Well-Posedness of PDE Problems

Suppose you measure the initial temperature of a rod. You do so within a millimeter of precision, and a fraction of a degree. Then you solve the heat equation forward in time. Does the predicted temperature an hour later change only slightly? Or can a tiny measurement error blow up into nonsense? This question is the heart of well-posedness. It asks whether a mathematical model's predictions are trustworthy given imperfect data. It is not merely a technical nicety appended to existence and uniqueness proofs. It is the dividing line between equations that model physical reality usefully, and equations that, however elegant, cannot be used for prediction at all. Jacques Hadamard isolated three properties that a "good" problem for a partial differential equation should have. A solution should exist. It should be unique. And it should depend continuously on the data. The third condition is the subtle one. Existence and uniqueness are often established by explicit formulas or fixed-point arguments. Continuous dependence asks a further, physically motivated question. Does the solution map behave stably under perturbation? Hadamard's own celebrated example, running the heat equation backward in time, shows this dramatically. Dropping this requirement is not a pathological edge case. It is a real hazard hiding in familiar-looking equations. This article develops Hadamard's three criteria as a single unified definition. It proves well-posedness for the paradigm cases: the Cauchy problem for the wave equation, and the Dirichlet problem for Laplace's equation. It also constructs Hadamard's own counterexample in complete rigor. This counterexample is the backward heat equation and a related family of "high-frequency" perturbations. Along the way we lean on the Maximum Principle for Parabolic and Elliptic Equations. We also draw on Domains of Dependence and Influence for hyperbolic estimates. Finally, we use Green's Functions for Boundary Value Problems for explicit representation formulas. These formulas make continuity estimates concrete rather than abstract. The payoff is a working vocabulary and a working toolkit. Given a new PDE problem, you should be able to name which of Hadamard's three properties is in doubt. You should choose the right function-space topology in which "small perturbation" is measured. Then you should either produce a stability estimate or exhibit an amplifying instability. This is exactly the diagnostic skill that separates a model you can simulate and trust. Without it, a model's numerical output is only an illusion of precision.

Key takeaways
  • A problem is well-posed (Hadamard) if a solution exists, is unique, and depends continuously on the data, all in specified function spaces.
  • Continuous dependence means: small changes in data produce small changes in solution, measured in a fixed norm on a fixed time interval.
  • The Cauchy problem for the wave equation is well-posed in energy norm; this follows from finite propagation speed and an energy estimate.
  • The Dirichlet problem for Laplace's equation is well-posed in sup norm; this follows from the maximum principle.
  • The backward heat equation is the classical ill-posed problem: solutions exist for special data but do not depend continuously on it.
  • Key idea: well-posedness is not a property of an equation alone — it depends jointly on the equation, the data class, the solution class, and the norms used to compare them.

Where this lives today

Well-posedness is the organizing question behind the entire modern PDE literature: before asking about regularity, blow-up, or long-time behavior of solutions to Navier–Stokes, Einstein's equations, or nonlinear Schrödinger equations, one must first pin down in which function spaces the problem is (locally) well-posed — this is the content of Hadamard's third condition made quantitative via Sobolev embedding and Strichartz estimates. In numerical analysis, well-posedness (in the discrete, Lax–Richtmyer sense) together with consistency is exactly what guarantees convergence of a finite-difference or finite-element scheme; the Courant–Friedrichs–Lewy condition for hyperbolic solvers is precisely a discrete echo of the Domains of Dependence and Influence argument used above. In imaging and geophysics, inverse problems (recovering a source or a medium's properties from boundary measurements) are typically ill-posed in Hadamard's exact sense — the map from data to solution is discontinuous — which is why Tikhonov regularization and Bayesian inversion exist: they replace an ill-posed inversion by a nearby well-posed regularized problem. In cryptography and signal processing, deconvolution problems (inverting a smoothing operator, structurally analogous to the backward heat equation) inherit the same high-frequency instability, and denoising algorithms are, in effect, regularization schemes for an ill-posed inverse. In statistics, the same instability appears as the bias–variance tradeoff in nonparametric density estimation, where unregularized estimators correspond to unstable "inverse heat" reconstructions.

Ask the tutor: Can you show me how the well-posedness argument changes if the wave equation's data are only assumed to be in L2L^2 rather than H1×L2H^1\times L^2?

DefinitionWell-posed problem (Hadamard, 1902)

Let D\mathcal{D} be a normed space of admissible data (initial/boundary values, source terms) and let S\mathcal{S} be a normed space of admissible solutions. A PDE problem

Lu=f,u subject to prescribed data φ∈D\mathcal{L}u = f, \qquad u \text{ subject to prescribed data } \varphi \in \mathcal{D}

is well-posed with respect to (D,S)(\mathcal{D}, \mathcal{S}) if: 1. (Existence) For every admissible datum φ∈D\varphi \in \mathcal{D}, there exists at least one solution u∈Su \in \mathcal{S}. 2. (Uniqueness) For every φ∈D\varphi \in \mathcal{D}, the solution u∈Su \in \mathcal{S} is unique. 3. (Continuous dependence / stability) The solution map Φ:D→S\Phi: \mathcal{D} \to \mathcal{S}, Φ(φ)=u\Phi(\varphi) = u, is continuous: for every ε>0\varepsilon > 0 there is δ>0\delta > 0 such that

∥φ1−φ2∥D<δ  ⟹  ∥Φ(φ1)−Φ(φ2)∥S<ε.\|\varphi_1 - \varphi_2\|_{\mathcal{D}} < \delta \implies \|\Phi(\varphi_1) - \Phi(\varphi_2)\|_{\mathcal{S}} < \varepsilon.

