On every \( \left\lbrack {a, b}\right\... | Proof The assertion about \( \mathcal{R} \) follows from Lemma 8.4, so we only need to prove that the \( 1/2 \) -dimensional Hausdorff measure of \( \mathcal{T} \) is zero.\n\nChoose any \( \delta > 0 \) . Since, according to previous results, the one-dimensional Hausdorff measure of the set \( \mathcal{T} \) is zero w... | Yes |
Theorem 8.17 Let \( u \) be a Leray-Hopf weak solution of the Navier-Stokes equations satisfying the 'Serrin condition'\n\n\[ u \in {L}^{r}\left( {0, T;{L}^{s}}\right) ,\;\text{ where }\;\frac{2}{r} + \frac{3}{s} = 1,\;r \geq 2, s > 3.\]\n\nThen \( u \) is smooth on the time interval \( (0, T\rbrack \) and for every \(... | We will now prove that Leray-Hopf weak solutions satisfying the Serrin condition (8.10) are unique in the class of all weak solutions satisfying the energy inequality. | No |
Lemma 8.18 If \( v \) is a Leray-Hopf weak solution satisfying the Serrin condition (8.10) on \( \left\lbrack {0, T}\right\rbrack \) then \( v \) can be used as a test function in (3.3) for a Leray-Hopf weak solution \( u \) . | Proof Let \( v \) be a Leray-Hopf weak solution of the Navier-Stokes equations that satisfies the Serrin condition. Since \( v \) is a smooth strong solution on \( \left\lbrack {\varepsilon, T}\right\rbrack \) for every \( \varepsilon > 0 \) it follows from Lemma 6.6 that\n\n\[ - {\int }_{\varepsilon }^{t}\left\langle ... | Yes |
Theorem 8.19 Leray-Hopf weak solutions satisfying the Serrin condition (8.10) are unique in the class of all weak solutions satisfying the energy inequality. | Proof The proof follows that of Theorem 6.10 (weak-strong uniqueness) with only two differences. First we use Lemma 8.18 instead of Lemma 6.6 to obtain for \( w = u - v \)\n\n\[ \n\frac{1}{2}\parallel w\left( t\right) {\parallel }^{2} + {\int }_{0}^{t}\parallel \nabla w\left( s\right) {\parallel }^{2}\mathrm{\;d}s \leq... | Yes |
Lemma 9.2 Take \( \\alpha > 0 \) and let \( {f}_{n} \\geq 0 \) be a sequence of non-negative functions on the time interval \( \\left\\lbrack {0, T}\\right\\rbrack \) . Suppose that \( \\left( {y}_{n}\\right) \) is a sequence of non-negative functions on \( \\left( {0, T}\\right) \) that satisfy\n\n\[ \n\\frac{\\mathrm... | Proof We compare \( {y}_{n}\\left( t\\right) \) with \( {Y}_{n}\\left( t\\right) \), the solution of\n\n\[ \n\\mathrm{d}{Y}_{n}/\\mathrm{d}t = \\alpha {Y}_{n}^{3},\\;{Y}_{n}\\left( 0\\right) = {\\eta }_{n}. \n\]\n\n(9.7)\n\nIt is easy to see (see Exercise 9.2) that \( {y}_{n}\\left( t\\right) \\leq {Y}_{n}\\left( t\\ri... | No |
Theorem 10.1 There exists an absolute constant \( {\varepsilon }_{h} > 0 \) such that whenever \( {u}_{0} \in {\dot{H}}^{1/2}\left( {\mathbb{T}}^{3}\right) \cap H \) and\n\n\[{\int }_{0}^{T}{\begin{Vmatrix}{\mathrm{e}}^{\Delta s}{u}_{0}\end{Vmatrix}}_{{\dot{H}}^{1}}^{4}\mathrm{\;d}s < {\varepsilon }_{h}\]\n\nfor some \... | Proof We consider the Galerkin approximations of \( u \), i.e. the solutions \( {u}_{n} \) of\n\n\[{\partial }_{t}{u}_{n} + A{u}_{n} + {P}_{n}\left\lbrack {\left( {{u}_{n} \cdot \nabla }\right) {u}_{n}}\right\rbrack = 0,\;{u}_{n}\left( 0\right) = {P}_{n}{u}_{0},\]\n\nwhere \( {P}_{n} \) is the projector on the space sp... | Yes |
