Exploration of Finite Time Singularities of the 3D Navier Stokes Equations over a Periodic Domain T3

Terry Moschandreou

doi:10.34257/GJSFRF256061

Conclusion
Appendix A
The General Ratio $\mathcal{R}_n$
LambertW Composition with Terminal $(1+W_j)$ Factors
Derivative of the LambertW function
First-level derivatives
Recursive derivatives
Proof of the Compact Product Formula for $\partial_z W_n$ and $\partial_t W_n$
Base case $n=1$
Recursive derivative relation
Induction hypothesis
Inductive step
Conclusion
Final Result
Mixed derivative
Derivation of the Mixed Derivative $\partial_{tz} W_n$
Apply the quotient rule
Substitute known derivative
Differentiate the product $P_n$
Substitute into mixed derivative
Telescoping structure
Final simplification
Final Result
Ratio of derivatives
Integrand for the velocity potential
Conclusion
Appendix B
1. Step 0: Setup
2. Integration by parts formula
3. First integration by parts
4. Repeat $n-1$ times
5. Final closed form after $n-1$ steps
6. Examples for small $n$
Proof of the Derivative Structure Used here
Proof of the Derivative Structure
1. Goal
2. Write the function as a Product
3. Differentiate using the Product Rule
4. Compute the Derivative of the Product
5. Differentiate the Square-Root Term
6. Substitute Both Derivatives
7. Insert the Factor $(1+W_n)^{3/2}$
8. Identify $H_0(z)$
9. Immediate Consequence
10. Structural Remark
Proof of and Repeated Integration by Parts
First Integration by Parts
Structure of the New Integrand
Repetition of the Procedure
Inductive Reduction
Termination After $(n-1)$ Steps
Final Result
Origin of the Division by $H_k(z)$ in the Final Expression
Integration by Parts Step
Key Observation
Recursive Structure
Final Expression
Conclusion
Full Proofs of Steps 2 and 3
Proof of and Step 3
Product Rule Differentiation
1. Theorem (Product Rule)
2. Application
Derivative of the Power Term
1. Chain Rule
Derivative of the Product $P(z)$
1. Finite Product Differentiation Rule
2. Proof
  1. Base Case: $m=2$
  2. Inductive Step
3. Apply to Our Product
Factorized Form
Appendix C
Multiplication of $\nu = \epsilon_3$ by the second derivative of $b_3$ in the viscosity term
Second Derivative and Scaled Limit for a Lambert $W$ Expression
Step 1 — Local Variable Near the Singular Point
Step 2 — Expand Numerator and Denominator
Step 3 — Limit as $z\to t+\pi/2$
Step 4 — Asymptotic Behavior of Lambert $W$
Step 5 — Behavior of Derivatives
Step 6 — Multiply by $\varepsilon_3$
Step 7 — Final Limit
Interpretation
Appendix D
Regularity of the Product $\displaystyle b(t)=u_x(t)\,u_y(t)$ with Lambert $W$ Structure
1. 1. Definition of the Velocity Components
2. 2. Product Structure
3. 3. Domain and Branch Point
4. 4. Local Expansion Near the Branch Point
5. 5. Derivatives of the Velocity Components
6. 6. Derivative of the Product
7. 7. Asymptotic behavior of the Product Derivative
8. 8. Regularity Conclusion
9. 9. Structural Interpretation
Appendix E
Appendix F
Theory of characteristics for a General Nonlinear Transport Equation
Appendix G
Local Well-Posedness from Smooth Initial data
1. Reformulation as a Quasilinear Transport Equation
2. Initial data
3. Characteristic system
4. Local Existence of characteristics
5. Smooth Dependence on Initial data
6. Invertibility of the Characteristic Map
7. Construction of the Eulerian solution
8. Uniqueness
9. Local Well-Posedness Theorem
10. Limitations of the Result
11. Relation to the Lambert $W$ Representation
Final Statement
Why the Characteristic System $(Z(t),B(t))$ is Introduced
1. General First-Order Quasilinear PDE
2. Definition of a Characteristic Curve
3. Total Derivative Along a Moving Curve
4. Reduction of the pde to an ode
5. Application to the Present Equation
6. Characteristic system
7. Initial conditions
8. Reconstruction of the pde solution
9. Geometric Interpretation
10. Relation to Shock and Singularity Formation
Final Statement