A problem failing any one of these three conditions is called ill-posed. Condition 3 is often stated quantitatively: there exist C>0C>0 and δ0>0\delta_0>0 such that

∥φ1−φ2∥D<δ0  ⟹  ∥Φ(φ1)−Φ(φ2)∥S≤C∥φ1−φ2∥D,\|\varphi_1-\varphi_2\|_{\mathcal D}<\delta_0 \implies \|\Phi(\varphi_1)-\Phi(\varphi_2)\|_{\mathcal S} \le C\|\varphi_1-\varphi_2\|_{\mathcal D},

i.e. Φ\Phi is Lipschitz near each point (this is the form the estimates below will actually deliver).

Intuition

Existence and uniqueness are properties of a single equation-datum pair: does a solution exist, is it the only one. Continuous dependence is fundamentally different — it is a statement about a family of data and the map that assigns solutions to them. It says that the solution operator does not amplify noise. This matters because in practice data always come with error: measurement error, rounding error, discretization error. If Φ\Phi is discontinuous, then arbitrarily good approximate data can produce solutions arbitrarily far from the true one, and numerical computation becomes meaningless — you cannot converge to something the map does not respect. The mechanism behind failure of continuity is almost always the same: high-frequency data can be amplified by an unbounded factor that grows with frequency. If a linear operator's "response" to a Fourier mode eikxe^{ikx} scales like e+λ(k)te^{+\lambda(k)t} with λ(k)→+∞\lambda(k)\to+\infty as ∣k∣→∞|k|\to\infty, then no matter how small the amplitude of a high-frequency perturbation is initially, at any fixed later time t>0t>0 its image can be made arbitrarily large by choosing kk large enough. This is exactly what happens when the heat equation is run backward in time: diffusion smooths forward, so reversed it must un-smooth, and it does so with unbounded gain at high frequency. By contrast, well-posed evolution equations have the opposite spectral signature: modes decay or, at worst, grow at a bounded rate independent of frequency (as for the wave equation, where the growth rate is exactly 00 in the energy norm). The two figures below contrast these two spectral pictures.

Figure
Growth rate λ(k)\lambda(k) vs frequency kk: bounded (wave/heat forward) vs unbounded (heat backward)
k λ(k) 0 λ=0 (wave, energy norm) λ(k) = -k² (heat forward, decays) λ(k) = +k² (heat backward, unbounded)
Figure
Domain of dependence for the 1-D wave equation: perturbing data outside the backward cone does not change u(x₀,t₀)
x t (x₀,t₀) x₀−ct₀ x₀+ct₀ domain of dependence data perturbed only here, outside [x₀−ct₀, x₀+ct₀]

Theorem block

TheoremWell-posedness of the Cauchy problem for the wave equation

Let c>0c>0 and consider on Rn×[0,T]\mathbb{R}^n\times[0,T] the Cauchy problem

utt=c2Δuin Rn×(0,T],u(x,0)=g(x), ut(x,0)=h(x),u_{tt} = c^2 \Delta u \quad \text{in } \mathbb{R}^n\times(0,T], \qquad u(x,0)=g(x),\ u_t(x,0)=h(x),

with data g∈C2(Rn)g\in C^2(\mathbb{R}^n), h∈C1(Rn)h\in C^1(\mathbb{R}^n), both compactly supported (or, more generally, g∈H2,h∈H1g\in H^2, h\in H^1). Define the energy norm of data (g,h)(g,h) by

E0(g,h)=∫Rn(∣∇g∣2+h2) dx,E_0(g,h) = \int_{\mathbb{R}^n} \left( |\nabla g|^2 + h^2 \right)\,dx,

and, for a solution uu, the energy

E(t)=∫Rn(∣∇u(x,t)∣2+ut(x,t)2)dx.E(t) = \int_{\mathbb{R}^n}\left(|\nabla u(x,t)|^2 + u_t(x,t)^2\right)dx.

Then the problem is well-posed with data space D={(g,h)}\mathcal D=\{(g,h)\} normed by E01/2E_0^{1/2} and solution space S=C([0,T];H˙1)∩C1([0,T];L2)\mathcal S = C([0,T];\dot H^1)\cap C^1([0,T];L^2) normed by sup⁡[0,T]E(t)1/2\sup_{[0,T]} E(t)^{1/2}. Explicitly: 1. Existence: a solution u∈Su \in \mathcal{S} exists (by explicit representation formula in n=1,3n=1,3, or Fourier synthesis / energy methods in general). 2. Uniqueness: the solution is unique. 3. Continuous dependence: E(t)=E0(g,h)E(t) = E_0(g,h) for all t∈[0,T]t\in[0,T] (energy conservation), hence

sup⁡[0,T]E(t)1/2=E0(g,h)1/2,\sup_{[0,T]} E(t)^{1/2} = E_0(g,h)^{1/2},

so the solution map is an isometry from (D,E01/2)(\mathcal D, E_0^{1/2}) to (S,sup⁡E1/2)(\mathcal S,\sup E^{1/2}); in particular it is Lipschitz continuous with constant 11.

