Examples are the working laboratory of analysis and partial differential equations. Theorems in PDE rarely say that every solution is smooth or that every coefficient behaves nicely. They say: under assumptions $A$, a phenomenon $P$ happens. To understand what matters, build examples that satisfy some assumptions but not others, and watch which conclusions survive.
A productive way to build PDE examples is to treat them as constrained design problems:
Smart TV Pick55-inch 4K Fire TVINSIGNIA 55-inch Class F50 Series LED 4K UHD Smart Fire TV
INSIGNIA 55-inch Class F50 Series LED 4K UHD Smart Fire TV
A general-audience television pick for entertainment pages, living-room guides, streaming roundups, and practical smart-TV recommendations.
- 55-inch 4K UHD display
- HDR10 support
- Built-in Fire TV platform
- Alexa voice remote
- HDMI eARC and DTS Virtual:X support
Why it stands out
- General-audience television recommendation
- Easy fit for streaming and living-room pages
- Combines 4K TV and smart platform in one pick
Things to know
- TV pricing and stock can change often
- Platform preferences vary by buyer
- Choose an equation class that exposes the feature you want.
- Use scaling to predict the natural regularity threshold.
- Pick data or coefficients that sit exactly at the edge of that threshold.
- Compute a quantity that remains controlled, then check which stronger quantities fail.
The recipes below are written so they can be applied repeatedly. Each recipe ends with a concrete example that can be checked by direct computation.
Recipe A: Start from a conserved or monotone quantity
If an equation has an energy identity, begin there. The energy dictates the natural function space, and the natural space dictates what kind of examples are meaningful.
Worked example: weak solutions for Poisson with rough data
Let $\Omega\subset\mathbb{R}^d$ be bounded. For the Dirichlet problem
the energy estimate suggests $u\in H^1_0(\Omega)$ if $f\in H^{-1}(\Omega)$. To build a basic example, choose $f\in L^2(\Omega)$. Lax–Milgram gives a unique $u\in H^1_0(\Omega)$ with
This $u$ is a weak solution even when $f$ has no pointwise meaning beyond $L^2$.
Now choose $f$ with a controlled $L^2$ norm but with sharp local oscillation. For instance, in a ball $B\subset\Omega$, set
Then $\|f_k\|_{L^2}$ is bounded uniformly in $k$, so the solutions $u_k$ are bounded in $H^1_0(\Omega)$. The recipe produces a family where the energy stays bounded while higher derivatives do not remain uniformly controlled. The sequence is perfect for testing compactness statements and for seeing why $H^1$ is the right level to expect stability.
The point is not the trigonometric function. The point is that the energy estimate makes it easy to generate bounded families that stress any stronger claim.
Recipe B: Use scaling to target a regularity threshold
Many PDE have a scaling symmetry. Scaling predicts which norms are critical, subcritical, or supercritical. Examples built at the critical scale are often the ones that separate true theorems from false generalizations.
Worked example: the heat equation and smoothing from rough initial data
On $\mathbb{R}^d$, the heat equation
has the scaling $u_\lambda(t,x)=u(\lambda^2 t,\lambda x)$. Under this scaling, $L^1$ norm is preserved up \to a factor, while $L^2$ behaves differently depending on $d$. This suggests that $L^1$ initial data is natural for constructing solutions via convolution with the heat kernel, while $L^2$ is the natural energy space on bounded domains.
A practical family of examples is the approximate identity sequence
where $\phi\ge 0$ is smooth with $\int \phi=1$. Then $u_0^{(k)}\rightharpoonup \delta_0$ as measures, while $\|u_0^{(k)}\|_{L^1}=1$ and $\|u_0^{(k)}\|_{L^2}\to\infty$. The corresponding solutions are explicit:
where $G_t$ is the heat kernel and $(G_t)_k$ is its rescaling. For any fixed $t>0$, $u^{(k)}(t,\cdot)$ becomes smooth and bounded uniformly in $k$ on compact sets away from $t=0$, while at $t=0$ the family has no $L^2$ control.
This example clarifies a frequent confusion: smoothing for $t>0$ is real, but it does not mean the initial trace lies in a nicer space than it actually does. Scaling makes the distinction unavoidable.
Recipe C: Build a boundary layer to test dependence on boundary conditions
When a PDE involves a small parameter or a singular perturbation, solutions can change rapidly near boundaries. Boundary layer examples test whether an estimate genuinely uses the boundary condition or only pretends \to.
Worked example: a simple boundary layer for a singularly perturbed ODE model
Consider the one-dimensional model
Solve explicitly: $u'(x)=C e^{x/\varepsilon}$, so $u(x)=A + C\varepsilon e^{x/\varepsilon}$. Using the boundary conditions gives
As $\varepsilon\to 0$, $u(x)\to 0$ for every fixed $x<1$, while $u(1)=1$. The change happens in a thin region near $x=1$ with width comparable \to $\varepsilon$.
Even though this is an ODE, it captures what boundary layers do in convection-diffusion PDE: interior estimates can be misleading if they ignore boundary structure, and uniform estimates require norms that see the layer.
Recipe D: Use a fundamental solution to design singularities
Fundamental solutions are ready-made singular functions. They provide examples that sit exactly at the borderline of integrability or regularity.