Conclusion

This work has developed a structured analytical framework for the study of the three-dimensional incompressible Navier–Stokes equations based on recursive functional compositions, algebraic transformations of nonlinear terms, and explicit characteristic solutions. The analysis demonstrates that a large class of nonlinear transport structures arising within the Navier–Stokes equations can be reduced to tractable scalar evolution equations whose solutions admit closed-form representations in terms of the Lambert $W$ function.

A central component of the framework is the recursive construction of composed Lambert $W$ functions and the derivation of compact formulas for their higher order derivatives. These derivatives factor into finite products of algebraic terms of the form $(1+W_j)$ , revealing an exact multiplicative structure that governs the growth and regularity of the resulting solutions. The recursive nature of these expressions provides a natural hierarchy of differential relations in which higher-order behavior is determined by explicit algebraic factors rather than uncontrolled nonlinear interactions.

Through successive row-operation transformations applied to the Navier-Stokes equations, a sequence of derived vector fields was constructed that preserves the differential structure of the original system. These transformations expose hidden symmetries in the nonlinear inertial terms and produce a hierarchy of transport equations that remain closed under differentiation and multiplication. The resulting system provides a transparent representation of nonlinear interactions among velocity components and clarifies the role of mixed-product terms in the evolution of gradients.

A key structural identity established in this analysis is the exact equality between nonlinear higher gradient production and viscous diffusion mechanisms. In the transformed variables, the nonlinear amplification of gradients is balanced pointwise by the smoothing action of the Laplacian operator, yielding a net divergence field. On periodic domains, this divergence structure integrates to zero, providing a global constraint on the evolution of gradient energy and demonstrating that the dominant nonlinear and diffusive mechanisms remain in precise algebraic balance.

Explicit solutions of the reduced scalar transport equations were obtained using the method of characteristics and expressed in closed form through the Lambert $W$ function. These solutions reveal a precise mathematical mechanism governing loss of regularity. In particular, the analysis shows that singular behavior is associated with the branch structure of the Lambert $W$ function rather than unbounded growth of the solution itself at time $t1<t2$ . When the argument of the Lambert function approaches its critical branch value, the derivative of the solution diverges while the solution amplitude remains finite. This establishes a clear distinction between boundedness of the velocity field and smoothness of its spatial derivatives.

The introduction of shifted trigonometric denominator structures provides an explicit criterion for preventing finite-time singularities. Because the sine function is uniformly bounded, the addition of a sufficiently large positive shift ensures that denominators remain strictly nonzero for all finite times. Under this condition, the coefficients of nonlinear quadratic terms remain smooth and bounded, guaranteeing global regularity of the corresponding velocity component. Conversely, when the effective shift parameter approaches the critical threshold at which the denominator vanishes, gradient amplification may occur and lead to loss of differentiability. It is of prime importance that I have shown that the vector $\bm{b} \in C^n$ for $n=0,1$ both spatially and in time. So any terms in the equation will be regular. This has been possible by transforming the original Navier Stokes equations using the matrices $M(\bm{u})$ and $M(\bm{b})$ . However there is a $||\bm{b}||^2$ term which leads to blowup in derivatives of higher derivatives as shown at $t=t1$ and for $t2(t2>t1)$ it was proven analytically and numerically that there is finite time amplitude blowup in the solution $b_3$ .

Taken together, these results establish a coherent analytical picture in which nonlinear amplification, diffusive smoothing, and algebraic functional structure are linked through explicit formulas. The recursive Lambert $W$ hierarchy provides a tractable representation of nonlinear transport dynamics, while the transformed Navier-Stokes system reveals the exact balance governing gradient evolution. The framework therefore offers a mathematically transparent mechanism for analyzing the formation of non-smooth behavior in nonlinear fluid equations.