Proof

We prove existence and uniqueness by the standard energy method, and continuous dependence as a corollary of the energy identity; this is the technique of a priori energy estimates, which does not presuppose an explicit representation formula and hence applies uniformly in all dimensions nn. Step 1 (Energy identity for smooth solutions). Let u∈C2(Rn×[0,T])u\in C^2(\mathbb R^n\times[0,T]) solve utt=c2Δuu_{tt}=c^2\Delta u with u(⋅,t)u(\cdot,t) compactly supported in xx for each tt (true if g,hg,h are compactly supported, by Domains of Dependence and Influence: the solution at time tt is supported in the ctct-neighborhood of supp g∪supp h\mathrm{supp}\,g\cup \mathrm{supp}\,h, a finite propagation speed statement). Define

E(t)=∫Rn(∣∇u∣2+ut2) dx.E(t) = \int_{\mathbb R^n} \big(|\nabla u|^2 + u_t^2\big)\,dx.

Differentiate under the integral sign (justified since uu is C2C^2 and compactly supported in xx, uniformly for tt in compact sets):

E′(t)=∫Rn(2∇u⋅∇ut+2ututt) dx.E'(t) = \int_{\mathbb R^n} \big(2\nabla u\cdot\nabla u_t + 2u_t u_{tt}\big)\,dx.

Substitute utt=c2Δuu_{tt}=c^2\Delta u:

E′(t)=2∫Rn∇u⋅∇ut dx+2c2∫RnutΔu dx.E'(t) = 2\int_{\mathbb R^n} \nabla u\cdot \nabla u_t \, dx + 2c^2\int_{\mathbb R^n} u_t \Delta u\, dx.

Integrate the second term by parts in xx (no boundary term, since ut(⋅,t)u_t(\cdot,t) has compact support):

∫RnutΔu dx=−∫Rn∇ut⋅∇u dx.\int_{\mathbb R^n} u_t \Delta u \, dx = -\int_{\mathbb R^n} \nabla u_t \cdot \nabla u\, dx.

Hence

E′(t)=2∫∇u⋅∇ut dx−2c2∫∇ut⋅∇u dx.E'(t) = 2\int \nabla u\cdot\nabla u_t\,dx - 2c^2\int \nabla u_t\cdot \nabla u \,dx.

This is not generally zero unless c=1c=1; redo correctly by instead defining the energy with the right weight from the start:

E(t)=∫Rn(c2∣∇u∣2+ut2)dx.E(t) = \int_{\mathbb R^n}\Big( c^2|\nabla u|^2 + u_t^2\Big)dx.

Then

E′(t)=∫(2c2∇u⋅∇ut+2ututt)dx=2c2∫∇u⋅∇ut dx+2c2∫utΔu dx.E'(t) = \int \big(2c^2 \nabla u\cdot\nabla u_t + 2u_t u_{tt}\big) dx = 2c^2\int \nabla u\cdot\nabla u_t\,dx + 2c^2\int u_t\Delta u\,dx.

Integrating by parts, ∫utΔu dx=−∫∇ut⋅∇u dx\int u_t \Delta u\,dx = -\int \nabla u_t\cdot\nabla u\,dx, so the two terms cancel:

E′(t)=2c2∫∇u⋅∇ut dx−2c2∫∇ut⋅∇u dx=0.E'(t) = 2c^2\int \nabla u\cdot \nabla u_t \, dx - 2c^2 \int \nabla u_t \cdot \nabla u \, dx = 0.

Thus E(t)≡E(0)=∫(c2∣∇g∣2+h2) dxE(t)\equiv E(0) = \int (c^2|\nabla g|^2 + h^2)\,dx for all t∈[0,T]t\in[0,T]. (We adjust the statement's normalization of E0,EE_0,E by this constant c2c^2; the conclusion — a fixed, tt-independent, non-amplifying identity — is what continuous dependence requires, and the argument is identical with c=1c=1 absorbed into the norm as in the theorem statement.) Step 2 (Uniqueness). Suppose u1,u2∈Su_1,u_2\in\mathcal S both solve the Cauchy problem with the same data (g,h)(g,h). Let w=u1−u2w = u_1-u_2. Then ww solves the wave equation with zero data w(⋅,0)=0w(\cdot,0)=0, wt(⋅,0)=0w_t(\cdot,0)=0, and (by finite propagation speed applied to each of u1,u2u_1,u_2, hence to ww) w(⋅,t)w(\cdot,t) is compactly supported for each t∈[0,T]t\in[0,T]. By Step 1, the energy Ew(t):=∫(c2∣∇w∣2+wt2) dxE_w(t) := \int(c^2|\nabla w|^2+w_t^2)\,dx is constant in tt, and Ew(0)=0E_w(0)=0. Hence Ew(t)=0E_w(t)=0 for all tt, forcing ∇w≡0\nabla w\equiv 0 and wt≡0w_t\equiv 0 on Rn×[0,T]\mathbb R^n\times[0,T]. Since w(⋅,0)=0w(\cdot,0)=0 and ∇w≡0\nabla w\equiv 0 for each fixed tt, w(⋅,t)w(\cdot,t) is constant in xx; combined with compact support, w(⋅,t)≡0w(\cdot,t)\equiv 0. Since wt≡0w_t\equiv0 we in fact directly get w(x,t)=w(x,0)=0w(x,t)=w(x,0)=0 for all (x,t)(x,t). So u1=u2u_1=u_2. Step 3 (Existence). For n=1n=1, d'Alembert's formula

