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A bootstrap argument assumes an a priori bound on an initial time interval and derives a strict improvement using an energy estimate or another inequality. Continuity extends the improved bound to the entire interval under consideration. For example $A(t)\le C\epsilon +C{\int }_0^t(1+s{)}^{-1}A(s{)}^{2}ds$ closes $A\le 2C\epsilon$ when $\epsilon \log (1+t)$ is sufficiently small. The strict improvement is what prevents a first failure time.

With zero endpoints, the centered first-difference matrix has opposite off-diagonal entries and is a skew-adjoint operator in the mesh-weighted Euclidean inner product. Adding a real multiple of it to the negative symmetric second-difference matrix does not change the real energy form. The semidiscrete convection-diffusion solution therefore obeys a mesh-independent energy estimate, even when its drift is large compared with its diffusion coefficient.

A real bilinear form $B$ on a normed vector space $X$ is coercive when there is one constant $\alpha >0$ such thatThe lower bound uniformly controls every direction in the chosen norm. Unlike the upper bound defining a bounded bilinear form, it forces positive diagonal values at every nonzero vector. No symmetry is assumed. For complex Hilbert spaces, the usual analogue uses a sesquilinear form and $\mathrm{Re}B(v,v)\ge \alpha \Vert v{\Vert }^2$; it is not the same assertion for a complex bilinear form.

A concrete bounded and coercive example on ${\mathbb{R}}^2$, with the Euclidean norm, isIts quadratic form isThe best lower constant is $\alpha =2$, attained at $(1,0)$. Nevertheless, $B(e_1,e_2)=1$ and $B(e_2,e_1)=-1$, so coercivity does not imply a symmetric bilinear form. The symmetric part of a matrix $S=(A+A^T)/2=\mathrm{diag}(2,3)$ determines $q$, while the skew-symmetric matrix part changes mixed pairings without changing $q$. This is the symmetric and alternating parts of a bilinear form decomposition.^[ref]

For the upper constant, direct multiplication gives $A^{T}A=\left(\begin{array}{cc}5& -1\\ -1& 10\end{array}\right)$. Its eigenvalues are $(15\pm \sqrt{29})/2$, so the optimal upper constant, the spectral norm, is $C=\sqrt{(15+\sqrt{29})/2}$. The two constants answer different questions: how large a mixed pairing can be, and how small a self-pairing can be relative to squared norm.

The plotted quadratic forms show the mechanism: $x^2-y^2$ descends in one direction, $x^2$ leaves one direction uncontrolled, and $2x^2+3y^2$ controls both. These are self-pairings; their level sets cannot reveal the skew-symmetric matrix part. Calling $q$ a quadratic energy here refers to this diagnostic quantity, not to a claim that a nonsymmetric equation comes from minimizing an energy functional.

On a real finite-dimensional vector space with any norm, strict positivity $B(v,v)>0$ for every nonzero $v$ does imply coercivity. Indeed the quadratic form is continuous, and its positive minimum on the compact unit sphere supplies $\alpha$. If $B$ is also a symmetric bilinear form, this is positive definiteness. For general $B$, it is the symmetric and alternating parts of a bilinear form decomposition that isolates the positive part. Switching between equivalent norms preserves coercivity, but usually changes its constant.

The infinite-dimensional vector space distinction is explicit on the real l2 sequence space:The Cauchy-Schwarz inequality gives boundedness with $C=1$, and $B(v,v)>0$ for every nonzero vector. But $B(e_n,e_n)=1/n$ with $\Vert e_n{\Vert }_2=1$, so no positive $\alpha$ works. On the span of $e_1,\dots ,e_N$, the best lower constant is $1/N$: each finite restriction is coercive, but the control disappears as $N$ grows. Strict positivity is therefore weaker than a uniform lower bound.