Future work may extend this approach to broader classes of nonlinear partial differential equations, investigate stability properties of the constructed solutions, and explore the role of recursive functional structures in turbulence modeling and multiscale flow dynamics. In particular, further investigation of branch-point dynamics and gradient growth mechanisms may provide deeper insight into the fundamental relationship between bounded solutions and loss of regularity in high-dimensional nonlinear systems.

Appendix A

The General Ratio $\mathcal{R}_n$

For an $n^{th}$ order composition $W_n$ , the dimensionalized viscosity related ratio of the temporal and mixed partial derivatives is defined as:

The integrand for the velocity potential $V_z$ is the inverse square root:

LambertW Composition with Terminal $(1+W_j)$ Factors

Let

Define the recursive LambertW composition

Note that the quantities $(1+W_j)$ appear only as multiplicative factors in the final expressions, not in the recursion itself.

Derivative of the LambertW function

For any argument $x$ ,

First-level derivatives

Let

Then

Since $W_1=\operatorname{LambertW}(x)$ and $x=W_1 e^{W_1}$ , we obtain

Recursive derivatives

For $j\ge2$ ,

so by the chain rule,

Iterating this recursion yields the compact products

Proof of the Compact Product Formula for $\partial_z W_n$ and $\partial_t W_n$

Let

Define the recursive LambertW composition

We prove the formula for $\partial_z W_n$ . The temporal derivative follows identically with $\omega$ replacing $\omega_2$ .

Base case $n=1$

Let

The derivative of the LambertW function is

Using the chain rule,

Since

and

we obtain

Because

this simplifies to

which matches the claimed formula for $n=1$ :

Thus the base case holds.

Recursive derivative relation

For $j\ge2$ ,

Applying the chain rule:

This is the fundamental recursion.

Induction hypothesis

Assume for some $k\ge1$ :

Inductive step

Using the recursion:

Substitute the induction hypothesis:

Cancel $W_k$ :

Combine denominators:

Thus the formula holds for $k+1$ .

Conclusion

By mathematical induction,

Similarly, since

the identical derivation yields

Final Result

The recursive differentiation of the $n^{\rm th}$ LambertW composition produces the compact product formulas:

which proves the statement:

Mixed derivative

Differentiate $\partial_t W_n$ with respect to $z$ :

Carrying out the derivative and simplifying the telescoping products gives

Derivation of the Mixed Derivative $\partial_{tz} W_n$

Let

Define recursively

Assume the proven compact derivative:

We compute the mixed derivative

where

Apply the quotient rule

Substitute known derivative

From the induction result:

Therefore

Differentiate the product $P_n$

Using the logarithmic derivative:

Substitute the derivative formula for each $W_k$ :

where

Thus

Hence

Substitute into mixed derivative

Return to

Substitute $\partial_z P_n$ :

Factor common terms:

Telescoping structure

Note that

Therefore the sum becomes

Now observe the identity

This is proved by expanding the product sequentially.

Therefore

Final simplification

Substitute this result:

Thus

Final Result

The mixed derivative of the $n^{\rm th}$ LambertW composition satisfies

and this result follows rigorously from the quotient rule, the recursive derivative identities, and the telescoping product identity.

Ratio of derivatives

Define

Substituting the expressions above:

Simplifying powers gives

Integrand for the velocity potential

The inverse square root is

Conclusion

For the $n^{\rm th}$ LambertW composition with

and

the ratio of derivatives and its inverse square root take the exact algebraic forms:

Appendix B

Step 0: Setup

Define

Then, using the derivative structure, we have

Hence

so the integral becomes

Integration by parts formula

Using

we have

Notice that $H_0'$ involves only derivatives of $W_j$ , which have the same structure as the original integrand.

First integration by parts

Define

Then

The integrand

Hence the product has been reduced by 1 factor.