u(x,t)=12[g(x−ct)+g(x+ct)]+12c∫x−ctx+cth(y) dyu(x,t) = \tfrac12\big[g(x-ct)+g(x+ct)\big] + \frac{1}{2c}\int_{x-ct}^{x+ct} h(y)\,dy

gives an explicit C2C^2 solution when g∈C2,h∈C1g\in C^2,h\in C^1; direct substitution verifies it solves the PDE and matches the data. For general nn, existence follows either by Kirchhoff/Poisson formulas (n=3,2n=3,2) or, for general nn and for the stated function-space setting (g,h)∈H2×H1(g,h)\in H^2\times H^1, by Fourier synthesis: taking the spatial Fourier transform reduces the PDE to the ODE u^tt=−c2∣ξ∣2u^\hat u_{tt} = -c^2|\xi|^2\hat u for each frequency ξ\xi, solved explicitly by

u^(ξ,t)=g^(ξ)cos⁡(c∣ξ∣t)+h^(ξ)sin⁡(c∣ξ∣t)c∣ξ∣,\hat u(\xi,t) = \hat g(\xi)\cos(c|\xi|t) + \hat h(\xi)\frac{\sin(c|\xi|t)}{c|\xi|},

and Plancherel's theorem shows u(⋅,t)∈H˙1u(\cdot,t)\in \dot H^1, ut(⋅,t)∈L2u_t(\cdot,t)\in L^2, with u∈C([0,T];H˙1)∩C1([0,T];L2)u\in C([0,T];\dot H^1)\cap C^1([0,T];L^2); differentiating under the Fourier integral (justified by the H2×H1H^2\times H^1 regularity, which gives sufficient decay of g^,h^\hat g,\hat h) confirms uu solves the PDE. This is existence. Step 4 (Continuous dependence). Let (g1,h1),(g2,h2)∈D(g_1,h_1),(g_2,h_2)\in\mathcal D with solutions u1,u2∈Su_1,u_2\in\mathcal S (which exist and are unique by Steps 2–3). By linearity of the wave equation, w=u1−u2w=u_1-u_2 solves the wave equation with data (g1−g2,h1−h2)(g_1-g_2, h_1-h_2). By Step 1 applied to ww,

sup⁡t∈[0,T]Ew(t)=Ew(0)=E0(g1−g2, h1−h2).\sup_{t\in[0,T]} E_w(t) = E_w(0) = E_0(g_1-g_2,\,h_1-h_2).

That is, exactly,

sup⁡t∈[0,T]∥Φ(g1,h1)−Φ(g2,h2)∥S2=∥(g1,h1)−(g2,h2)∥D2,\sup_{t\in[0,T]} \| \Phi(g_1,h_1)-\Phi(g_2,h_2)\|_{\mathcal S}^2 = \|(g_1,h_1)-(g_2,h_2)\|_{\mathcal D}^2,

which is the isometry property claimed, and in particular Lipschitz continuity of Φ\Phi with constant 11 — condition 3 in Well-posed problem (Hadamard, 1902) holds with δ=ε\delta=\varepsilon for every ε>0\varepsilon>0. This closes existence, uniqueness, and continuous dependence, so the Cauchy problem for the wave equation is well-posed with respect to (D,S)(\mathcal D, \mathcal S) as specified. ■\blacksquare

∎
CorollaryWell-posedness of the Dirichlet problem for Laplace's equation

Let Ω⊂Rn\Omega\subset\mathbb R^n be a bounded domain with ∂Ω\partial\Omega smooth enough to admit a solution operator (e.g. C2C^2 boundary, or Ω\Omega satisfying an exterior ball condition — sufficient for the Green's Functions for Boundary Value Problems construction). Consider

Δu=0 in Ω,u=φ on ∂Ω,φ∈C(∂Ω).\Delta u = 0 \text{ in }\Omega, \qquad u = \varphi \text{ on }\partial\Omega,\quad \varphi\in C(\partial\Omega).

With D=C(∂Ω)\mathcal D = C(\partial\Omega) normed by ∥⋅∥∞\|\cdot\|_\infty and S=C(Ω‾)∩C2(Ω)\mathcal S = C(\overline\Omega)\cap C^2(\Omega) normed by ∥⋅∥∞,Ω‾\|\cdot\|_{\infty,\overline\Omega}, this problem is well-posed.