The practical payoff is a stability estimate for a weak formulation. Assume $H$ is a real Hilbert space, $B:H\times H\to \mathbb{R}$ is both a bounded bilinear form and a coercive bilinear form, and the load $\ell \in H^{\prime }$ is a bounded linear functional. The Lax-Milgram theorem gives a unique $u\in H$ satisfying $B(u,v)=\ell (v)$ for all $v\in H$, including nonsymmetric $B$.^[ref] The dual norm $\Vert \ell {\Vert }_{H^{\prime }}={\sup }_{v\ne 0}|\ell (v)|/\Vert v\Vert$ measures load size. Testing with $v=u$ gives the short energy estimateFor $u=0$ the bound holds directly; otherwise divide by $\Vert u\Vert$. Subtracting the equations for two loads and using the same argument gives $\Vert u_1-u_2\Vert \le \Vert {\ell }_1-{\ell }_2{\Vert }_{H^{\prime }}/\alpha$. Thus a positive lower constant limits the response to load errors. The upper bound and completeness of the Hilbert space enter the existence theorem; the estimate itself uses the lower bound and the load's dual norm.

For the displayed matrix $A$, take $\ell (v)=f^{T}v$. Then $B(u,v)=\ell (v)$ for every $v$ means $A^{T}u=f$, not $Au=f$, with this ordering of the bilinear form. Since $\det A=7$,For $f=t(1,0{)}^T$, the exact response is $u=t(3,-1{)}^T/7$ and $\Vert u{\Vert }_2=\sqrt{10}|t|/7<|t|/2$. For $f=t(0,1{)}^T$, it is $u=t(1,2{)}^T/7$, with norm $\sqrt{5}|t|/7$. The actual maximal amplification is the operator norm $\Vert (A^T{)}^{-1}{\Vert }_2=\sqrt{2/(15-\sqrt{29})}<1/2$: the coercivity estimate is valid but need not be sharp for a nonsymmetric bilinear form.

For zero endpoint values and real constant drift, integration by parts gives $\frac{1}{2}d\Vert u{\Vert }_{L^2}^2/dt=-\Vert u_x{\Vert }_{L^2}^2$. The drift contributes only a zero boundary term in the real part. Thus the solution is contractive, uniformly in the drift. Multiplication by $e^{\kappa x}$ converts the equation to a heat equation with constant killing rate ${\kappa }^2$, providing existence and uniqueness as well as the direct energy estimate.

For smooth localized compatible wave map Cauchy data and compact Riemannian target, one-dimensional wave maps stay smooth globally. In null coordinates, the equation is $D_v{\varphi }_u=D_u{\varphi }_v=0$. Compatibility of the target metric with the covariant derivative makes $|{\varphi }_u{|}^2$ independent of $v$ and $|{\varphi }_v{|}^2$ independent of $u$. These transported derivative bounds and higher energy estimates prevent finite-time breakdown.

On $0<x<L$, $t>0$, an initial condition $u(x,0)=g(x)$ gives the starting profile. For the heat equation, Dirichlet boundary conditions $u(0,t)=a(t)$, $u(L,t)=b(t)$ can model ends held at prescribed temperatures. Alternatively, zero Neumann boundary conditions describe insulated ends in the constant-conductivity model. These choices can produce different evolutions from the same initial condition.

For a continuous classical solution reaching the corners, the Dirichlet boundary data must satisfy $g(0)=a(0)$ and $g(L)=b(0)$; higher smoothness can require further compatibility. In dimensionless variables, a complete worked heat equation problem isIts exact classical solution is $u(x,t)=x+e^{-{\pi }^{2}t}\sin (\pi x)$. The straight profile $x$ is the equilibrium; the extra bump decays. Substituting verifies the equation and every datum, and the energy estimate proves uniqueness among smooth solutions with these data.

For the transport equation with $c>0$ on the same interval, prescribe an inflow transport boundary condition only at $x=0$. Backtracking a characteristic curve gives $g(x-ct)$ if $x\ge ct$, and $a(t-x/c)$ otherwise. Values at $x=L$ are determined by this formula; independently prescribing them can make the problem inconsistent.^[ref]

Stability of a numerical method controls how perturbations in initial data, roundoff and local residuals propagate. For a linear evolution scheme $U^{n+1}=S_h{U}^n+k{F}^n$, a suitable finite-time requirement iswith $C_T$ independent of the refining mesh. The norm should approximate the norm of the continuous problem, such as an area-weighted L2 norm. The bound need not imply decay: $C_T=e^{CT}$ is appropriate when the equation itself permits finite-time growth. For a multilevel scheme the state includes all required time levels.

