Well-Posedness of PDEs

graph TD D1["D1 Well-posed problem\nExistence, uniqueness, continuity"] D2["D2 Ill-posed problem\nViolates one of Hadamard conditions"] D3["D3 Cauchy problem\nInitial data on non-characteristic surface"] D4["D4 Continuous dependence\nSmall data change ⇒ small solution change"] T1["T1 Cauchy-Kowalevski\nAnalytic data ⇒ analytic solution"] T2["T2 Hadamard example\nLaplace eq. Cauchy ill-posed"] T3["T3 Energy estimates\nStability for parabolic/hyperbolic"] T4["T4 Lax-Milgram\nExistence for elliptic in Hilbert space"] T5["T5 Semigroup theory\nWell-posedness for evolution equations"] D1 --> T3 D2 --> T2 D3 --> T1 D4 --> T3 D3 --> T2 D1 --> T4 D1 --> T5 T1 --> D1 T3 --> D4 T4 --> D1 T5 --> D1 classDef definition fill:#3498db,color:#fff,stroke:#2980b9 classDef theorem fill:#1abc9c,color:#fff,stroke:#16a085 class D1,D2,D3,D4 definition class T1,T2,T3,T4,T5 theorem