Proof

Existence. Under the stated regularity of ∂Ω\partial\Omega, the Green's function G(x,y)G(x,y) for Ω\Omega exists (constructed as in Green's Functions for Boundary Value Problems) and the Poisson kernel P(x,y)=−∂νyG(x,y)P(x,y) = -\partial_{\nu_y}G(x,y) for y∈∂Ωy\in\partial\Omega gives, for any φ∈C(∂Ω)\varphi\in C(\partial\Omega),

u(x)=∫∂ΩP(x,y)φ(y) dS(y),u(x) = \int_{\partial\Omega} P(x,y)\varphi(y)\,dS(y),

a function harmonic in Ω\Omega (since G(x,⋅)G(x,\cdot) is harmonic in xx away from yy, differentiation under the integral preserves harmonicity) with u→φ(y0)u\to \varphi(y_0) as x→y0∈∂Ωx\to y_0\in\partial\Omega (a standard property of the Poisson kernel: it is an approximate identity concentrating at y0y_0 as x→y0x\to y_0). So u∈Su\in\mathcal S solves the problem: existence holds.

Uniqueness. Suppose u1,u2∈Su_1,u_2\in\mathcal S both solve the problem with the same φ\varphi. Then w=u1−u2w=u_1-u_2 is harmonic in Ω\Omega, continuous on Ω‾\overline\Omega, with w=0w=0 on ∂Ω\partial\Omega. By the Maximum Principle for Parabolic and Elliptic Equations (elliptic case), a harmonic function on a bounded domain, continuous up to the boundary, attains both its max and min on ∂Ω\partial\Omega. Hence max⁡Ω‾w=max⁡∂Ωw=0\max_{\overline\Omega} w = \max_{\partial\Omega} w = 0 and min⁡Ω‾w=min⁡∂Ωw=0\min_{\overline\Omega} w = \min_{\partial\Omega} w = 0, so w≡0w\equiv 0 on Ω‾\overline\Omega, i.e. u1=u2u_1=u_2.

Continuous dependence. Let φ1,φ2∈C(∂Ω)\varphi_1,\varphi_2\in C(\partial\Omega) with solutions u1,u2u_1,u_2 (existing and unique by the above). By linearity of Laplace's equation, w=u1−u2w = u_1-u_2 is harmonic in Ω\Omega with boundary data φ1−φ2\varphi_1-\varphi_2. Apply the maximum principle to ww:

max⁡Ω‾∣w∣=max⁡∂Ω∣w∣≤∥φ1−φ2∥∞.\max_{\overline\Omega}|w| = \max_{\partial\Omega}|w| \le \|\varphi_1-\varphi_2\|_\infty.

(The middle equality: apply the maximum principle to ww and to −w-w separately, each attaining its extremum on ∂Ω\partial\Omega, and combine.) Hence

∥u1−u2∥∞,Ω‾≤∥φ1−φ2∥∞,\|u_1-u_2\|_{\infty,\overline\Omega} \le \|\varphi_1-\varphi_2\|_\infty,

which is Lipschitz continuity of the solution map with constant 11 in the sup norm, i.e. condition 3 with δ=ε\delta=\varepsilon. Existence, uniqueness, and continuous dependence together give well-posedness. ■\blacksquare

∎

Worked examples

Example 1 — quantitative stability for the wave equation with genuinely asymmetric, non-compactly-adapted data. Take c=2c=2, n=1n=1, and two data pairs on R\mathbb R:

g1(x)=e−x2,h1(x)=xe−x2,g2(x)=e−x2+0.01sin⁡(5x)e−x2/4,h2(x)=xe−x2.g_1(x) = e^{-x^2}, \quad h_1(x) = x e^{-x^2}, \qquad g_2(x) = e^{-x^2} + 0.01\sin(5x)e^{-x^2/4},\quad h_2(x) = xe^{-x^2}.

These are not symmetric under any obvious reflection, and g2g_2 perturbs g1g_1 by a bounded, smooth, but non-monotone wiggle. We are not asked to solve the PDE in closed form; the point of the well-posedness theorem is that we don't have to, in order to bound the difference of solutions at any later time. The perturbation is δg=g2−g1=0.01sin⁡(5x)e−x2/4\delta g = g_2-g_1 = 0.01\sin(5x)e^{-x^2/4}, δh=0\delta h = 0. Compute E0(δg,δh)E_0(\delta g,\delta h) using the energy from the theorem (with c=2c=2, so E0=∫(c2(δg′)2+(δh)2)dx=4∫(δg′)2dxE_0 = \int (c^2 (\delta g')^2 + (\delta h)^2)dx = 4\int (\delta g')^2 dx):

δg′(x)=0.01(5cos⁡(5x)e−x2/4−x2sin⁡(5x)e−x2/4).\delta g'(x) = 0.01\Big(5\cos(5x)e^{-x^2/4} - \tfrac{x}{2}\sin(5x)e^{-x^2/4}\Big).

Rather than evaluate the exact Gaussian integral, we bound: ∣δg′(x)∣≤0.01(5+∣x∣2)e−x2/4|\delta g'(x)| \le 0.01\big(5 + \tfrac{|x|}{2}\big)e^{-x^2/4}, so using ∫R(5+∣x∣2)2e−x2/2dx=:M<∞\int_{\mathbb R} (5+\tfrac{|x|}2)^2 e^{-x^2/2}dx =: M < \infty (a finite, explicitly computable constant — numerically M≈252π+lower order≈62.7+…M\approx 25\sqrt{2\pi} + \text{lower order} \approx 62.7 + \dots; for our purposes we only need finiteness), we get

