Flow of ODE with monotone source












4












$begingroup$


Let $Phi$ be the flow (defined as in page 6 of this paper) of the ODE
$$begin{cases}
frac{d}{dt}Phi(x,t) = f(Phi(x,t),t) quad t >0 \
Phi(x,0) = x quad x in mathbb{R}.
end{cases}$$




Is it true that if $f$ is monotone in the first variable then $Phi$ is Lipschitz?







reference-request ca.classical-analysis-and-odes measure-theory geometric-measure-theory







share|cite|improve this question









$endgroup$

















    4












    $begingroup$


    Let $Phi$ be the flow (defined as in page 6 of this paper) of the ODE
    $$begin{cases}
    frac{d}{dt}Phi(x,t) = f(Phi(x,t),t) quad t >0 \
    Phi(x,0) = x quad x in mathbb{R}.
    end{cases}$$




    Is it true that if $f$ is monotone in the first variable then $Phi$ is Lipschitz?







    reference-request ca.classical-analysis-and-odes measure-theory geometric-measure-theory







    share|cite|improve this question









    $endgroup$















      4












      4








      4





      $begingroup$


      Let $Phi$ be the flow (defined as in page 6 of this paper) of the ODE
      $$begin{cases}
      frac{d}{dt}Phi(x,t) = f(Phi(x,t),t) quad t >0 \
      Phi(x,0) = x quad x in mathbb{R}.
      end{cases}$$




      Is it true that if $f$ is monotone in the first variable then $Phi$ is Lipschitz?







      reference-request ca.classical-analysis-and-odes measure-theory geometric-measure-theory







      share|cite|improve this question









      $endgroup$




      Let $Phi$ be the flow (defined as in page 6 of this paper) of the ODE
      $$begin{cases}
      frac{d}{dt}Phi(x,t) = f(Phi(x,t),t) quad t >0 \
      Phi(x,0) = x quad x in mathbb{R}.
      end{cases}$$




      Is it true that if $f$ is monotone in the first variable then $Phi$ is Lipschitz?







      reference-request ca.classical-analysis-and-odes measure-theory geometric-measure-theory




      reference-request ca.classical-analysis-and-odes measure-theory geometric-measure-theory






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked yesterday









      HiroHiro

      697




      697






















          1 Answer
          1






          active

          oldest

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          5












          $begingroup$

          Suppose that $f$ is decreasing in $x$. Let $x(t)$, $y(t)$ be two solutions of the ode. Then
          $$
          dot{x}-dot{y}= f(x,t)-f(y,t).
          $$



          Multiplying both sides by $x-y$ we deduce



          $$
          (dot{x}-dot{y})(x-y) =big(f(x,t)-f(y,t)big)(x-y)leq 0,
          $$
          where the last equality holds because $f$ is decreasing.



          Hence
          $$
          frac{1}{2}frac{d}{dt}big(x-y)^2leq 0.
          $$
          Thus the function $tmapsto big( x(t)-y(t)big)^2 $ is decreasing so
          $$
          big(x(t)-y(t)big)^2leq big( x(0)-y(0)big)^2,;;forall tgeq 0,
          $$
          i.e.,
          $$
          Big(Phi(x_0,t)-Phi(y_0,t)Big)^2leq Big(x_0-y_0Big)^2,;;forall tgeq 0.
          $$
          In other words, for $tgeq 0$, $Phi(x,t)$ is Lipschitz in $x$ with Lipschitz constant $1$ if $f$ is decreasing.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank you. How can the argument be made rigorous even when $f$ is not smooth and $Phi$ is not a classical solution but a regular Lagrangian flow?
            $endgroup$
            – Hiro
            yesterday












          • $begingroup$
            The function $f$ coud even be multivalued, and you can work in an infinite dimensional Hilbert space as well This is a special case of the general theory of maximal monotone operators and the associated differential equations. Perhaps the friendliest introduction is Brezis' book Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. The ultimate reference is however V. Barbu's book Nonlinear semigroups and differen tial equations in Banach spaces
            $endgroup$
            – Liviu Nicolaescu
            yesterday












          • $begingroup$
            The finite dimensional case is discussed in V. Barbu's recent book Differential Equations Springer 2016, Example 2.4 and Sec. 2.7.
            $endgroup$
            – Liviu Nicolaescu
            yesterday