Worked example: Laplace fundamental solution and borderline integrability
In $\mathbb{R}^d$ with $d\ge 3$, the fundamental solution of Laplace is
Compute its gradient:
Near the origin, $|\nabla \Phi|^2\sim |x|^{2-2d}$. The integral over a ball of radius $r$ behaves like
This diverges as $r\to 0$. So $\Phi\notin H^1_{\text{loc}}$ near the origin. Yet $\Phi\in L^p_{\text{loc}}$ for many $p$, and $\Phi$ is harmonic away from the origin.
This single computation generates many sharp examples. Any claim that a harmonic function with a point singularity must lie in $H^1$ is false. Any theorem that assumes $H^1$ can be tested by cutting off $\Phi$ near the origin and watching what breaks.
Recipe E: Use compactness failure to build counterexamples
Compactness is the invisible step in many existence proofs. Examples that fail compactness show why weak convergence is not enough for nonlinearities.
Worked example: concentration in Sobolev embedding
In $\mathbb{R}^d$ with $d\ge 3$, the Sobolev embedding $H^1(\mathbb{R}^d)\hookrightarrow L^{2d/(d-2)}(\mathbb{R}^d)$ is continuous but not compact. A classical concentrating sequence is
for a fixed $u\in C_c^\infty$. Then $\|\nabla u_k\|_{L^2}=\|\nabla u\|_{L^2}$, so the sequence is bounded in $H^1$, and $u_k\rightharpoonup 0$ weakly. But $\|u_k\|_{L^{2d/(d-2)}}=\|u\|_{L^{2d/(d-2)}}$ does not go to zero.
This kind of example shows why nonlinear terms at the critical exponent require additional structure, such as concentration-compactness or profile decompositions, rather than naive weak convergence.
A short checklist for building PDE examples
When building examples, a few checkpoints prevent wasted effort.
- Decide which norm you want to control and which norm you want to fail.
- Use scaling or dimension counting to predict where the borderline lies.
- Choose data that are smooth away from one designed defect: a singularity, an oscillation, a thin layer, or a concentration bubble.
- Verify the claims by explicit computation in polar coordinates or by a direct inequality estimate.
Examples are not secondary to the theory. They are how the theory stays honest, how assumptions become visible, and how a proof reveals its real dependence on the hypotheses.
Recipe F: Put roughness into coefficients, not into the forcing
Many elliptic and parabolic results depend more sensitively on coefficient regularity than on the smoothness of the \right-hand side. A good example family keeps $f$ simple and moves all complexity into the coefficient field.
Worked example: a divergence-form operator with a sharp interface
Consider in two dimensions a divergence-form elliptic equation
with $u$ prescribed on $\partial\Omega$. Let $a(x)$ be piecewise constant:
with $a_1\neq a_2$. Solutions are harmonic on each half, but the transmission condition across the interface forces continuity of $u$ and of the normal flux $a\,\partial_{x_2}u$.
Choose boundary data so that the solution is affine in each half-plane:
with $\alpha_1$ and $\alpha_2$ chosen so that $a_1\alpha_1=a_2\alpha_2$. This $u$ is a weak solution. Its gradient jumps across the interface unless $a_1=a_2$. So even with perfectly smooth boundary data away from corners, the best regularity one can demand in general is limited by coefficient discontinuities. The example is a template: \to test whether a theorem truly needs Hölder continuity of coefficients, replace smooth coefficients by a sharp interface and see which regularity conclusions fail.
Recipe G: Use characteristics to build finite-time singular structure in first-order PDE
First-order nonlinear PDE often generate steep gradients from smooth initial data. The cleanest construction uses characteristics, which turn the PDE into ODE along curves.
Worked example: inviscid Burgers and gradient blow-up
The inviscid Burgers equation on the line is
Along a characteristic $x(t)$ satisfying $x'(t)=u(t,x(t))$, the quantity $u$ is constant:
So $u(t,x(t))=u_0(\xi)$ if $x(0)=\xi$. The characteristic then solves $x(t)=\xi + t u_0(\xi)$. The map $\xi\mapsto x(t)$ ceases to be one-\to-one when
for some $\xi$. If $u_0'$ is negative somewhere, the first time this happens is
Before $T_*$, there is a classical solution. At $T_*$, the slope becomes unbounded even though $u$ itself remains bounded. This is a sharp example that separates uniform bounds from derivative bounds and shows why weak and entropy formulations are necessary after $T_*$.
This recipe is broadly useful: if a theorem claims global smoothness for a first-order nonlinear PDE without extra structure, characteristics provide a fast way to test the claim.
Recipe H: Impose a self-similar ansatz to expose scaling laws
Self-similar constructions are controlled experiments: you reduce a PDE to an ODE by forcing the solution to respect the scaling symmetry.
Worked example: a self-similar profile for the heat equation
On $\mathbb{R}^d$, seek a solution of the heat equation in the form
Set $y=x/\sqrt{t}$. A direct computation shows that the PDE reduces to the ODE in $y$:
The Gaussian $F(y)=C e^{-|y|^2/4}$ solves this equation, producing the fundamental solution. The point of the recipe is not the Gaussian itself. It is that the ansatz translates scaling into a solvable constraint and makes it easy to generate families that probe borderline behavior, such as large-time decay rates and the minimal integrability needed to define a solution at $t=0$.
Self-similarity is also the entry point to constructing special solutions in nonlinear PDE, where it often reveals which exponents are critical for global bounds.
Books by Drew Higgins
Prophecy and Its Meaning for Today
New Testament Prophecies and Their Meaning for Today
A focused study of New Testament prophecy and why it still matters for believers now.
Leave a Reply