∫(δg′)2 dx≤(0.01)2M≈(0.0001)(65)=0.0065,\int (\delta g')^2\,dx \le (0.01)^2 M \approx (0.0001)(65) = 0.0065,

using the rough bound M≲65M\lesssim 65. Hence E0(δg,δh)≤4(0.0065)=0.026E_0(\delta g,\delta h) \le 4(0.0065) = 0.026, so E01/2≲0.161E_0^{1/2}\lesssim 0.161. By the isometry proved in the theorem, for every t∈[0,T]t\in[0,T] (any TT at all — the estimate is uniform in time, which is the whole content of energy conservation),

E(t)1/2=(∫(c2∣∂xw∣2+wt2) dx)1/2≤0.161,E(t)^{1/2} = \Big(\int (c^2 |\partial_x w|^2 + w_t^2)\,dx\Big)^{1/2} \le 0.161,

where w=u1−u2w = u_1-u_2 is the difference of the two exact solutions. This is a genuine, computable, a priori guarantee — obtained without ever writing down u1u_1 or u2u_2 explicitly (though d'Alembert's formula could produce them) — that a 1%1\%-amplitude high-frequency wiggle in the initial displacement produces a solution difference of energy-norm size at most ≈0.16\approx 0.16, for all time, not just briefly. This is precisely what "continuous dependence" buys a modeler: an error bound that does not degrade as t→∞t\to\infty, in sharp contrast with Example 2 below.

Example 2 — the same style of perturbation for the backward heat equation, showing the bound cannot exist. Consider the backward heat equation on [0,π][0,\pi] with Dirichlet boundary conditions,

ut=−uxx,u(0,t)=u(π,t)=0,u(x,0)=φ(x),u_t = -u_{xx}, \quad u(0,t)=u(\pi,t)=0, \quad u(x,0)=\varphi(x),

(equivalently, v(x,t):=u(x,T−t)v(x,t) := u(x,T-t) solves the ordinary forward heat equation, so this is exactly "run the heat equation backward from time TT to recover data at time 00"). Separation of variables gives, for φ(x)=∑n≥1bnsin⁡(nx)\varphi(x)=\sum_{n\ge1} b_n \sin(nx),

u(x,t)=∑n≥1bnen2tsin⁡(nx).u(x,t) = \sum_{n\ge 1} b_n e^{n^2 t}\sin(nx).

Take two data functions, asymmetric and unrelated in form: φ1(x)=sin⁡(x)+sin⁡(2x)\varphi_1(x) = \sin(x) + \sin(2x) (a genuine two-mode profile, not a pure tone), and φ2(x)=φ1(x)+ϵNsin⁡(Nx)\varphi_2(x) = \varphi_1(x) + \epsilon_N \sin(Nx) for a large integer NN and small ϵN>0\epsilon_N>0, chosen as ϵN=e−N2\epsilon_N = e^{-N^2}. In sup norm, ∥φ1−φ2∥∞=ϵN=e−N2→0\|\varphi_1-\varphi_2\|_\infty = \epsilon_N = e^{-N^2} \to 0 as N→∞N\to\infty: the data converge in C[0,π]C[0,\pi]-norm, arbitrarily fast. But the solution difference at a fixed time t0>0t_0>0, again by the explicit series (each mode evolves independently, so the difference of solutions is exactly the image of the difference of data), is

w(x,t0)=ϵNeN2t0sin⁡(Nx),∥w(⋅,t0)∥∞=ϵNeN2t0=e−N2eN2t0=eN2(t0−1).w(x,t_0) = \epsilon_N e^{N^2 t_0}\sin(Nx), \qquad \|w(\cdot,t_0)\|_\infty = \epsilon_N e^{N^2 t_0} = e^{-N^2}e^{N^2t_0} = e^{N^2(t_0-1)}.

Fix any t0>0t_0>0: if we had instead chosen ϵN=e−N2t0/2\epsilon_N = e^{-N^2 t_0/2} (still →0\to 0), we would get ∥w(⋅,t0)∥∞=eN2t0/2→∞\|w(\cdot,t_0)\|_\infty = e^{N^2 t_0/2}\to \infty as N→∞N\to\infty. So data converging to 00 in sup norm produce solutions at time t0t_0 diverging to ∞\infty. No Lipschitz (or even continuous) bound ∥u1(⋅,t0)−u2(⋅,t0)∥∞≤δ(∥φ1−φ2∥∞)\|u_1(\cdot,t_0)-u_2(\cdot,t_0)\|_\infty \le \delta(\|\varphi_1-\varphi_2\|_\infty) with δ(s)→0\delta(s)\to 0 as s→0s\to0 can hold, because we have just exhibited, for every η>0\eta>0, data within η\eta of each other in sup norm (take NN large enough that e−N2t0/2<ηe^{-N^2t_0/2}<\eta) whose solutions at time t0t_0 differ by an arbitrarily large amount. Continuous dependence fails: this is a fully quantitative, asymmetric-data instance of Hadamard's instability, not merely an abstract existence claim.

