What does the third condition mean in the definition of a well-posed PDE (in the sense of Hadamard)?A PDE is called well-posed (in the sense of Hadamard), if1. A solution exists.2. The solution is unique.3. The solution should depend continuously on the initial and/or boundary data.As far as I have understood, it means that the solution does not undergo a sudden deviation as we increment/decrement the initial/boundary data by a little amount.Does it mean the same? How can I interpret this mathematically?

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odvucimo1pp17 · Accepted Answer

For practical purposes, it means that the solution can be estimated in terms of the data. Consider the PDE      u    t    =  P  u  +  F  (  x  ,  t  )  ,where P is some differential operator and F is a forcing function, and where the problem is defined in some domain   Ω. Say that we additionally have the initial data  u  (  x  ,  0  )  =  f  (  x  )  ,and that  L  u  =  g  (  x  ,  t  )  ,should be satisfied at the boundary Γ of the domain   Ω. Here, f and g are known functions and L is some boundary operator. Then, the problem is said to be (strongly) well posed if an estimate can be obtained along the lines of  ‖  u  (  ⋅  ,  t  )      ‖    2    ≤  K      {    ‖    f          ‖      2        +          ∫      0      t        ‖    F          ‖      2        d    τ    +          ∫      0      t        ‖    g          ‖      Γ      2        }    ,where K is a constant independent of F, f and g.Such an estimate tells us that the solution cannot be &quot;arbitrarily erratic&quot;, e.g. it cannot run away to infinity in any finite time, unless the data already does this.Sometimes other estimates than the one above are used, usually by removing one or both of the integral terms in the right-hand side. The type of estimate needed is largely problem dependent. The problem at hand will in general also determine the norms used in the estimate.

What does the third condition mean in the definition of a well-posed PDE (in the sense of Hadamard)?

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