Corollary 10.2 Suppose that \( {u}_{0} \in H \cap {\dot{H}}^{1/2}\left( {\mathbb{T}}^{3}\right) \). Then\n\n(i) there exists a time \( T = T\left( {u}_{0}\right) > 0 \) such that the Navier-Stokes equations have a solution\n\n\[ u \in {L}^{\infty }\left( {0, T;{\dot{H}}^{1/2}}\right) \cap {L}^{2}\left( {0, T;{\dot{H}}^... | Proof We have shown in (10.13) that the solution \( v\left( t\right) \) of \( {\partial }_{t}v + {Av} = 0 \) with \( v\left( 0\right) = {u}_{0} \) satisfies\n\n\[ \frac{1}{2}\parallel v\left( t\right) {\parallel }_{{\dot{H}}^{1/2}}^{2} + {\int }_{0}^{t}\parallel v\left( s\right) {\parallel }_{{\dot{H}}^{3/2}}^{2}\mathr... | Yes |
Lemma 10.3 Suppose that \( x, y,\delta \) are real-valued, non-negative functions that are continuous on \( \lbrack 0, T), x \) is differentiable, and for some \( c > 0 \)\n\n\[ \n\frac{\mathrm{d}x}{\mathrm{\;d}t} + y \leq {cxy} + \delta \left( t\right) ,\;t \in \lbrack 0, T),\;\text{ and }\;x\left( 0\right) = 0.\n\]\n... | Proof If \( D = 0 \) then the assertion follows easily from Gronwall’s Lemma. So let us assume that \( D > 0 \) and let\n\n\[ \nY\left( t\right) = {\int }_{0}^{t}y\left( s\right) \mathrm{d}s\n\]\n\nThe function \( Y \) is continuous and non-decreasing with \( Y\left( 0\right) = 0 \) . To prove that \( Y\left( t\right) ... | Yes |
Lemma 11.1 If \( u \in {C}^{\infty }\left( {\mathbb{T}}^{3}\right) \) or \( {C}_{c}^{\infty }\left( {\mathbb{R}}^{3}\right) \) then\n\n\[ \int - {\Delta u} \cdot u\left| u\right| \geq \int {\left| \nabla u\right| }^{2}\left| u\right| \] | Proof Integrating by parts and using (11.2) we obtain\n\n\[ - \int \left( {{\partial }_{i}{\partial }_{i}{u}_{j}}\right) {u}_{j}\left| u\right| = \int \left( {{\partial }_{i}{u}_{j}}\right) \left( {{\partial }_{i}{u}_{j}}\right) \left| u\right| + \int \left( {{\partial }_{i}{u}_{j}}\right) {u}_{j}{u}_{k}\left( {{\parti... | Yes |
Lemma 11.2 There exists a constant \( c > 0 \) such that\n\n\[ \parallel u{\parallel }_{{L}^{9}}^{3} \leq c{\int }_{{\mathbb{R}}^{3}}{\left| \nabla u\right| }^{2}\left| u\right| \]\n\n(11.5)\n\nfor any \( u \in {W}^{1,9/4}\left( {\mathbb{R}}^{3}\right) \) . The same is valid for functions on the torus provided that \( ... | Proof We give the proof of (11.5) on \( {\mathbb{R}}^{3} \), which is simple, deferring the proof on \( {\mathbb{T}}^{3} \) to the end of the chapter. First take \( u \in {C}_{c}^{\infty }\left( {\mathbb{R}}^{3}\right) \) . Certainly \( {\left| u\right| }^{3/2} \in {\dot{H}}^{1}\left( {\mathbb{R}}^{3}\right) \), and fr... | Yes |
Theorem 11.4 There is an absolute constant \( {\varepsilon }_{3} > 0 \) with the following property: if \( {u}_{0} \in {L}^{3} \cap {L}^{2} \) is divergence free and \( v\left( t\right) \) is the solution of the heat equation with initial data \( {u}_{0} \) then whenever\n\n\[ \n{\begin{Vmatrix}{u}_{0}\end{Vmatrix}}_{{... | Proof Given \( {u}_{0} \in {L}^{3} \cap {L}^{2} \), let \( v \) be the solution of\n\n\[ \n{\partial }_{t}v - {\Delta v} = 0,\;v\left( 0\right) = {u}_{0} \n\]\n\nand let \( {v}_{\alpha } \) be the solution of\n\n\[ \n{\partial }_{t}{v}_{\alpha } - \Delta {v}_{\alpha } = 0,\;{v}_{\alpha }\left( 0\right) = {u}_{0,\alpha ... | Yes |
Lemma 11.5 There exists a constant \( c > 0 \) such that if \( u \in {W}^{1,9/4}\left( {\mathbb{T}}^{3}\right) \) with \( {\int }_{{\mathbb{T}}^{3}}u = 0 \) then\n\n\[ \parallel u{\parallel }_{{L}^{9}}^{3} \leq c{\int }_{{\mathbb{T}}^{3}}{\left| \nabla u\right| }^{2}\left| u\right| \] | Proof If we apply the embedding \( {H}^{1}\left( {\mathbb{T}}^{3}\right) \subset {L}^{6}\left( {\mathbb{T}}^{3}\right) \) to the function \( {\left| u\right| }^{3/2} \), then\n\n\[ \parallel u{\parallel }_{{L}^{9}}^{3} = {\begin{Vmatrix}{\left| u\right| }^{3/2}\end{Vmatrix}}_{{L}^{6}}^{2} \leq C{\begin{Vmatrix}{\left| ... | Yes |