CounterexampleDropping continuous dependence: Hadamard's original example

The classical historical example strips the problem down further to show that dropping only the continuity requirement — while existence and uniqueness formally still hold on a restricted data class — produces a genuinely unusable problem. Consider the Cauchy problem for Laplace's equation in the upper half-plane (an elliptic equation given Cauchy data on a line, rather than boundary data on a closed curve — note this already departs from the Dirichlet problem in the theorem above, which used data on all of ∂Ω\partial\Omega):

uxx+uyy=0 in {y>0},u(x,0)=0,uy(x,0)=φn(x):=sin⁡(nx)n,n=1,2,3,…u_{xx}+u_{yy}=0 \text{ in } \{y>0\}, \qquad u(x,0) = 0,\quad u_y(x,0) = \varphi_n(x) := \frac{\sin(nx)}{n}, \quad n=1,2,3,\dots

One directly checks that

un(x,y)=sin⁡(nx)sinh⁡(ny)n2u_n(x,y) = \frac{\sin(nx)\sinh(ny)}{n^2}

solves Δun=0\Delta u_n = 0 (since sin⁡(nx)sinh⁡(ny)\sin(nx)\sinh(ny) is the real part of −i ein(x+iy)-i\,e^{in(x+iy)}, harmonic as a holomorphic function's real part) and matches the data: un(x,0)=0u_n(x,0)=0 and ∂yun(x,0)=sin⁡(nx)n⋅ncosh⁡(0)n\partial_y u_n(x,0) = \frac{\sin(nx)}{n}\cdot\frac{n\cosh(0)}{n} , let us verify precisely:

∂yun(x,y)=sin⁡(nx) ncosh⁡(ny)n2=sin⁡(nx)cosh⁡(ny)n,∂yun(x,0)=sin⁡(nx)n=φn(x).\partial_y u_n(x,y) = \frac{\sin(nx)\, n\cosh(ny)}{n^2} = \frac{\sin(nx)\cosh(ny)}{n}, \qquad \partial_y u_n(x,0) = \frac{\sin(nx)}{n} = \varphi_n(x).

Good — this matches the prescribed Neumann data exactly. As n→∞n\to\infty,

∥φn∥∞=1n→0,∥un(⋅,0)∥∞=0 for all n,\|\varphi_n\|_\infty = \frac1n \to 0, \qquad \|u_n(\cdot,0)\|_\infty = 0 \text{ for all } n,

so the data (Dirichlet part 00, Neumann part φn\varphi_n) converge to the zero data in every reasonable norm. But for any fixed y0>0y_0>0,

∥un(⋅,y0)∥∞=sinh⁡(ny0)n2⟶∞(n→∞),\|u_n(\cdot,y_0)\|_\infty = \frac{\sinh(ny_0)}{n^2} \longrightarrow \infty \quad (n\to\infty),

since sinh⁡(ny0)\sinh(ny_0) grows exponentially in nn while the denominator n2n^2 is merely polynomial. So the solutions do not converge to the solution (≡0\equiv 0) corresponding to zero data; in fact they blow up. Existence and uniqueness hold for each fixed nn (the unu_n above are literally exhibited, and uniqueness for the Cauchy problem for elliptic equations holds by Holmgren's uniqueness theorem, a fact beyond our present scope), but continuous dependence — the third Hadamard condition — fails completely. The lesson: dropping continuous dependence from the definition of well-posedness (equivalently: asking only for existence and uniqueness) admits problems whose solution, though "correctly defined" mode by mode, is useless for approximation, simulation, or physical prediction, because no amount of improved measurement precision in the data controls the error in the solution. This is exactly why Hadamard insisted on all three conditions together, and exactly why the Cauchy problem for Laplace's equation (elliptic, data on a line rather than a closed boundary) is the standard textbook example of an ill-posed problem, in contrast to the Dirichlet problem proved well-posed above.

Scope, edge cases & common traps

SituationWhat can go wrongCorrect statement
Backward heat equationExistence fails for generic data; when it exists, continuity failsIll-posed in CC or L2L^2; well-posed only on a highly restricted analytic data class with weighted norms
Cauchy problem for elliptic PDE (data on a curve, not full boundary)Continuity fails even though existence/uniqueness may hold on special dataIll-posed (Hadamard's example above); contrast with the boundary-value Dirichlet problem, which is well-posed
Wave equation, non-compactly-supported unbounded dataEnergy E0E_0 may be infinite; finite propagation speed argument for uniqueness (Step 2) still applies locallyRestrict to finite-energy or locally finite-energy data; state norm explicitly
Changing the topology on D\mathcal D or S\mathcal SA problem well-posed in one norm can be ill-posed in another (e.g. sup norm vs. HsH^s with large ss for evolution equations with derivative loss)Always name the topologies; "well-posed" is meaningless without them
Nonlinear PDEsExistence/uniqueness may only be local in time; "well-posed" often means local well-posedness plus continuous dependence on that time intervalState the maximal time of existence as part of the solution space

The mathematician behind it

Origins · 1 minJacques Salomon Hadamard and how this was discovered
JS
Jacques Salomon Hadamard
1865–1963 · Paris, France

Hadamard was one of the most influential analysts of his generation, working across analysis, number theory (he gave the first proof of the Prime Number Theorem, independently of de la Vallée Poussin, in 1896), and the theory of partial differential equations. His 1902 lectures on the Cauchy problem, later expanded into Lectures on Cauchy's Problem in Linear Partial Differential Equations (1923), introduced the tripartite existence–uniqueness–continuous-dependence framework that now bears his name, motivated directly by his study of hyperbolic versus elliptic Cauchy problems and by his construction of the backward-heat and Cauchy-for-Laplace instabilities used above.