          • $begingroup$
            In the scalar case all you need for existence and uniqueness is that $f$ is decreasing and the function $mathbb{R}ni xmapsto f(x)-xinmathbb{R}$ is onto.
            $endgroup$
            – Liviu Nicolaescu
            yesterday












          • $begingroup$
            Thank you. What if $f$ is increasing?
            $endgroup$
            – Hiro
            yesterday












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          5












          $begingroup$

          Suppose that $f$ is decreasing in $x$. Let $x(t)$, $y(t)$ be two solutions of the ode. Then
          $$
          dot{x}-dot{y}= f(x,t)-f(y,t).
          $$



          Multiplying both sides by $x-y$ we deduce



          $$
          (dot{x}-dot{y})(x-y) =big(f(x,t)-f(y,t)big)(x-y)leq 0,
          $$
          where the last equality holds because $f$ is decreasing.



          Hence
          $$
          frac{1}{2}frac{d}{dt}big(x-y)^2leq 0.
          $$
          Thus the function $tmapsto big( x(t)-y(t)big)^2 $ is decreasing so
          $$
          big(x(t)-y(t)big)^2leq big( x(0)-y(0)big)^2,;;forall tgeq 0,
          $$
          i.e.,
          $$
          Big(Phi(x_0,t)-Phi(y_0,t)Big)^2leq Big(x_0-y_0Big)^2,;;forall tgeq 0.
          $$
          In other words, for $tgeq 0$, $Phi(x,t)$ is Lipschitz in $x$ with Lipschitz constant $1$ if $f$ is decreasing.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank you. How can the argument be made rigorous even when $f$ is not smooth and $Phi$ is not a classical solution but a regular Lagrangian flow?
            $endgroup$
            – Hiro
            yesterday












          • $begingroup$
            The function $f$ coud even be multivalued, and you can work in an infinite dimensional Hilbert space as well This is a special case of the general theory of maximal monotone operators and the associated differential equations. Perhaps the friendliest introduction is Brezis' book Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. The ultimate reference is however V. Barbu's book Nonlinear semigroups and differen tial equations in Banach spaces
            $endgroup$
            – Liviu Nicolaescu
            yesterday












          • $begingroup$
            The finite dimensional case is discussed in V. Barbu's recent book Differential Equations Springer 2016, Example 2.4 and Sec. 2.7.
            $endgroup$
            – Liviu Nicolaescu
            yesterday










          • $begingroup$
            In the scalar case all you need for existence and uniqueness is that $f$ is decreasing and the function $mathbb{R}ni xmapsto f(x)-xinmathbb{R}$ is onto.
            $endgroup$
            – Liviu Nicolaescu
            yesterday












          • $begingroup$
            Thank you. What if $f$ is increasing?
            $endgroup$
            – Hiro
            yesterday
















          5












          $begingroup$

          Suppose that $f$ is decreasing in $x$. Let $x(t)$, $y(t)$ be two solutions of the ode. Then
          $$
          dot{x}-dot{y}= f(x,t)-f(y,t).
          $$



          Multiplying both sides by $x-y$ we deduce



          $$
          (dot{x}-dot{y})(x-y) =big(f(x,t)-f(y,t)big)(x-y)leq 0,
          $$
          where the last equality holds because $f$ is decreasing.



          Hence
          $$
          frac{1}{2}frac{d}{dt}big(x-y)^2leq 0.
          $$
          Thus the function $tmapsto big( x(t)-y(t)big)^2 $ is decreasing so
          $$
          big(x(t)-y(t)big)^2leq big( x(0)-y(0)big)^2,;;forall tgeq 0,
          $$
          i.e.,
          $$
          Big(Phi(x_0,t)-Phi(y_0,t)Big)^2leq Big(x_0-y_0Big)^2,;;forall tgeq 0.
          $$
          In other words, for $tgeq 0$, $Phi(x,t)$ is Lipschitz in $x$ with Lipschitz constant $1$ if $f$ is decreasing.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank you. How can the argument be made rigorous even when $f$ is not smooth and $Phi$ is not a classical solution but a regular Lagrangian flow?
            $endgroup$
            – Hiro
            yesterday