Theorem 12.1 Let \( u \) be a weak solution of the Navier-Stokes equations. Then the pair \( \left( {u,\omega }\right) \), where \( \omega = \operatorname{curl}u \), satisfies (12.4) for every choice of test function \( \phi \in {C}_{c}^{\infty }\left( {{\mathbb{R}}^{n} \times \left( {0,\infty }\right) }\right) \) . | Proof Let \( u \) be a weak solution of the Navier-Stokes equations in the sense of Definition 3.3. Given \( \phi \in {C}_{c}^{\infty }\left( {{\mathbb{R}}^{n} \times \left( {0,\infty }\right) }\right) \), we observe that \( \operatorname{curl}\phi \) is divergence free and as a consequence can be used as a test functi... | Yes |
Theorem 12.2 Let\n\n\\[ \nT\\left( \\omega \\right) \\mathrel{\\text{:=}} - \\frac{1}{4\\pi }{\\int }_{{\\mathbb{R}}^{3}}\\frac{x - y}{{\\left| x - y\\right| }^{3}} \\times \\omega \\left( y\\right) \\mathrm{d}y \n\\]\n\nbe the operator on the right-hand side of (12.7). Then \\( T \\) is a bounded linear operator from ... | Sketch of the proof It is straightforward to see that \\( T \\) is well defined for smooth, compactly supported functions. Since these functions are dense in \\( {L}^{p} \\) for \\( 1 < p < 3, T \\) will extend to \\( {L}^{p} \\) provided we show that \\( \\parallel T\\left( w\\right) {\\parallel }_{{L}^{q}} \\leq c\\p... | No |
Theorem 12.3 Let \( {u}_{0} \in V\left( {\mathbb{R}}^{3}\right) \) and let \( u \) be the corresponding solution of the Navier-Stokes equations. If for every time \( T > 0 \) we have\n\n\[ \n{\int }_{0}^{T}\parallel \omega \left( {\cdot, s}\right) {\parallel }_{{L}^{\infty }}\mathrm{d}s < \infty \n\]\n\n(12.9)\n\nthen ... | Proof It suffices to show that the solution can be continued past the time \( t = T \) whenever \( {\int }_{0}^{T}\parallel \omega {\parallel }_{{L}^{\infty }}\mathrm{d}s < \infty \), and in order to prove this we show that \( \parallel u\left( {\cdot, T}\right) {\parallel }_{{H}^{1}} < \infty \) and then use the exist... | Yes |
Theorem 12.4 Let \( {\omega }_{0} \in {L}^{2} \) . Then there exists a unique solution of the \( {2D} \) Navier-Stokes equation that remains strong for all time. | Proof Since \( \begin{Vmatrix}{\omega }_{0}\end{Vmatrix} = \begin{Vmatrix}{\nabla {u}_{0}}\end{Vmatrix} \) our existence result guarantees that we have a unique strong solution for a short time. Notice that by taking the inner product of the equation (12.13) with \( \omega \) we obtain\n\n\[ \n\frac{1}{2}\frac{\mathrm{... | Yes |
Theorem 12.5 (Biot-Savart Law, local version) Assume that \( u \) and \( \omega \) are defined in an open domain \( B \), satisfy \( \omega = \operatorname{curl}u \), and that \( u \in {L}^{q}\left( B\right) \) and \( \nabla u \in {L}^{r}\left( B\right) \) for some \( 1 \leq q, r \leq \infty \). Then for any open set \... | Proof Notice that without any requirements on \( A \) equation (12.14) is trivially true, taking it as the definition of \( A \). So all we need to prove is that \( A \) is harmonic. Notice that it is enough to show that \[ u\left( x\right) + \frac{1}{4\pi }{\int }_{\Omega }\left( {{\nabla }_{y}\frac{1}{\left| x - y\ri... | Yes |
