An estimate for a 1d hyperbolic PDE












1












$begingroup$


Let $L, T, lambda> 0$ be fixed, and let $f in C^1([0,T];H^1(0,L))$, $g in C^1([0,T];H^1(0,L)) cap C^2([0,T];L^2(0,L))$ and $v^0 in H^1(0,L)$. Consider the problem
$$
begin{cases}
partial_t v + lambda partial_x v = f(x,t) & text{in }(0,L)times(0,T)\
v(0,t) = g(0,t) & text{on }(0,T)\
v(x,0) = v^0(x) & text{on }(0,L),
end{cases}
$$
where the unknown is $v colon (0,L)times (0,T) rightarrow mathbb{R}$. My question is:




Is it possible, or not, to obtain an inequality of the form
$$
|v(cdot, t)|_{H^1(0,L)} leq C big( |v^0|_{H^1(0,L)} + max_{tau in [0,T]}|g(cdot, tau)|_{H^1(0,L)} big),
$$
for $C>0$ independant of $v$, where $v in C^0([0,T]; H^1(0,L)) cap C^1([0,T]; L^2(0,L))$ is the solution to above problem?




Having in mind to latter use Gronwall's lemma, for $t in [0,T]$ fixed, I compute
begin{align*}
frac{mathrm{d}}{mathrm{d}t} int_0^L v(x,t)^2 dt &= 2 int_0^L v partial_tv dt \
&= 2 int_0^L v(- lambda partial_x v + f)dt \
&= - lambda int_0^L partial_x(v^2)dt + 2 int_0^L vf dt\
&= - lambda (v(L,t)^2 - v(0,t)^2) + 2 int_0^L vf dt\
&leq lambda g(0,t)^2 + |v(cdot, t)|_{L^2(0,L)}^2 + |f(cdot, t)|_{L^2(0,L)}^2 \
&leq lambda C_0 |g(cdot, t)|_{H^1(0,L)}^2 + |v(cdot, t)|_{L^2(0,L)}^2 + |f(cdot, t)|_{L^2(0,L)}^2,
end{align*}
where $C_0>0$ and Poincaré inequality is used for the last estimation. However, I do not know if it is possible to use a similar procedure for the $L^2$-norm of $partial_x v$. I tried the following computations:
begin{align*}
frac{mathrm{d}}{mathrm{d}t} int_0^L (partial_x v(x,t))^2 dt &= 2 int_0^L partial_x v partial_{xt }v dt \
&= 2 int_0^L partial_x v(- lambda partial_{xx} v + partial_x f)dt \
&= - lambda int_0^L partial_x(partial_x v^2)dt + 2 int_0^L partial_x v partial_x f dt\
&= - lambda (partial_x v(L,t)^2 - partial_x v(0,t)^2) + 2 int_0^L partial_x v partial_x f dt\
&leq partial_x v(0,t)^2 + |partial_x v(cdot, t)|_{L^2(0,L)}^2 + |f(cdot, t)|_{H^1(0,L)}^2,
end{align*}
and I do not know how to estimate the term $partial_x v(0,t)^2$ without making appear the time derivative of $g$.






pde estimation hyperbolic-equations







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    Let $L, T, lambda> 0$ be fixed, and let $f in C^1([0,T];H^1(0,L))$, $g in C^1([0,T];H^1(0,L)) cap C^2([0,T];L^2(0,L))$ and $v^0 in H^1(0,L)$. Consider the problem
    $$
    begin{cases}
    partial_t v + lambda partial_x v = f(x,t) & text{in }(0,L)times(0,T)\
    v(0,t) = g(0,t) & text{on }(0,T)\
    v(x,0) = v^0(x) & text{on }(0,L),
    end{cases}
    $$
    where the unknown is $v colon (0,L)times (0,T) rightarrow mathbb{R}$. My question is:




    Is it possible, or not, to obtain an inequality of the form
    $$
    |v(cdot, t)|_{H^1(0,L)} leq C big( |v^0|_{H^1(0,L)} + max_{tau in [0,T]}|g(cdot, tau)|_{H^1(0,L)} big),
    $$
    for $C>0$ independant of $v$, where $v in C^0([0,T]; H^1(0,L)) cap C^1([0,T]; L^2(0,L))$ is the solution to above problem?




