In my PDE class we are following Evans PDE book, we were reading about Regularity of weak solutions for Hyperbolic equations, more specific in the proof theorem 5 section 7.2.3., the author states that we have d d t ( ‖ u ~ m ′ ‖ L 2 ( U ) 2 + A [ u ~ m , u ~ m ] ) ≤ C ( ‖ u ~ m ′ ‖ L 2 ( U ) 2 + A [ u ~ m , u ~ m ] + ‖ f ′ ‖ L 2 ( U ) 2 )where u ~ m = u m ′ , also the estimate ‖ u m ‖ H 2 ( U ) 2 ≤ C ( ‖ f ‖ L 2 ( U ) 2 + ‖ u m ″ ‖ L 2 ( U ) 2 + ‖ u m ‖ L 2 ( U ) 2 )Evans says that using this last inequality in the first and aplying Gronwall's Inequality we deduce that sup 0 ≤ t ≤ T ( ‖ u m ( t ) ‖ H 2 ( U ) 2 + ‖ u m ′ ( t ) ‖ H 1 ( U ) 2 + ‖ u m ″ ( t ) ‖ L 2 ( U ) 2 ) ≤ C ( ‖ f ‖ H 1 ( 0 , T ; L 2 ( U ) ) 2 + ‖ g ‖ H 2 ( U ) 2 + ‖ h ‖ H 1 ( U ) 2 )My problem is that I don't understand how this last expression is obtained, can anyone help me?

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In my PDE class we are following Evans PDE book, we were reading about Regularity of weak solutions for Hyperbolic equations, more specific in the proof theorem 5 section 7.2.3., the author states that we have      d          d      t        (  ‖                    u        ~              m                          ′                  ‖                  L        2            (      U      )        2    +  A  [                    u        ~              m    ,                    u        ~              m    ]  )  ≤  C  (  ‖                    u        ~              m                          ′                  ‖                  L        2            (      U      )        2    +  A  [                    u        ~              m    ,                    u        ~              m    ]  +  ‖      f                          ′                  ‖                  L                  2                    (      U      )        2    )where                     u        ~              m    =      u    m                          ′             , also the estimate  ‖      u    m        ‖                  H        2            (      U      )        2    ≤  C  (  ‖  f      ‖                  L        2            (      U      )        2    +  ‖      u    m                          ″                  ‖                  L        2            (      U      )        2    +  ‖      u    m        ‖                  L        2            (      U      )        2    )Evans says that using this last inequality in the first and aplying Gronwall&#039;s Inequality we deduce that      sup          0      ≤      t      ≤      T        (  ‖      u    m    (  t  )      ‖                  H        2            (      U      )        2    +  ‖      u    m                          ′              (  t  )      ‖                  H        1            (      U      )        2    +  ‖      u    m                          ″              (  t  )      ‖                  L        2            (      U      )        2    )  ≤  C  (  ‖  f      ‖                  H        1            (      0      ,      T      ;              L        2            (      U      )      )        2    +  ‖  g      ‖                  H        2            (      U      )        2    +  ‖  h      ‖                  H        1            (      U      )        2    )My problem is that I don&#039;t understand how this last expression is obtained, can anyone help me?

Mackenzie Zimmerman · Accepted Answer

Step 1I am dropping the subscript m which is used to indicate approximating solutions.The first inequality (with the time derivative on the left) comes from considering the pde that is satisfied by             u      ~        =      u    ′   and applying the usual energy estimate. Apply a Gronwall argument here to obtain an estimate      sup    t        (    ‖                            u          ~                    ′        (    t    )          ‖                        L          2                    2        +    A    (                  u        ~              (    t    )    ,                  u        ~              (    t    )    )    )        ≤  C      (    ‖                            u          ~                    ′        (    0    )          ‖                        L          2                    2        +    A    (                  u        ~              (    0    )    ,                  u        ~              (    0    )    )    +          ∫      0      T        ‖          f      ′              ‖                        L          2                    2        )  You read off from the pde for             u      ~       what             u      ~        (  0  ) and                     u        ~              ′    (  0  ) must be. This implies estimates for      sup    t        (    ‖          u              t        t              (    t    )          ‖                        L          2                      +    ‖          u      t        (    t    )          ‖                        H          1                      )  since the form A is (essentially) coercive.The second inequality follows from the pde itself plus elliptic regularity theory for the operator L. Just write   L  u  =  −      u          t      t        +  f and use an estimate like  ‖  u      ‖                  H        2              ≤  C  (  ‖  L  u      ‖                  L        2              +  ‖  u      ‖                  L        2              )which surely appears in an earlier chapter of the book.Since you already have an estimate for   ‖      u          t      t            ‖                  L        2             , the desired estimate now can be derived. Just keep track of where norms of g and h enter the estimates.

In my PDE class we are following Evans PDE book, we were reading about Regularity of weak solutions

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