Example 13.2 (Serrin's example) Let us consider the following local solution of the Navier-Stokes equations in a smooth bounded domain \( U \) in \( {\mathbb{R}}^{3} \) , \[ u\left( {x, t}\right) = a\left( t\right) \nabla \Psi \left( x\right) \] where \( \Psi \) is a harmonic function and \( a \in {L}^{1}\left( {0,\inf... | We will show that \( u \) satisfies the local weak formulation of the Navier-Stokes equations from Definition 13.1, because every term in (13.2) vanishes. Indeed, \( u\left( t\right) \) is the gradient of a smooth function for all \( t \), while \( {\partial }_{t}\varphi \) is divergence free, and so the first term of ... | Yes |
Theorem 13.4 Let \( u \) be a local weak solution of the Navier-Stokes equations (as in Definition 13.1) on some parabolic cylinder \( {Q}_{R}^{ * }\left( {{x}_{0},{t}_{0}}\right) \), and suppose that\n\n\[ u \in {L}_{t}^{{q}^{\prime }}{L}_{x}^{q}\left( {{Q}_{R}^{ * }\left( {{x}_{0},{t}_{0}}\right) }\right) ,\;\frac{2}... | Note that the case of strict inequality in Theorem 13.4, i.e.\n\n\[ \frac{2}{{q}^{\prime }} + \frac{3}{q} < 1 \]\n\nis a simple consequence of the result with equality, by an application of Hölder's inequality. This case was originally proved by Serrin (1962), and we will see later that this result is a key ingredient ... | No |
Theorem 13.7 Let \( u \) be a weak solution of (13.2) on some centred parabolic cylinder \( {Q}_{R}^{ * } = {Q}_{R}^{ * }\left( {{x}_{0},{t}_{0}}\right) \), and suppose that\n\n\[ u \in {L}_{t}^{{q}^{\prime }}{L}_{x}^{q}\left( {Q}_{R}^{ * }\right) ,\;\frac{2}{{q}^{\prime }} + \frac{3}{q} < 1, \]\n\nwhere \( 3 < q \leq ... | Sketch of the proof It is sufficient to prove that under the conditions of the theorem \( u \) is smooth in \( {Q}_{R/2}^{ * } \) : any point \( \left( {x, t}\right) \in {Q}_{R}^{ * }\left( {{x}_{0},{t}_{0}}\right) \) belongs to a cylinder \( {Q}_{\rho }^{ * }\left( {x, t}\right) \), such that \( {Q}_{2\rho }^{ * }\lef... | No |
Lemma 13.8 Suppose that \( U \subset {\mathbb{R}}^{3} \) is a smooth bounded domain and that \( {\partial }_{t}g \in {L}^{q}\left( {0, T;{L}^{r}\left( U\right) }\right) \) . Then \[ {\begin{Vmatrix}g\left( {t}_{2}\right) - g\left( {t}_{1}\right) \end{Vmatrix}}_{{L}^{r}\left( U\right) } \leq C{\left| {t}_{2} - {t}_{1}\r... | Proof We have \[ {\begin{Vmatrix}g\left( {t}_{2}\right) - g\left( {t}_{1}\right) \end{Vmatrix}}_{{L}^{r}\left( U\right) } \leq {\int }_{{t}_{1}}^{{t}_{2}}{\begin{Vmatrix}{\partial }_{t}g\left( \tau \right) \end{Vmatrix}}_{{L}^{r}\left( U\right) }\mathrm{d}\tau \] \[ \leq {\left( {\int }_{{t}_{1}}^{{t}_{2}}{\begin{Vmatr... | Yes |
Lemma 13.9 Let \( U \subset {\mathbb{R}}^{3} \) be a smooth bounded domain and suppose that \( f \in {H}^{k}\left( U\right) \) for some integer \( k \geq 2 \), and that \( 1 \leq r \leq 2 \) . Then\n\n\[ \parallel f{\parallel }_{{L}^{\infty }\left( U\right) } \leq {C}_{k, r}\parallel f{\parallel }_{{L}^{r}\left( U\righ... | Proof If we interpolate the \( {H}^{1} \) and \( {H}^{2} \) norms between \( {L}^{2} \) and \( {H}^{k} \) using Lemma 1.17,\n\n\[ \parallel f{\parallel }_{{H}^{1}} \leq c\parallel f{\parallel }_{{L}^{2}}^{1 - 1/k}\parallel f{\parallel }_{{H}^{k}}^{1/k},\;\parallel f{\parallel }_{{H}^{2}} \leq c\parallel f{\parallel }_{... | Yes |
Proposition 13.10 Suppose that \( u \) is a weak solution of the Navier-Stokes initial-value problem on any three-dimensional domain \( \Omega \) (a smooth bounded domain, \( {\mathbb{R}}^{3} \), or \( {\mathbb{T}}^{3} \) ), and let \( U \) be a smooth bounded subdomain of \( \Omega \) . If \( u \in {L}^{\infty }\left(... | Proof Corollary 5.2 or its extension to bounded domains (Theorem 5.7) shows that\n\n\[ \left( {u \cdot \nabla }\right) u + \nabla p \in {L}^{{2r}/\left( {{4r} - 3}\right) }\left( {0, T;{L}^{r}\left( \Omega \right) }\right) ,\;1 < r \leq 3/2, \]\n\nfor any (global) weak solution of the Navier-Stokes equations. Since by ... | Yes |