The error recurrence shows why stability matters. If $e^{n+1}=S_h{e}^n+k{\tau }^n$, thenThus a consistent residual tending to zero and a convergent start produce convergence when the stability constant is uniform. Tiny defects need not remain tiny without that bound. The Lax equivalence theorem makes this precise: for a well-posed linear initial value problem and a consistent linear approximation, stability is equivalent to convergence in the compatible solution norms. It is not a theorem for arbitrary nonlinear schemes, and a high formal order does not substitute for stability.

For constant coefficients on a periodic or infinite grid, von Neumann stability analysis diagonalizes spatial translations using Fourier modes. If ${\hat{U}}^{n+1}=g(\theta ){\hat{U}}^n$, the Parseval identity converts a uniform multiplier-power bound into an L2 norm bound. The contraction condition $|g(\theta )|\le 1$ is sufficient; a more general $|g(\theta )|\le 1+Ck$ also yields a finite-time bound. For the centered-space explicit heat method,so $|g|\le 1$ exactly for $0\le \mu \le 1/2$. Larger fixed Courant numbers amplify high frequencies. The Backward Euler method gives instead $g=[1+4\mu {\sin }^2(\theta /2){]}^{-1}$ and is contractive for every $\mu \ge 0$.

For $u_t=u_x$, forward time and centered space give $g=1+i\nu \sin \theta$, with $\nu =k/h$. Hence $|g{|}^2=1+{\nu }^2{\sin }^2\theta >1$ for nonzero frequencies, causing instability at a fixed nonzero Courant number. The directionally correct upwind finite difference scheme uses the neighbor at $m+1$ and hasIt is contractive exactly for $0\le \nu \le 1$. This also has a direct maximum-norm proof: the update is a convex combination of two old values. The continuum domain of dependence must fit inside the numerical one for a local explicit hyperbolic method to converge, giving the Courant–Friedrichs–Lewy condition. That necessary geometric restriction is not by itself a sufficient stability proof.

For systems and multilevel methods, each Fourier multiplier becomes a matrix. All amplification roots must obey the appropriate root bound, but their moduli alone are insufficient. A repeated unit root in a companion matrix gives a Jordan block; for instance $J=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ has $J^n=\left(\begin{array}{cc}1& n\\ 0& 1\end{array}\right)$. Even distinct roots require control of the diagonalizing matrices uniformly in frequency and mesh. The uniform power bound from separated amplification roots supplies such control for a bounded two-root family with a positive gap. Question5 illustrates both sides: strict $0<\mu <1$ gives a uniform gap, whereas the endpoints have a double unit root and are unstable despite unit moduli.

Energy estimates are especially useful when coefficients vary or boundaries prevent Fourier diagonalization. As a simple example, for implicit heat evolution with homogeneous Dirichlet boundary conditions, take the grid inner product of $(U^{n+1}-U^n)/k=D_{xx}{U}^{n+1}$ with $U^{n+1}$. Discrete integration by parts givesThe boundary term vanishes because the boundary values are zero, proving contraction. For variable positive diffusion coefficients the corresponding flux-weighted squared differences remain nonnegative, so the argument extends without a constant-coefficient Fourier symbol. With forcing or lower-order terms, a discrete energy inequality and Gronwall inequality yield a mesh-independent finite-time bound.

A related technique is eigenvalue stability analysis of a finite difference method. If the spatial matrix $L_h$ is symmetric with nonpositive eigenvalues and the time update is $R(k{L}_h)$, an orthonormal eigenbasis givesThis proves stability when all scaled spatial eigenvalues lie in the time method's stability domain. For a nonnormal spatial or amplification matrix, the eigenvalues alone do not control powers: ill-conditioned eigenvectors or Jordan terms can cause transient growth. Direct operator-norm estimates, symmetrizers or energy estimates then provide the missing information. In particular an A-stable scalar time formula does not automatically give mesh-uniform bounds for every ill-conditioned family of spatial matrices.

Boundary conditions and starting procedures are part of the discretization. Interior Fourier analysis may fail to detect a growing boundary mode, and a multilevel method can excite a parasitic mode through a poor start. One must therefore analyze the actual finite-domain operator or include its boundary terms in an energy proof. Stability is a uniform error-propagation bound; Fourier symbols, energy identities, matrix norms and spectral analysis are complementary ways of establishing that bound.