    Having in mind to latter use Gronwall's lemma, for $t in [0,T]$ fixed, I compute
    begin{align*}
    frac{mathrm{d}}{mathrm{d}t} int_0^L v(x,t)^2 dt &= 2 int_0^L v partial_tv dt \
    &= 2 int_0^L v(- lambda partial_x v + f)dt \
    &= - lambda int_0^L partial_x(v^2)dt + 2 int_0^L vf dt\
    &= - lambda (v(L,t)^2 - v(0,t)^2) + 2 int_0^L vf dt\
    &leq lambda g(0,t)^2 + |v(cdot, t)|_{L^2(0,L)}^2 + |f(cdot, t)|_{L^2(0,L)}^2 \
    &leq lambda C_0 |g(cdot, t)|_{H^1(0,L)}^2 + |v(cdot, t)|_{L^2(0,L)}^2 + |f(cdot, t)|_{L^2(0,L)}^2,
    end{align*}
    where $C_0>0$ and Poincaré inequality is used for the last estimation. However, I do not know if it is possible to use a similar procedure for the $L^2$-norm of $partial_x v$. I tried the following computations:
    begin{align*}
    frac{mathrm{d}}{mathrm{d}t} int_0^L (partial_x v(x,t))^2 dt &= 2 int_0^L partial_x v partial_{xt }v dt \
    &= 2 int_0^L partial_x v(- lambda partial_{xx} v + partial_x f)dt \
    &= - lambda int_0^L partial_x(partial_x v^2)dt + 2 int_0^L partial_x v partial_x f dt\
    &= - lambda (partial_x v(L,t)^2 - partial_x v(0,t)^2) + 2 int_0^L partial_x v partial_x f dt\
    &leq partial_x v(0,t)^2 + |partial_x v(cdot, t)|_{L^2(0,L)}^2 + |f(cdot, t)|_{H^1(0,L)}^2,
    end{align*}
    and I do not know how to estimate the term $partial_x v(0,t)^2$ without making appear the time derivative of $g$.






    pde estimation hyperbolic-equations







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      Let $L, T, lambda> 0$ be fixed, and let $f in C^1([0,T];H^1(0,L))$, $g in C^1([0,T];H^1(0,L)) cap C^2([0,T];L^2(0,L))$ and $v^0 in H^1(0,L)$. Consider the problem
      $$
      begin{cases}
      partial_t v + lambda partial_x v = f(x,t) & text{in }(0,L)times(0,T)\
      v(0,t) = g(0,t) & text{on }(0,T)\
      v(x,0) = v^0(x) & text{on }(0,L),
      end{cases}
      $$
      where the unknown is $v colon (0,L)times (0,T) rightarrow mathbb{R}$. My question is:




      Is it possible, or not, to obtain an inequality of the form
      $$
      |v(cdot, t)|_{H^1(0,L)} leq C big( |v^0|_{H^1(0,L)} + max_{tau in [0,T]}|g(cdot, tau)|_{H^1(0,L)} big),
      $$
      for $C>0$ independant of $v$, where $v in C^0([0,T]; H^1(0,L)) cap C^1([0,T]; L^2(0,L))$ is the solution to above problem?




      Having in mind to latter use Gronwall's lemma, for $t in [0,T]$ fixed, I compute
      begin{align*}
      frac{mathrm{d}}{mathrm{d}t} int_0^L v(x,t)^2 dt &= 2 int_0^L v partial_tv dt \
      &= 2 int_0^L v(- lambda partial_x v + f)dt \
      &= - lambda int_0^L partial_x(v^2)dt + 2 int_0^L vf dt\
      &= - lambda (v(L,t)^2 - v(0,t)^2) + 2 int_0^L vf dt\
      &leq lambda g(0,t)^2 + |v(cdot, t)|_{L^2(0,L)}^2 + |f(cdot, t)|_{L^2(0,L)}^2 \
      &leq lambda C_0 |g(cdot, t)|_{H^1(0,L)}^2 + |v(cdot, t)|_{L^2(0,L)}^2 + |f(cdot, t)|_{L^2(0,L)}^2,
      end{align*}
      where $C_0>0$ and Poincaré inequality is used for the last estimation. However, I do not know if it is possible to use a similar procedure for the $L^2$-norm of $partial_x v$. I tried the following computations:
      begin{align*}
      frac{mathrm{d}}{mathrm{d}t} int_0^L (partial_x v(x,t))^2 dt &= 2 int_0^L partial_x v partial_{xt }v dt \
      &= 2 int_0^L partial_x v(- lambda partial_{xx} v + partial_x f)dt \
      &= - lambda int_0^L partial_x(partial_x v^2)dt + 2 int_0^L partial_x v partial_x f dt\
      &= - lambda (partial_x v(L,t)^2 - partial_x v(0,t)^2) + 2 int_0^L partial_x v partial_x f dt\
      &leq partial_x v(0,t)^2 + |partial_x v(cdot, t)|_{L^2(0,L)}^2 + |f(cdot, t)|_{H^1(0,L)}^2,
      end{align*}
      and I do not know how to estimate the term $partial_x v(0,t)^2$ without making appear the time derivative of $g$.