Theorem 14.1 Fix \( \varepsilon > 0 \) . Then for each \( {u}_{0} \in H \) the regularised equation (14.6) has a unique weak solution \( {u}^{\varepsilon } \) such that for every \( T > 0 \n\n\[ \n{u}^{\varepsilon } \in {L}^{\infty }\left( {0, T;H}\right) \cap {L}^{2}\left( {0, T;V}\right) \;\text{ and }\;{\partial }_{... | Sketch proof Consider the Galerkin approximations\n\n\[ \n{\partial }_{t}{u}_{n} - \Delta {u}_{n} + {P}_{n}\left\{ {\left\lbrack {\left( {{\psi }_{\varepsilon } * {u}_{n}}\right) \cdot \nabla }\right\rbrack {u}_{n}}\right\} = 0,\;{u}_{n}\left( 0\right) = {P}_{n}{u}_{0} \n\]\n\n(14.11)\n\n(recall that \( {P}_{n} \) deno... | Yes |
Theorem 14.2 For any \( {u}_{0} \in H \) there exists a Leray-Hopf weak solution \( u \) of the Navier-Stokes equations that satisfies the local energy inequality\n\n\[ 2{\int }_{0}^{T}{\int }_{{\mathbb{T}}^{3}}{\left| \nabla u\right| }^{2}\varphi \mathrm{d}x\mathrm{\;d}s \]\n\n\[ \leq {\int }_{0}^{T}{\int }_{{\mathbb{... | Proof As in the previous section we consider the Leray regularisations\n\n\[ {\partial }_{t}{u}^{\varepsilon } - \Delta {u}^{\varepsilon } + \left\lbrack {\left( {{\psi }_{\varepsilon } * {u}^{\varepsilon }}\right) \cdot \nabla }\right\rbrack {u}^{\varepsilon } + \nabla {p}^{\varepsilon } = 0,\;\nabla \cdot {u}^{\varep... | Yes |
Theorem 14.4 Given \( {u}_{0} \in H\left( {\mathbb{R}}^{3}\right) \) there exists a weak solution \( u \) of the Navier-Stokes equations that satisfies the strong energy inequality, i.e.\n\n\[ \frac{1}{2}\parallel u\left( t\right) {\parallel }^{2} + {\int }_{s}^{t}\parallel \nabla u{\parallel }^{2} \leq \frac{1}{2}\par... | Proof First observe that if \( f \in {L}^{2}\left( {\mathbb{R}}^{3}\right) \), \n\n(i) \( {f}^{\varepsilon } \rightarrow f \) strongly in \( {L}^{2}\left( K\right) \) for every compact \( K \subset {\mathbb{R}}^{3} \), and \n\n(ii) for every \( \eta > 0 \) there exists \( R\left( \eta \right) \) such that \n\n\[ {\int ... | Yes |
Theorem 15.4 (First Local Regularity Theorem) There exist absolute constants \( {\varepsilon }_{0}^{ * } > 0 \) and \( {c}_{M} > 0 \) such that if \( \left( {u, p}\right) \) is a suitable weak solution of the Navier-Stokes equations on \( {Q}_{r}\left( {0,0}\right) \) and for some \( {\varepsilon }_{0} \leq {\varepsilo... | Proof The rescaled function \( {u}_{r}\left( {x, t}\right) = {ru}\left( {{rx},{r}^{2}t}\right) \) solves the Navier-Stokes equations on \( {Q}_{1} \) (see Section 15.1). Using Theorem 15.3, if\n\n\[ {\int }_{{Q}_{1}\left( {0,0}\right) }{\left| {u}_{r}\right| }^{3} + {\left| {p}_{r}\right| }^{3/2} < {\varepsilon }_{0} \... | Yes |
Proposition 15.7 Let \( u \) be a suitable Leray-Hopf weak solution of the Navier-Stokes equations. Then the set \( S \) of space-time singularities is a bounded subset of \( \Omega \times \left( {0,\infty }\right) \) . | Proof Theorem 8.1 guarantees that any Leray-Hopf solution is regular for \( t \geq T\left( {u}_{0}\right) \), so the theorem follows immediately on a bounded domain; we only need consider the case \( \Omega = {\mathbb{R}}^{3} \). We show that there exists an \( R > 0 \) such that for any parabolic cylinder \( {Q}_{1}\l... | Yes |