Write $h=\Delta x$, impose $u_0=u_{M+1}=0$, and use the discrete inner product $⟨u,v{⟩}_h=h{\sum }_{m=1}^M{\overline{u}}_m{v}_m$. Discrete integration by parts for the second difference givesFor the centered drift, reindexing with the zero endpoints givessince the two sums are complex conjugates. This is the fact that centered discrete advection is skew-adjoint. The semidiscrete energy estimate is thereforeThe finite-dimensional linear system has a unique solution for all times, and $\Vert u(t){\Vert }_h\le \Vert u(0){\Vert }_h$, uniformly in the mesh and in real $\kappa$. Differences of two semidiscrete solutions obey the same bound, which is the required stability.

One may also use the smallest eigenvalue ${\lambda }_h=4h^{-2}{\sin }^2(\pi /[2(M+1)])$ of the negative second-difference operator to obtain $\Vert u(t){\Vert }_h\le e^{-{\lambda }_{h}t}\Vert u(0){\Vert }_h$. This conclusion concerns the method of lines; a later choice of time integrator has its own stability conditions.

The eigenvalue stability analysis of a finite difference method reduces a linear evolution discretization to its modal behavior, but it must retain the norm and the dependence on the mesh. For a homogeneous system obtained by the method of lines, $U^{\prime }=A_{h}U$, the solution is $e^{t{A}_h}U(0)$. A one-step time discretization gives $U^{n+1}=G_{h,k}{U}^n$; for a Runge-Kutta method, where its stage matrices are invertible, $G_{h,k}=R(k{A}_h)$. With a multistep method one instead uses an amplification matrix on the augmented vector of several time levels.

The relevant finite-time stability estimate iswith $C_T$ independent of the refining spatial mesh and allowed time steps. Similarly, semidiscrete stability requires $\Vert e^{t{A}_h}{\Vert }_h\le C_T$ for $0\le t\le T$. The norm should represent the continuum problem, such as the discrete L2 norm $\Vert U{\Vert }_{2,h}^2=h{\sum }_j|U_j{|}^2$ in one dimension. A fixed finite matrix having decaying solutions as $t\to \infty$ is not by itself a mesh-uniform result for the partial differential equation.

If $G=X\Lambda X^{-1}$, thenThus eigenvalues of modulus at most one give a uniform bound when the diagonalizing bases have uniformly bounded condition numbers. For a normal matrix in the chosen inner product, the spectral theorem for normal operators supplies an orthonormal basis, so the condition number is one and the norm of $G^n$ is exactly ${\max }_j|{\lambda }_j{|}^n$. For a self-adjoint or skew-adjoint spatial matrix, applying a scalar stability function preserves this favorable modal structure. This is the setting where eigenvalue calculations provide especially clean, reliable step restrictions.

A precise fixed-matrix statement is also useful: a matrix is power bounded for all nonnegative integers $n$ if and only if all its eigenvalues lie in the closed unit disk and every Jordan block at a unit-modulus eigenvalue has size one. Indeed, powers of an interior Jordan block contain a polynomial in $n$ times a decaying geometric factor and are bounded; a larger block on the unit circle produces unbounded polynomial growth. For a family of matrices, the resulting bounds must still be uniform. Moreover, finite-time stability versus power boundedness distinguishes this all-time condition from the physical-time estimate above. For example $G_k=I+kN$, $N^2=0$, has $G_k^n=I+nkN$, bounded for $nk\le T$ despite a nontrivial Jordan block at one. Even the scalar update $1+k$ is stable on fixed intervals, since $(1+k{)}^n\le e^T$, although it represents a growing equation rather than a contraction.

For a constant-coefficient periodic difference operator, Fourier modes diagonalize the spatial matrix. The discrete Fourier transform is a unitary operator, so Parseval's identity identifies its modal maximum with the operator norm. The eigenvalue stability analysis of a finite difference method then becomes von Neumann stability analysis. For the Dirichlet discrete Laplacian, a discrete sine transform plays the same role. These facts explain its advantages: a large matrix calculation becomes a scalar symbol or a known spectral interval, and the extremal eigenvalue directly determines a safe time step.