Repeat $n-1$ times

At each step $k$ , define

After $k$ integrations by parts, the integral becomes a sum of algebraic terms

Final closed form after $n-1$ steps

After $n-1$ steps, all factors $(1+W_1), \dots, (1+W_{n-1})$ have been reduced, leaving only derivatives $W_j'$ and $W_n$ . The integral collapses completely, giving the final closed form

Here we adopt the convention $\prod_{j=1}^{0} W_j' = 1$ . All terms are fully algebraic in $W_j$ and $W_n$ , and no integral signs remain.

Examples for small $n$

$n=2$ :

$I_{2} = 2 [\frac{\sqrt{1 + W_{2}}}{W_{2}^{'}} (1 + W_{1}) + \frac{\sqrt{1 + W_{2}}}{W_{2}^{'}} W_{1}^{'}] .$
$n=3$ :

$I_{3} = 2 \frac{\sqrt{1 + W_{3}}}{W_{3}^{'}} [(1 + W_{1}) (1 + W_{2}) + W_{1}^{'} (1 + W_{2}) + W_{1}^{'} W_{2}^{'}] .$

Clearly, the pattern generalizes to arbitrary $n$ .

Proof of the Derivative Structure Used here

Proof of the Derivative Structure

Goal

Given

we prove that

where

Write the function as a Product

Define

Then

Differentiate using the Product Rule

Compute the Derivative of the Product

Using the logarithmic derivative identity,

we obtain

Differentiate the Square-Root Term

Substitute Both Derivatives

Factor the common term $P(z) W_n^{-1/2}$ :

Insert the Factor $(1+W_n)^{3/2}$

Multiply and divide by the same quantity:

Thus

Identify $H_0(z)$

By definition,

Therefore we obtain the exact identity

Immediate Consequence

Dividing both sides by $H_0(z)$ yields

Hence the integral transformation follows directly:

Structural Remark

This identity holds because the logarithmic derivative converts the product into a sum, allowing the derivative to factor exactly into the required integrand structure. The inserted factor $(1+W_n)^{3/2}$ restores the denominator $\sqrt{W_n(1+W_n)^3}$ , enabling repeated integration by parts to reduce the product sequentially.

Proof of and Repeated Integration by Parts

We consider the integral

Define

and suppose we have already established the identity

where

Therefore

First Integration by Parts

We use the integration-by-parts identity

Let

Then

Therefore

This is the exact first integration-by-parts step.

Structure of the New Integrand

Recall

The derivative $H_0'(z)$ is a linear combination of derivatives of the terms

Thus every term in $H_0'(z)$ contains a factor of the form

Therefore the new integrand has the structure

Hence one factor $(1+W_{n-1})$ has effectively been replaced by its derivative.

This establishes that the number of multiplicative factors is reduced by one.

Repetition of the Procedure

We now define recursively, for each integer $k$ with

Similarly define

Then we have the identity

Therefore the remaining integral at step $k$ becomes

Applying integration by parts again gives

Inductive Reduction

At each step:

One factor $(1+W_j)$ disappears from the product
A derivative factor $W_j'$ appears
The structure of the integral remains the same

Thus after $k$ repetitions, the expression becomes a finite sum of terms of the form

Termination After $(n-1)$ Steps

When

the product is empty, and by convention

Therefore

No further product factors remain.
Thus the repeated integration-by-parts process terminates after exactly $(n-1)$ steps.

Final Result

The integral becomes a finite algebraic sum:

which contains no remaining integrals.

Origin of the Division by $H_k(z)$ in the Final Expression

We start from the identity

which implies

Therefore the remaining integral at step $k$ is

Integration by Parts Step

We use the standard formula

Choose

Then

and

Substituting into the integration-by-parts formula gives

Key Observation

The term

is the boundary term produced by integration by parts.

Thus the division by $H_k(z)$ arises directly from the choice

It is not introduced artificially.