Corollary 15.9 The space-time singular set \( S \) of a suitable weak solution has Lebesgue measure zero. | Proof Take a sequence of compact subsets \( {K}_{j} \) of \( {\mathbb{R}}^{4} \) such that\n\n\[ \Omega \times \left( {0,\infty }\right) = \mathop{\bigcup }\limits_{{j = 1}}^{\infty }{K}_{j} \]\n\nThen by Theorem \( {15.8}{\dim }_{B}\left( {{K}_{j} \cap S}\right) \leq 5/3 \) for each \( j \), and it follows from Coroll... | Yes |
Theorem 16.2 (Partial Regularity II) Let \( S \) denote the singular set of a suitable weak solution of the Navier-Stokes equations. Then \( {\mathcal{P}}^{1}\left( S\right) = 0 \), and in particular \( {\dim }_{\mathrm{H}}\left( S\right) \leq 1 \) . | Proof We have already shown in Corollary 15.9 that \( \mu \left( S\right) = 0 \), and we know that \( \nabla u \in {L}^{2}\left( {\Omega \times \left( {0, T}\right) }\right) \) . So given any \( \delta > 0 \), we can choose a neighbourhood \( U \) of \( S \) such that\n\n\[ \frac{1}{{\varepsilon }_{1}}{\int }_{U}{\left... | Yes |
Lemma 16.6 For any fixed \( r > 0 \) and \( 0 < \theta \leq 1/2 \) there exists a function \( \phi \in {C}_{c}^{\infty }\left( {\mathbb{R}}^{4}\right) \) such that \( \left( {\operatorname{supp}\phi }\right) \cap {Q}_{1} \subseteq {Q}_{r} \), \[ \phi \geq {C}_{3}^{-1}{\left( \theta r\right) }^{-1}\;\text{ for }\;\left(... | Proof As in our previous construction (Lemma 15.11) we take \( \phi \) of the form \( \phi \left( {x, t}\right) = {c\psi }\left( {x, t}\right) \vartheta \left( {x, t}\right) \), where \( c \) depends on \( {\theta r},\psi \) is a solution of the backwards heat equation, and \( \vartheta \left( {x, t}\right) = \chi \lef... | Yes |
Lemma 16.8 (Vitali Covering Lemma for parabolic cylinders) Given any family \( \mathfrak{Q} \) of centred parabolic cylinders contained in a bounded subset of \( {\mathbb{R}}^{3} \times \mathbb{R} \), there exists a finite or countable disjoint subfamily\n\n\[ \n{\mathfrak{Q}}^{ \dagger } = \left\{ {{Q}_{i}^{ \dagger }... | Proof We use an inductive construction, choosing the first element of \( {\mathfrak{Q}}^{ \dagger } \) to be \( {Q}_{1}^{ \dagger } = {Q}_{{r}_{1}}^{ * }\left( {{x}_{1},{t}_{1}}\right) \in \mathfrak{Q} \) such that \( r \leq 2{r}_{1} \) for all \( {Q}_{r}^{ * }\left( {x, t}\right) \in \mathfrak{Q} \).\n\nNow if we have... | Yes |
Corollary 17.2 If\n\n\[ \n u \in {L}^{\infty }\left( {0, T;{\dot{H}}^{1/2}}\right) \;\text{ and }\;\sqrt{t}u \in {L}^{2}\left( {0, T;{\dot{H}}^{5/2}}\right) \n\]\n\nthen \( \parallel u\left( t\right) {\parallel }_{{L}^{\infty }} \leq \theta \left( t\right) \), where \( \theta \in {L}^{1}\left( {0, T}\right) \) and for ... | Proof We start with Agmon's inequality (see Theorem 1.20) in the form\n\n\[ \n \parallel u\left( t\right) {\parallel }_{{L}^{\infty }} \leq \theta \left( t\right) \mathrel{\text{:=}} c\parallel u{\parallel }_{{\dot{H}}^{1/2}}^{1/2}\parallel u{\parallel }_{{\dot{H}}^{5/2}}^{1/2} \n\]\n\n(17.7)\n\n(see Exercise 1.10) and... | No |
Theorem 17.6 For any Leray-Hopf weak solution u there exists at least one ’solution mapping’ \( \Phi : {\mathbb{T}}^{3} \times \lbrack 0,\infty ) \rightarrow {\mathbb{T}}^{3} \) such that\n\n(i) for each \( a \in {\mathbb{T}}^{3},{X}_{a}\left( \cdot \right) \mathrel{\text{:=}} \Phi \left( {a, \cdot }\right) \) is absol... | We prove this theorem in a number of steps. First we show in Proposition 17.7 that for every \( a \in {\mathbb{T}}^{3} \) there is at least one solution of (17.18); the argument is very similar to that used already to prove Lemma 17.3 (existence of trajectories when \( {u}_{0} \in H \cap {\dot{H}}^{1/2} \) ).\n\nBy cho... | Yes |