History
- 1865 — Hadamard born in Versailles, France.
- 1892 — awarded the Grand Prix des Sciences Mathématiques for work related to the Riemann zeta function's zero-free region.
- 1896 — proves the Prime Number Theorem.
- 1902–1923 — develops and publishes the well-posedness framework (existence, uniqueness, continuous dependence) for the Cauchy problem, including the elliptic Cauchy-data instability example.
- 1963 — Hadamard dies in Paris, having also shaped functional analysis via the notion of well-posed operator problems later generalized by Tikhonov and others in the theory of ill-posed and inverse problems.
Is well-posedness the same as "the equation has a formula for its solution"?

No. Well-posedness is compatible with having no closed-form solution at all — the wave equation's well-posedness proof above used an energy identity, not d'Alembert's formula, precisely so the argument generalizes to dimensions and equations without simple formulas. Conversely, having an explicit formula does not guarantee well-posedness: Hadamard's elliptic Cauchy-data example above has an entirely explicit formula, un(x,y)=sin⁡(nx)sinh⁡(ny)/n2u_n(x,y)=\sin(nx)\sinh(ny)/n^2, and is still ill-posed, because the family of solutions across the data does not depend continuously on the data. Well-posedness is a statement about a map from data to solutions, not about whether any single instance can be written down.

Why do we need to fix a norm before saying a problem is well-posed?

Continuity is a topological notion: "small perturbation in data implies small change in solution" is meaningless until "small" is measured by specific norms on specific spaces. The same evolution equation can be well-posed when measured in an energy norm (as the wave equation is, exactly, by the isometry proof above) but only conditionally stable, or even unstable, if measured in a norm involving more derivatives than the equation naturally controls. This is why serious statements of well-posedness always specify the data space D\mathcal D and solution space S\mathcal S, exactly as in Hadamard's definition.

Does uniqueness of a solution imply continuous dependence?

No, and Hadamard's elliptic-Cauchy-data example is a clean illustration: uniqueness for that problem holds (via Holmgren's theorem, applicable to elliptic Cauchy problems with analytic coefficients), yet continuous dependence fails outright, as shown by the sinh⁡(ny)/n2\sinh(ny)/n^2 blow-up. Uniqueness only says the solution map is well-defined (single-valued); it says nothing about whether that map is continuous. These are logically independent properties, which is exactly why Hadamard listed them as three separate conditions rather than folding continuity into uniqueness.

Is the backward heat equation always ill-posed, with no exceptions?

It is ill-posed on the natural data classes C(Ω‾)C(\overline\Omega) or L2(Ω)L^2(\Omega): as shown above, arbitrarily small sup-norm perturbations produce unboundedly large solution differences at any fixed later time. However, if one restricts the admissible data to a much smaller class — for instance, boundary values of functions analytic in a strip, or data with Fourier coefficients decaying faster than any e−cn2e^{-cn^2} — the backward problem can be shown to be well-posed on that restricted class, with a correspondingly weak (logarithmic-type) continuity estimate. This is the basis of "conditional well-posedness" and Tikhonov regularization for inverse heat problems: well-posedness is recovered by shrinking D\mathcal D, not by changing the equation.

How does finite propagation speed enter the wave equation's well-posedness proof?

It enters in two places. First, it guarantees that if the initial data are compactly supported, the solution at any later time remains compactly supported (in an enlarged but still bounded region), which is exactly what licenses integration by parts with no boundary terms in the energy identity. Second, referenced via Domains of Dependence and Influence, it is the qualitative reason the wave equation is "causal": the value of uu at (x0,t0)(x_0,t_0) depends only on data in [x0−ct0,x0+ct0][x_0-ct_0,x_0+ct_0], so perturbing data far away cannot instantaneously affect nearby solution values — a stronger, pointwise version of the continuous-dependence conclusion.

Can a nonlinear PDE be well-posed?

Yes, but usually only locally in time, and the definition must be adapted: existence and uniqueness on a maximal interval [0,T∗)[0,T^*) where T∗T^* may depend on the size of the data, plus continuous dependence of the solution on the data on any compact subinterval [0,T]⊂[0,T∗)[0,T]\subset[0,T^*). Classical examples include the Cauchy problem for the incompressible Navier–Stokes equations in three dimensions (locally well-posed, globally open) and for many quasilinear hyperbolic systems (locally well-posed, with possible finite-time breakdown via shock formation). The energy-estimate technique used for the linear wave equation above is the direct ancestor of the local-existence arguments used in these nonlinear settings.

What role do Green's functions play in checking well-posedness?

Green's functions convert an abstract PDE problem into an explicit integral representation of the solution in terms of the data, as in the Poisson-kernel formula used for the Dirichlet problem above. This is useful for well-posedness in two ways: it directly supplies existence (the integral is a solution, checked by differentiating under the integral sign), and it often makes continuous dependence estimates concrete, because bounding ∥u1−u2∥\|u_1-u_2\| reduces to bounding an integral operator's norm acting on φ1−φ2\varphi_1-\varphi_2. When such kernels are singular or unbounded in a way that grows with frequency (as an inverse-heat "kernel" formally would), that same representation exposes the instability directly, rather than the pointwise bounds having to be found by other means.

Before this