Recursive Structure

At the next step, the same structure holds:

Therefore each integration-by-parts step contributes one algebraic term of the form

Final Expression

After $(n-1)$ repetitions, the integral becomes a finite sum of such boundary terms:

where

and

Conclusion

The division by $H_k(z)$ appears naturally as the boundary term from each integration-by-parts step when choosing

Full Proofs of Steps 2 and 3

Proof of and Step 3

We consider the function

where

We now prove rigorously:

The derivative of $F_0(z)$ using the product rule.
The explicit derivative formula for the product $P(z)$ .

Product Rule Differentiation

Theorem (Product Rule)

Let $f(z)$ and $g(z)$ be differentiable functions. Then

Application

Let

Then by the product rule,

This completes rigorously.

Derivative of the Power Term

We now compute

Chain Rule

Let $u(z)=W_n(z)$ . Then

Therefore

Derivative of the Product $P(z)$

We now compute the derivative of

Finite Product Differentiation Rule

Let

where each $f_j(z)$ is differentiable.

Then the derivative is

Proof

We prove by induction.

Base Case: $m=2$

Let

Then by the product rule,

which matches the formula.

Inductive Step

Assume the formula holds for $m-1$ functions. Write

Define

Then

Differentiate using the product rule:

By the induction hypothesis,

Substitute into the expression:

This becomes

Thus the formula holds for $m$ , completing the proof.

Apply to Our Product

Let

Then

Therefore

Factorized Form

We now factor out the full product

Observe that

Therefore

Hence

which is the exact identity used in the derivative structure.

This expression shows how the gradients of the velocity components interact through the nonlinear vector field $\vec{b}$ , which is common in analyzing vortex stretching and enstrophy production.

Appendix C

Multiplication of $\nu = \epsilon_3$ by the second derivative of $b_3$ in the viscosity term

Second Derivative and Scaled Limit for a Lambert $W$ Expression

Let

We compute:

The second derivative with respect to $z$ ,
The limit as $z\to t+\tfrac{\pi}{2}$ ,
Multiply by $\varepsilon_3$ and take $\varepsilon_3\to0$ .

Step 1 — Local Variable Near the Singular Point

Let

Then the evaluation point

Use the Taylor expansion:

Step 2 — Expand Numerator and Denominator

Denominator:

Numerator:

Therefore

Hence the Lambert $W$ argument is

Step 3 — Limit as $z\to t+\pi/2$

Set $\delta=0$ :

Thus

Step 4 — Asymptotic Behavior of Lambert $W$

For $x\to-\infty$ ,

Apply to

Then

Therefore

Hence

Step 5 — Behavior of Derivatives

Near $\delta=0$ :

Since

we obtain the scaling:

Therefore at the evaluation point:

Differentiating again gives

Hence

for some finite constant $C$ .

Step 6 — Multiply by $\varepsilon_3$

Step 7 — Final Limit

Since

we obtain

Interpretation

The second derivative diverges like

$\frac{1}{ε_{3} \ln^{2} (1 / ε_{3})} .$
After multiplying by $\varepsilon_3$ , the logarithmic denominator dominates.
Therefore the scaled curvature vanishes in the limit:

$ε_{3} b_{z z} ⟶ 0.$

Appendix D

Regularity of the Product $\displaystyle b(t)=u_x(t)\,u_y(t)$ with Lambert $W$ Structure

We analyze the regularity of the time derivative of the product

when each component is defined in terms of the Lambert $W$ function.

1. Definition of the Velocity Components

Let

where $W$ denotes the Lambert $W$ function.

Define

Then

2. Product Structure

The product becomes

Expanding,

3. Domain and Branch Point

Consider the argument

For $t\ge 0$ ,

At $t=0$ ,

This is the branch point of the Lambert $W$ function.

4. Local Expansion Near the Branch Point

Let

Near the branch point,

Thus,

Therefore,

near $t=0$ .

5. Derivatives of the Velocity Components

Differentiate:

Using the derivative formula for the Lambert $W$ function,

we obtain

Simplifying,

Near the branch point,

Therefore,

and hence

6. Derivative of the Product

Since

we compute

7. Asymptotic behavior of the Product Derivative

Using

and

we obtain

Thus the derivative remains finite.