As a first example, centered space discretization of the heat equation $u_t=u_{xx}$ has eigenvaluesForward Euler method gives $R(k\lambda )=1-4r{\sin }^2(\theta /2)$ with $r=k/h^2$. Requiring all factors in $[-1,1]$ yields $0\le r\le 1/2$ for all periodic meshes. In two dimensions the corresponding uniform bound is $r\le 1/4$. Because these are normal matrices, these scalar bounds prove contraction rather than merely suggest it.

Backward Euler method has $R(z)=1/(1-z)$, so the centered discretization of the heat equation is contractive for every $k>0$. The Crank-Nicolson method has $R(z)=(1+z/2)/(1-z/2)$ and is likewise contractive for these negative real modes. However, its factor tends to $-1$ for very stiff modes, leaving high-frequency oscillations weakly damped; Backward Euler method tends to zero. The eigenvalues therefore expose the presence of stiff differential equations and damping as well as binary stability. Unconditional stability does not remove accuracy requirements.

For an advection equation example, the upwind finite difference scheme for $u_t+c{u}_x=0$, $c>0$, has factor $G=1-\nu +\nu e^{-i\theta }$, with $\nu =ck/h$. Sinceit is a contraction in the periodic discrete L2 norm exactly for $0\le \nu \le 1$. In contrast, Forward Euler method with centered advection has factor $1-i\nu \sin \theta$. Its modulus exceeds one at nonzero active frequencies, and repeated amplification makes it unstable under fixed nonzero Courant refinement. A much smaller scaling $k=O(h^2)$ can bound its finite-time growth, demonstrating again that exact contraction and the most general stability estimate are different claims.

The major limitation is the possible presence of a nonnormal matrix. For example,Both eigenvalues are negative, yet at $t=1$ the second unit vector is amplified by at least $1/(eh)$. This proves that negative spectra do not imply uniform semidiscrete stability. Ill-conditioned eigenvectors, Jordan normal form and transient amplification can invalidate a spectral-only argument. Direct energy estimates or estimates for the resolvent can supply the missing norm control.

Boundary conditions are another limitation. A periodic symbol proves a periodic or whole-line result, not an arbitrary initial-boundary-value problem. Boundary closure of a difference scheme can introduce growing modes or mesh-dependent amplification. A spatial operator with variable coefficients can still be studied through its assembled matrix, but then whether it is a normal matrix, its condition number and the physical inner product must be checked rather than borrowed from a Fourier stability analysis. For the finite element method the natural mass matrix inner product often turns a generalized eigenvalue problem into a self-adjoint one.

For time-dependent or split updates, separate spectra are insufficient to control products. For instance, $G_1=\left(\begin{array}{cc}0& K\\ 0& 0\end{array}\right)$ and $G_2=\left(\begin{array}{cc}0& 0\\ K& 0\end{array}\right)$ each have only zero eigenvalues, while $G_2{G}_1$ has eigenvalue $K^2$. For $K>1$, alternating the two is unstable. A common contractive norm would control the products, but their individual eigenvalues do not.

Finally, stability and consistency of a numerical method have different roles. The Lax equivalence theorem states that, for a well-posed linear initial-value problem and a consistent finite difference method on compatible grid spaces, stability on every fixed time interval is equivalent to convergence of a numerical method as the admissible meshes refine. Bounded grid representation and the correct initial and boundary setup are part of that framework. An eigenvalue estimate can establish its stability hypothesis in the favorable settings described above; it neither proves consistency of a numerical method nor bypasses its uniformity requirements. The method is therefore a powerful modal tool, provided its spectral calculation is connected to a genuine norm estimate.

The local wave energy estimate is, for $T\ge 0$, $r>0$, and any center $x_0$,To prove it, let $R=r+T$ and integrate the local energy estimate identity ${\partial }_{t}e=\mathrm{div}(u_t\nabla u)$, where $e=(u_t^2+|\nabla u{|}^2)/2$, over the shrinking ball $B_{R-t}(x_0)$. Differentiation of this moving-domain integral yieldsIntegrating in time proves the local wave energy estimate, without assumptions at spatial infinity.