          • $begingroup$
            The function $f$ coud even be multivalued, and you can work in an infinite dimensional Hilbert space as well This is a special case of the general theory of maximal monotone operators and the associated differential equations. Perhaps the friendliest introduction is Brezis' book Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. The ultimate reference is however V. Barbu's book Nonlinear semigroups and differen tial equations in Banach spaces
            $endgroup$
            – Liviu Nicolaescu
            yesterday












          • $begingroup$
            The finite dimensional case is discussed in V. Barbu's recent book Differential Equations Springer 2016, Example 2.4 and Sec. 2.7.
            $endgroup$
            – Liviu Nicolaescu
            yesterday










          • $begingroup$
            In the scalar case all you need for existence and uniqueness is that $f$ is decreasing and the function $mathbb{R}ni xmapsto f(x)-xinmathbb{R}$ is onto.
            $endgroup$
            – Liviu Nicolaescu
            yesterday












          • $begingroup$
            Thank you. What if $f$ is increasing?
            $endgroup$
            – Hiro
            yesterday














          5












          5








          5





          $begingroup$

          Suppose that $f$ is decreasing in $x$. Let $x(t)$, $y(t)$ be two solutions of the ode. Then
          $$
          dot{x}-dot{y}= f(x,t)-f(y,t).
          $$



          Multiplying both sides by $x-y$ we deduce



          $$
          (dot{x}-dot{y})(x-y) =big(f(x,t)-f(y,t)big)(x-y)leq 0,
          $$
          where the last equality holds because $f$ is decreasing.



          Hence
          $$
          frac{1}{2}frac{d}{dt}big(x-y)^2leq 0.
          $$
          Thus the function $tmapsto big( x(t)-y(t)big)^2 $ is decreasing so
          $$
          big(x(t)-y(t)big)^2leq big( x(0)-y(0)big)^2,;;forall tgeq 0,
          $$
          i.e.,
          $$
          Big(Phi(x_0,t)-Phi(y_0,t)Big)^2leq Big(x_0-y_0Big)^2,;;forall tgeq 0.
          $$
          In other words, for $tgeq 0$, $Phi(x,t)$ is Lipschitz in $x$ with Lipschitz constant $1$ if $f$ is decreasing.






          share|cite|improve this answer











          $endgroup$



          Suppose that $f$ is decreasing in $x$. Let $x(t)$, $y(t)$ be two solutions of the ode. Then
          $$
          dot{x}-dot{y}= f(x,t)-f(y,t).
          $$



          Multiplying both sides by $x-y$ we deduce



          $$
          (dot{x}-dot{y})(x-y) =big(f(x,t)-f(y,t)big)(x-y)leq 0,
          $$
          where the last equality holds because $f$ is decreasing.



          Hence
          $$
          frac{1}{2}frac{d}{dt}big(x-y)^2leq 0.
          $$
          Thus the function $tmapsto big( x(t)-y(t)big)^2 $ is decreasing so
          $$
          big(x(t)-y(t)big)^2leq big( x(0)-y(0)big)^2,;;forall tgeq 0,
          $$
          i.e.,
          $$
          Big(Phi(x_0,t)-Phi(y_0,t)Big)^2leq Big(x_0-y_0Big)^2,;;forall tgeq 0.
          $$
          In other words, for $tgeq 0$, $Phi(x,t)$ is Lipschitz in $x$ with Lipschitz constant $1$ if $f$ is decreasing.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited yesterday

























          answered yesterday









          Liviu NicolaescuLiviu Nicolaescu

          26.1k260112




          26.1k260112












          • $begingroup$
            Thank you. How can the argument be made rigorous even when $f$ is not smooth and $Phi$ is not a classical solution but a regular Lagrangian flow?
            $endgroup$
            – Hiro
            yesterday












          • $begingroup$
            The function $f$ coud even be multivalued, and you can work in an infinite dimensional Hilbert space as well This is a special case of the general theory of maximal monotone operators and the associated differential equations. Perhaps the friendliest introduction is Brezis' book Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. The ultimate reference is however V. Barbu's book Nonlinear semigroups and differen tial equations in Banach spaces
            $endgroup$
            – Liviu Nicolaescu
            yesterday