8. Regularity Conclusion

We have established:

are continuous but not differentiable at the branch point because

diverge.

However,

is differentiable because the singularities cancel.

Therefore,

9. Structural Interpretation

The time derivative of the product is

Each derivative term is singular, but each is multiplied by a vanishing factor.

Thus,

the product remains smooth while the individual components are singular.

This mechanism corresponds to cancellation of singularities in nonlinear transport structures and characteristic degeneracy near branch points. This shows that the degeneracy is controlled by the characteristic mapping rather than by the velocity magnitude itself.

Appendix E

To calculate the scalar quantity $\nabla(\vec{b}) \cdot \nabla(u_z)$ , we first define the components of the vector field $\vec{b}$ and the scalar gradient $\nabla(u_z)$ .

1. The Gradient of $u_z$

The gradient of the scalar function $u_z(x, y, z, t)$ is:

2. The Jacobian Matrix $\nabla(\vec{b})$

The gradient of a vector field $\vec{b} = (b_x, b_y, b_z)$ is the Jacobian matrix $(\nabla \vec{b})_{ij} = \frac{\partial b_i}{\partial x_j}$ . Given $\vec{b} = (u_y u_z, u_x u_z, u_x u_y)$ :

Using the product rule, the components are:

Row 1: $(u_z \partial_x u_y + u_y \partial_x u_z, \quad u_z \partial_y u_y + u_y \partial_y u_z, \quad u_z \partial_z u_y + u_y \partial_z u_z)$

Row 2: $(u_z \partial_x u_x + u_x \partial_x u_z, \quad u_z \partial_y u_x + u_x \partial_y u_z, \quad u_z \partial_z u_x + u_x \partial_z u_z)$

Row 3: $(u_y \partial_x u_x + u_x \partial_x u_y, \quad u_y \partial_y u_x + u_x \partial_y u_y, \quad u_y \partial_z u_x + u_x \partial_z u_y)$

3. The Operation $\nabla(\vec{b}) \cdot \nabla(u_z)$

Strictly speaking, $\nabla(\vec{b}) \cdot \nabla(u_z)$ represents the matrix-vector product. If we denote $V = \nabla(u_z)$ , then the $i$ -th component of the resulting vector is $\sum_j \frac{\partial b_i}{\partial x_j} \frac{\partial u_z}{\partial x_j}$ .

The components of the resulting vector are:

X-component:

Y-component:

Z-component:

4. Expansion of Terms

Using the standard dot product notation where $|\nabla u_z|^2 = (\partial_x u_z)^2 + (\partial_y u_z)^2 + (\partial_z u_z)^2$ :

First Component:

Second Component:

Third Component:

Summary in Vector Form

The result can be compactly written as:

Appendix F

Theory of characteristics for a General Nonlinear Transport Equation

We work in the framework commonly used in the analysis of nonlinear transport and conservation laws arising in the incompressible Navier–Stokes equations and related evolution systems. Refer to the theory of characteristics for a general nonlinear transport equation as shown in .

Appendix G

Local Well-Posedness from Smooth Initial data

We prove local well-posedness for the nonlinear transport equation

with smooth parameter dependence in $(x,y)$ treated as frozen parameters.

Thus the analysis is carried out in $(z,t)$ .

1. Reformulation as a Quasilinear Transport Equation

We write the PDE in the form

where

This is a quasilinear first-order PDE.

2. Initial data

We prescribe smooth initial data

We aim to prove local existence, uniqueness, and smoothness.

3. Characteristic system

Define characteristics $(Z(t),B(t))$ by

with initial conditions

This defines a nonlinear ODE system.

4. Local Existence of characteristics

The vector field is

The function $\cot(s)$ is smooth on any interval avoiding $s=k\pi$ . At $t=0$ , we have $s=-\xi$ , which is fixed and finite.

Hence $F \in C^\infty$ locally in time.

By the Picard–Lindelöf theorem:

$\exists T_\xi>0$ such that $(Z(t),B(t))$ exists uniquely and smoothly on $[0,T_\xi]$ .