      pde estimation hyperbolic-equations







      share|cite|improve this question









      $endgroup$




      Let $L, T, lambda> 0$ be fixed, and let $f in C^1([0,T];H^1(0,L))$, $g in C^1([0,T];H^1(0,L)) cap C^2([0,T];L^2(0,L))$ and $v^0 in H^1(0,L)$. Consider the problem
      $$
      begin{cases}
      partial_t v + lambda partial_x v = f(x,t) & text{in }(0,L)times(0,T)\
      v(0,t) = g(0,t) & text{on }(0,T)\
      v(x,0) = v^0(x) & text{on }(0,L),
      end{cases}
      $$
      where the unknown is $v colon (0,L)times (0,T) rightarrow mathbb{R}$. My question is:




      Is it possible, or not, to obtain an inequality of the form
      $$
      |v(cdot, t)|_{H^1(0,L)} leq C big( |v^0|_{H^1(0,L)} + max_{tau in [0,T]}|g(cdot, tau)|_{H^1(0,L)} big),
      $$
      for $C>0$ independant of $v$, where $v in C^0([0,T]; H^1(0,L)) cap C^1([0,T]; L^2(0,L))$ is the solution to above problem?




      Having in mind to latter use Gronwall's lemma, for $t in [0,T]$ fixed, I compute
      begin{align*}
      frac{mathrm{d}}{mathrm{d}t} int_0^L v(x,t)^2 dt &= 2 int_0^L v partial_tv dt \
      &= 2 int_0^L v(- lambda partial_x v + f)dt \
      &= - lambda int_0^L partial_x(v^2)dt + 2 int_0^L vf dt\
      &= - lambda (v(L,t)^2 - v(0,t)^2) + 2 int_0^L vf dt\
      &leq lambda g(0,t)^2 + |v(cdot, t)|_{L^2(0,L)}^2 + |f(cdot, t)|_{L^2(0,L)}^2 \
      &leq lambda C_0 |g(cdot, t)|_{H^1(0,L)}^2 + |v(cdot, t)|_{L^2(0,L)}^2 + |f(cdot, t)|_{L^2(0,L)}^2,
      end{align*}
      where $C_0>0$ and Poincaré inequality is used for the last estimation. However, I do not know if it is possible to use a similar procedure for the $L^2$-norm of $partial_x v$. I tried the following computations:
      begin{align*}
      frac{mathrm{d}}{mathrm{d}t} int_0^L (partial_x v(x,t))^2 dt &= 2 int_0^L partial_x v partial_{xt }v dt \
      &= 2 int_0^L partial_x v(- lambda partial_{xx} v + partial_x f)dt \
      &= - lambda int_0^L partial_x(partial_x v^2)dt + 2 int_0^L partial_x v partial_x f dt\
      &= - lambda (partial_x v(L,t)^2 - partial_x v(0,t)^2) + 2 int_0^L partial_x v partial_x f dt\
      &leq partial_x v(0,t)^2 + |partial_x v(cdot, t)|_{L^2(0,L)}^2 + |f(cdot, t)|_{H^1(0,L)}^2,
      end{align*}
      and I do not know how to estimate the term $partial_x v(0,t)^2$ without making appear the time derivative of $g$.






      pde estimation hyperbolic-equations




      pde estimation hyperbolic-equations






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 10 '18 at 9:31









      user344045user344045

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