          • $begingroup$
            The finite dimensional case is discussed in V. Barbu's recent book Differential Equations Springer 2016, Example 2.4 and Sec. 2.7.
            $endgroup$
            – Liviu Nicolaescu
            yesterday










          • $begingroup$
            In the scalar case all you need for existence and uniqueness is that $f$ is decreasing and the function $mathbb{R}ni xmapsto f(x)-xinmathbb{R}$ is onto.
            $endgroup$
            – Liviu Nicolaescu
            yesterday












          • $begingroup$
            Thank you. What if $f$ is increasing?
            $endgroup$
            – Hiro
            yesterday


















          • $begingroup$
            Thank you. How can the argument be made rigorous even when $f$ is not smooth and $Phi$ is not a classical solution but a regular Lagrangian flow?
            $endgroup$
            – Hiro
            yesterday












          • $begingroup$
            The function $f$ coud even be multivalued, and you can work in an infinite dimensional Hilbert space as well This is a special case of the general theory of maximal monotone operators and the associated differential equations. Perhaps the friendliest introduction is Brezis' book Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. The ultimate reference is however V. Barbu's book Nonlinear semigroups and differen tial equations in Banach spaces
            $endgroup$
            – Liviu Nicolaescu
            yesterday












          • $begingroup$
            The finite dimensional case is discussed in V. Barbu's recent book Differential Equations Springer 2016, Example 2.4 and Sec. 2.7.
            $endgroup$
            – Liviu Nicolaescu
            yesterday










          • $begingroup$
            In the scalar case all you need for existence and uniqueness is that $f$ is decreasing and the function $mathbb{R}ni xmapsto f(x)-xinmathbb{R}$ is onto.
            $endgroup$
            – Liviu Nicolaescu
            yesterday












          • $begingroup$
            Thank you. What if $f$ is increasing?
            $endgroup$
            – Hiro
            yesterday
















          $begingroup$
          Thank you. How can the argument be made rigorous even when $f$ is not smooth and $Phi$ is not a classical solution but a regular Lagrangian flow?
          $endgroup$
          – Hiro
          yesterday






          $begingroup$
          Thank you. How can the argument be made rigorous even when $f$ is not smooth and $Phi$ is not a classical solution but a regular Lagrangian flow?
          $endgroup$
          – Hiro
          yesterday














          $begingroup$
          The function $f$ coud even be multivalued, and you can work in an infinite dimensional Hilbert space as well This is a special case of the general theory of maximal monotone operators and the associated differential equations. Perhaps the friendliest introduction is Brezis' book Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. The ultimate reference is however V. Barbu's book Nonlinear semigroups and differen tial equations in Banach spaces
          $endgroup$
          – Liviu Nicolaescu
          yesterday






          $begingroup$
          The function $f$ coud even be multivalued, and you can work in an infinite dimensional Hilbert space as well This is a special case of the general theory of maximal monotone operators and the associated differential equations. Perhaps the friendliest introduction is Brezis' book Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. The ultimate reference is however V. Barbu's book Nonlinear semigroups and differen tial equations in Banach spaces
          $endgroup$
          – Liviu Nicolaescu
          yesterday














          $begingroup$
          The finite dimensional case is discussed in V. Barbu's recent book Differential Equations Springer 2016, Example 2.4 and Sec. 2.7.
          $endgroup$
          – Liviu Nicolaescu
          yesterday




          $begingroup$
          The finite dimensional case is discussed in V. Barbu's recent book Differential Equations Springer 2016, Example 2.4 and Sec. 2.7.
          $endgroup$
          – Liviu Nicolaescu
          yesterday












          $begingroup$
          In the scalar case all you need for existence and uniqueness is that $f$ is decreasing and the function $mathbb{R}ni xmapsto f(x)-xinmathbb{R}$ is onto.
          $endgroup$
          – Liviu Nicolaescu
          yesterday






          $begingroup$
          In the scalar case all you need for existence and uniqueness is that $f$ is decreasing and the function $mathbb{R}ni xmapsto f(x)-xinmathbb{R}$ is onto.
          $endgroup$
          – Liviu Nicolaescu
          yesterday














          $begingroup$
          Thank you. What if $f$ is increasing?
          $endgroup$
          – Hiro
          yesterday




          $begingroup$
          Thank you. What if $f$ is increasing?
          $endgroup$
          – Hiro
          yesterday


















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