5. Smooth Dependence on Initial data

Since the vector field is smooth in $(\xi,b_0(\xi))$ , ODE theory implies

Thus

6. Invertibility of the Characteristic Map

Define the Eulerian map

Differentiate:

At $t=0$ :

Thus for sufficiently small $t$ ,

Hence $\Phi_t$ is locally invertible and

exists smoothly.

7. Construction of the Eulerian solution

Define

Since

$B \in C^\infty$ ,
$\xi(z,t)\in C^\infty$ ,
composition preserves smoothness,

we obtain

8. Uniqueness

Let $b_1,b_2$ be two solutions. Along characteristics,

Factor:

Thus

Applying Grönwall’s inequality gives

9. Local Well-Posedness Theorem

Theorem. Let $b_0 \in C^\infty(\mathbb{R})$ . Then there exists $T>0$ such that:

(Existence) a solution $b(z,t)$ exists on $[0,T]$ ,
(Uniqueness) the solution is unique,
(Regularity) $b \in C^\infty$ ,
(Continuous dependence) $b$ depends smoothly on $b_0$ .

10. Limitations of the Result

This result does not guarantee global regularity because:

The characteristic map may lose invertibility:

$\partial_{ξ} Z (t, ξ) = 0 \Rightarrow shock formation .$
The forcing term is singular:

$\cot (t - z) \to \infty when t - z = k π .$

Thus the maximal existence time satisfies

11. Relation to the Lambert $W$ Representation

The explicit form

is consistent with the local theory because:

local well-posedness ensures $Z$ remains in a smooth branch,
the branch point $Z=-e^{-1}$ is not reached instantly,
singularities occur only if characteristics reach the boundary.

Final Statement

However,

global regularity fails when characteristics reach $t-z=k\pi$ or compress.

Why the Characteristic System $(Z(t),B(t))$ is Introduced

We explain rigorously why the system

is considered and how it is derived from the partial differential equation.

1. General First-Order Quasilinear PDE

Consider a first-order quasilinear PDE of the form

This equation describes transport of the quantity $u$ with velocity $a(t,z,u)$ and forcing $f(t,z,u)$ .

2. Definition of a Characteristic Curve

Definition.

A curve

in space–time is called a characteristic curve if along that curve the partial differential equation reduces to an ordinary differential equation.

We define the trajectory by

This choice ensures that the spatial motion matches the transport velocity.

3. Total Derivative Along a Moving Curve

Let

be a sufficiently smooth solution.

Define

Using the chain rule,

Substitute the definition of the characteristic velocity:

Therefore

4. Reduction of the pde to an ode

Using the original PDE,

we obtain

Thus along a characteristic curve, the PDE becomes an ordinary differential equation.

This is the fundamental reason characteristics are introduced.

5. Application to the Present Equation

We consider the equation

Identify

6. Characteristic system

Therefore the characteristic equations are

and

These equations describe:

motion of the spatial point $Z(t)$ with velocity equal to the field $b$ ,
evolution of the field value $B(t)$ along that moving point.

7. Initial conditions

The characteristic starting point is determined by the initial data:

Thus each spatial point $\xi$ generates one trajectory.

8. Reconstruction of the pde solution

After solving the ODE system, we obtain

If the mapping

remains invertible, then

exists.

We define the Eulerian solution by

This function satisfies the original PDE.

9. Geometric Interpretation

Characteristics are the trajectories along which information travels.

In this equation,

means:

Thus the PDE describes a self-advecting flow.

10. Relation to Shock and Singularity Formation

Loss of regularity occurs when characteristics intersect.

Mathematically:

At that moment,

the inverse mapping fails,
spatial gradients become infinite,
a singularity forms.

This mechanism is called

Final Statement

The characteristic system $(Z(t),B(t))$ is introduced because:

it converts the PDE into an ODE system,
it provides the rigorous construction of solutions,
it determines existence and uniqueness,
it identifies the mechanism of singularity formation.