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Well-posedness in the sense of Hadamard
Existence, uniqueness, continuous dependence, and the classic examples that fail.
Hadamard called a problem well-posed when (1) a solution exists, (2) it is unique, and (3) it depends continuously on the data: small changes in the data produce small changes in the solution. The third condition is what makes a model usable, because real data are never exact. All three are relative to a choice of spaces, for the data and for the solution, and much of the theory consists of finding spaces in which a problem is well-posed.
The backward heat equation fails (3). For \( t \ge 0 \) the function \( u_n = \frac1ne^{n^2t}\sin nx \) solves \( u_t = -u_{xx} \) with data \( u_n(x,0) = \frac1n\sin nx \), which tends to zero uniformly. But at \( t = 1 \) its amplitude is \( e^{n^2}/n \), which the example shows tends to infinity. Arbitrarily small data give arbitrarily large solutions, so no estimate \( \|u(1)\| \le C\|u(0)\| \) can hold.
Laplace's equation with initial data fails (3). Hadamard's example: \( u_{xx} + u_{yy} = 0 \) with \( u(x,0) = 0 \), \( u_y(x,0) = \frac1n\sin nx \) has the solution \( u = \frac1{n^2}\sin nx\sinh ny \). The data tend to zero, but for any \( y > 0 \) the solution grows like \( e^{ny}/n^2 \). Elliptic equations want boundary data, not Cauchy data. The same data are perfectly fine for the wave equation \( u_{tt} = u_{xx} \), whose solution \( \frac1{n^2}\sin nx\sin nt \) stays small.
Picture it: a well-posed problem is a map from data to solutions with no cliffs. The backward heat equation is a map with a cliff at every point: a ripple too small to see in the data is enough to change everything.
Think it: continuous dependence is usually proved as an estimate, \( \|u_1 - u_2\|_Y \le C\|f_1 - f_2\|_X \), and the energy and maximum principle bounds from earlier lessons are exactly such estimates. The Navier-Stokes Millennium problem is a well-posedness question: for smooth, rapidly decaying, divergence-free data in \( \mathbb R^3 \), does a smooth solution exist for all time? Local well-posedness is known; global existence of smooth solutions is not.
Gumagana halimbawa · limit of exp(n^2)/n as n -> oo
Hakbang-hakbang
- \lim_{n \to \infty^+-} \frac{e^{n^{2}}}{n}
Try direct substitution first.
- \
As x grows without bound, compare the fastest-growing terms (or divide top and bottom by the highest power).
- = \infty
Take the limit.
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Paano: Well-posedness in the sense of Hadamard
- Fix the spaces: in which norm are the data measured, and in which norm the solution?
- Existence: produce a solution (formula, series, fixed point or compactness).
- Uniqueness: show the difference of two solutions with the same data vanishes (energy or maximum principle).
- Stability: prove an estimate of the difference of solutions by the difference of data.
- To show ill-posedness, find a sequence of data tending to zero whose solutions do not.
Mga katanungan na tanong ng mga tao
Are ill-posed problems useless?
No. Recovering an image from a blurred photograph is a backward heat problem. It is solved by regularisation: restricting to data with extra smoothness or adding a penalty, which restores a weaker form of stability.
Can a problem be well-posed in one space and ill-posed in another?
Yes, and this is common: many nonlinear dispersive and fluid equations are well-posed in H^s for s above a threshold and ill-posed below it.
What should I know before starting?
Partial derivatives, the divergence theorem and multiple integrals from multivariable calculus; linear second-order ODEs; eigenvalues from linear algebra; and enough analysis to be comfortable with uniform convergence and integrals over infinite intervals.
Why are there so few formulas for solutions?
Explicit solutions exist for linear equations with constant coefficients on simple domains. Almost everything else (curved domains, variable coefficients, nonlinear terms) has none, so the modern subject proves that a solution exists and estimates its size and smoothness without ever writing it down.
How does this course lead to the Navier-Stokes problem?
The Navier-Stokes equations are a heat equation for the velocity with a transport term and a pressure that is found from a Poisson equation. Reading the Millennium problem needs energy estimates, weak solutions, Sobolev spaces, local existence with a blow-up criterion, and the scaling argument that explains why three dimensions is hard. The last six lessons build those.
What do elliptic, parabolic and hyperbolic mean?
Elliptic equations (Laplace) describe equilibrium and smooth everything; parabolic equations (heat) describe diffusion forward in time; hyperbolic equations (waves) carry signals at finite speed and keep their sharp edges. The type decides which data make a sensible problem.
Higit pa sa Partial Differential Equations
What a PDE is, and how to classify oneFirst-order linear equations and characteristicsNonlinear first-order equations and shocksThe heat equation: where it comes fromSeparation of variablesFourier series: coefficients, convergence and ParsevalThe heat equation on an intervalThe wave equation and d'Alembert's formulaThe vibrating string: modes and energyLaplace's equation and harmonic functionsMaximum principles and uniquenessGreen's functions and the fundamental solutionThe Fourier transform and the heat kernelEnergy methods: uniqueness, stability and decay