Proposition 17.7 If \( u \) is divergence free and\n\n\[ u \in {L}^{2}\left( {0, T;{H}^{1}}\right) \cap {L}^{2/3}\left( {0, T;{H}^{2}}\right) \]\n\nthen for every \( a \in {\mathbb{T}}^{3} \) there is at least one solution \( {X}_{a} \in {W}^{1,1}\left( {0, T;{\mathbb{T}}^{3}}\right) \) of the integral equation\n\n\[ X... | Proof The proof of the existence of solutions of (17.20) given a weak solution follows almost exactly the proof in Lemma 17.3, except that instead of (17.7) we use the inequality\n\n\[ \parallel u\left( t\right) {\parallel }_{{L}^{\infty }} \leq \theta \left( t\right) \mathrel{\text{:=}} c\parallel u\left( t\right) {\p... | Yes |
Lemma 17.9 If \( \Psi : {\mathbb{T}}^{3} \rightarrow {\mathbb{T}}^{3} \) is a measurable mapping such that\n\n\[ \n{\int }_{{\mathbb{T}}^{3}}g\left( {\Psi \left( x\right) }\right) \mathrm{d}x = {\int }_{{\mathbb{T}}^{3}}g\left( a\right) \mathrm{d}a\;\text{ for all }\;g \in {C}^{\infty }\left( {\mathbb{T}}^{3}\right) \n... | Proof We can extend (17.24) to every bounded Borel function on \( {\mathbb{T}}^{3} \) by continuity. Then taking \( g \) to be the characteristic function of \( A \) we obtain\n\n\[ \n\mu \left( A\right) = {\int }_{A}1\mathrm{\;d}a = {\int }_{{\mathbb{T}}^{3}}{\chi }_{A}\left( a\right) \mathrm{d}a \n\]\n\n\[ \n= {\int ... | Yes |
Proposition 17.10 The solution mapping \( \Phi \) obtained via the measurable selection theorem of the previous section is volume preserving, i.e. for any Borel set \( B \subset {\mathbb{T}}^{3} \) we have\n\n\[ \mu \left\lbrack {\Phi {\left( \cdot, t\right) }^{-1}\left( B\right) }\right\rbrack = \mu \left( B\right) \;... | Proof Given any \( g \in {C}^{\infty }\left( {\mathbb{T}}^{3}\right) \), define\n\n\[ J\left( t\right) = {\int }_{{\mathbb{T}}^{3}}g\left( {\Phi \left( {a, t}\right) }\right) \mathrm{d}a. \]\n\n\nWe will show that \( J \) is constant by showing that \( J \) is absolutely continuous with time derivative \( j = 0 \) almo... | Yes |
Lemma 17.11 Let \( K \) be a bounded subset of \( {\mathbb{R}}^{n} \) and denote by \( O\left( {K,\varepsilon }\right) \) the \( \varepsilon \) -neighbourhood of \( K \), i.e.\n\n\[ O\left( {K,\varepsilon }\right) = \left\{ {z \in {\mathbb{R}}^{n} : z = x + y, x \in K,\left| y\right| < \varepsilon }\right\} .\n\]\n\nTh... | Proof For any \( d > {\dim }_{\mathrm{B}}\left( K\right) \) and \( {\varepsilon }_{0} > 0 \) there exists a \( {C}_{d}^{\prime } > 0 \) such that for any \( 0 < \varepsilon < {\varepsilon }_{0} \) the set \( K \) can be covered by \( {N}_{\varepsilon } \leq {C}_{d}^{\prime }{\varepsilon }^{-d} \) balls of radius \( \va... | Yes |
Theorem 17.13 If \( u \) is a Leray-Hopf suitable weak solution and \( \Phi \) is a corresponding solution mapping then for almost every \( a \in {\mathbb{T}}^{3},\Phi \left( {a, t}\right) \notin S \) for all \( t \geq 0 \) . | Proof Fix \( T > 0 \) and let \( {K}_{n} \) be a sequence of compact subsets of \( {\mathbb{T}}^{3} \times \left( {0, T}\right) \) such that \( {\mathbb{T}}^{3} \times \left( {0, T}\right) = \mathop{\bigcup }\limits_{{n = 1}}^{\infty }{K}_{n} \), e.g. \( {K}_{n} = {\mathbb{T}}^{3} \times \left\lbrack {\frac{1}{n}, T - ... | Yes |
If \( u \) is a Leray-Hopf suitable weak solution arising from the initial condition \( {u}_{0} \in H \cap {\dot{H}}^{1/2}\left( {\mathbb{T}}^{3}\right) \) then the equation \n\n\[ \n\dot{X} = u\left( {X, t}\right) ,\;X\left( 0\right) = a \n\] \nhas a unique solution for almost every \( a \in {\mathbb{T}}^{3} \), i.e. ... | Proof Since \( {u}_{0} \in H \cap {\dot{H}}^{1/2} \) it follows (using Corollary 10.2) that \n\n\[ \nu \in {L}^{\infty }\left( {0, T;{\dot{H}}^{1/2}}\right) \cap {L}^{2}\left( {0, T;{\dot{H}}^{3/2}}\right) \n\] \n\nfor some \( T > 0 \), and hence Proposition 17.5 guarantees that trajectories are unique on \( \left\lbra... | Yes |
Lemma 17.16 There exists a constant \( C > 0 \) such that if \( u \in {\dot{H}}^{5/2}\left( {\mathbb{T}}^{3}\right) \) then \( u \in {C}^{0}\left( {\mathbb{T}}^{3}\right) \) and\n\n\[ \left| {u\left( x\right) - u\left( y\right) }\right| \leq C\parallel u{\parallel }_{{\dot{H}}^{5/2}}\left| {x - y}\right| {\left| \log \... | Proof We expand \( u \) as a Fourier series and split the sum into low modes and high modes:\n\n\[ u\left( x\right) = \mathop{\sum }\limits_{{k \in {\dot{\mathbb{Z}}}^{3}}}{\widehat{u}}_{k}{\mathrm{e}}^{\mathrm{i}k \cdot x} \]\n\n\[ = \mathop{\sum }\limits_{{1 \leq \left| k\right| \leq N}}{\widehat{u}}_{k}{\mathrm{e}}^... | Yes |
Proposition 1.1 Let \( {\left( {X}_{n}\right) }_{n \geq 1} \) be a sequence of real random variables such that, for every \( n \geq 1,{X}_{n} \) follows the \( \mathcal{N}\left( {{m}_{n},{\sigma }_{n}^{2}}\right) \) -distribution. Suppose that \( {X}_{n} \) converges in \( {L}^{2} \) to X. Then:\n\n(i) The random varia... | (i) The convergence in \( {L}^{2} \) implies that \( {m}_{n} = E\left\lbrack {X}_{n}\right\rbrack \) converges to \( E\left\lbrack X\right\rbrack \) and \( {\sigma }_{n}^{2} = \) \( \operatorname{var}\left( {X}_{n}\right) \) converges to \( \operatorname{var}\left( X\right) \) as \( n \rightarrow \infty \) . Then, sett... | Yes |
Proposition 1.2 Under the preceding assumptions, the random variables \( {X}_{1},\ldots ,{X}_{d} \) are independent if and only if the covariance matrix \( {\left( \operatorname{cov}\left( {X}_{j},{X}_{k}\right) \right) }_{1 \leq j, k \leq d} \) is diagonal or equivalently if and only if \( {q}_{X} \) is of diagonal fo... | Proof If the random variables \( {X}_{1},\ldots ,{X}_{d} \) are independent, the covariance matrix \( {\left( \operatorname{cov}\left( {X}_{j},{X}_{k}\right) \right) }_{j, k = 1,\ldots d} \) is diagonal. Conversely, if this matrix is diagonal, we have for every \( u = \mathop{\sum }\limits_{{j = 1}}^{d}{u}_{j}{e}_{j} \... | Yes |
Proposition 1.7 If \( {\left( {X}_{t}\right) }_{t \in T} \) is a Gaussian process, the closed linear subspace of \( {L}^{2} \) spanned by the variables \( {X}_{t}, t \in T \), is a Gaussian space, which is called the Gaussian space generated by the process \( X \) . | Proof It suffices to observe that an \( {L}^{2} \) -limit of centered Gaussian variables is still centered Gaussian, by Proposition 1.1. | No |
Theorem 1.9 Let \( H \) be a centered Gaussian space and let \( {\left( {H}_{i}\right) }_{i \in I} \) be a collection of linear subspaces of \( H \) . Then the subspaces \( {H}_{i}, i \in I \), are (pairwise) orthogonal in \( {L}^{2} \) if and only the \( \sigma \) -fields \( \sigma \left( {H}_{i}\right), i \in I \), a... | Proof Suppose that the \( \sigma \) -fields \( \sigma \left( {H}_{i}\right) \) are independent. Then, if \( i \neq j \), if \( X \in {H}_{i} \) and \( Y \in {H}_{j} \) ,\n\n\[ E\left\lbrack {XY}\right\rbrack = E\left\lbrack X\right\rbrack E\left\lbrack Y\right\rbrack = 0, \]\n\nso that the linear spaces \( {H}_{i} \) a... | Yes |
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