Let the union of the initial supports be a compact set $K$. If $\mathrm{dist}(x_0,K)>T$, choose $r>0$ with $r+T<\mathrm{dist}(x_0,K)$. The initial energy on $B_{r+T}(x_0)$ vanishes. Applying the shrinking-ball identity up to every intermediate time shows $u_t$ and $\nabla u$ vanish throughout that cone. In particular, along the vertical segment through $x_0$, $u_t=0$; its initial value is also zero, so $u(T,x_0)=0$. This last value check removes the constant ambiguity invisible to gradient energy.

Consequently the finite propagation speed conclusion isThis set is compact for each finite $t$. The speed is at most one in these units, or $c$ for $u_{tt}-c^2\Delta u=0$. Time reversal gives the same conclusion for negative time.

For the sharp energy estimate, $C_0=0$ stays uniform while the available bound $C_1=M^2/\epsilon$ diverges as $\epsilon \downarrow 0$ when $M>0$. If the coarser estimate is used for the first part, both displayed bounds $M^2/\epsilon$ diverge, but the first divergence is only an artifact of discarding an exact cancellation.

The method of characteristics explains why uniform control of the gradient cannot generally persist for the inviscid scalar conservation law. Before characteristic crossing, with initial data $u_0$,If $F^{\prime \prime }(u_0(\xi ))u_0^{\prime }(\xi )<0$ somewhere, the denominator reaches zero in finite positive time. The solution steepens and the smooth description breaks down; an entropy solution can subsequently contain shocks. Positive viscosity replaces such a discontinuity by a thin smooth layer, which can have large gradient even while its $L^2$ norm remains controlled. The divergent bound does not assert that every flux and every initial datum form a shock; a linear flux, for example, has no such steepening.

Multiply the viscous scalar conservation law by $-u_{xx}$, rather than estimating $F^{\prime \prime }$ after differentiating. Integration by parts yieldsThus $d\Vert u_x{\Vert }_2^2/dt\le (M^2/\epsilon )\Vert u_x{\Vert }_2^2$. Applying the Gronwall inequality givesOnly the assumed bound on $F^{\prime }$ is used; a global bound on $F^{\prime \prime }$ is not needed for this energy estimate.

Let $M=\Vert F^{\prime }{\Vert }_{\infty }$. The sharper energy estimate uses the divergence structure of the viscous scalar conservation law. Multiply by $u$ and integrate over the line. With $G(s)={\int }_0^{s}r{F}^{\prime }(r)dr$,because the decay makes $G(u)$ tend to zero at both ends. Integration by parts in the diffusion term therefore givesOne may take $C_0=0$, uniformly in $\epsilon$. This does not require $F(0)=0$.

If a bound explicitly involving $M$ and $\epsilon$ is desired, retaining the transport term and using the Cauchy-Schwarz inequality and the elementary inequality $ab\le (a^2+b^2)/2$ givesThe Gronwall inequality then gives the valid but weaker choice $C_0=M^2/\epsilon$. The exact cancellation explains why its divergence is unnecessary.

We prove finite propagation speed using a shrinking cone energy argument. Fix $T>0$ and $x_{\ast }$, and suppose the Cauchy data vanish on $B(x_{\ast },T)$. For $0\le s<T$, define the local wave energyThe homogeneous wave equation gives the local conservation lawDifferentiate the integral over the moving ball. Its boundary moves inward with speed one, so the divergence theorem givesHere $n$ is the outward unit normal, ${\partial }_n$ the normal derivative, and ${\nabla }_{\tan }$ the component of the gradient tangent to the boundary. Since $E(0)=0$ and $e\ge 0$, we have $E(s)=0$. Hence both ${\varphi }_t$ and $\nabla \varphi$ vanish inside the backward light cone. Integrating ${\varphi }_t(s,x_{\ast })=0$ from the zero initial displacement gives $\varphi (s,x_{\ast })=0$; continuity then gives $\varphi (T,x_{\ast })=0$.

Applying the same energy estimate to the difference of two solutions proves the domain of dependence assertion. If $\mathrm{dist}(x_{\ast },K)>T$, the initial ball misses $K$, and the preceding argument proves the stated support bound. No disturbance propagates faster than one.

We establish a quantitative almost global existence for wave equations estimate. Take $0<\epsilon \le 1$, fix an integer $m\ge 6$, and use the commutation vector fields for the wave equation from the preceding part. Put $\partial =({\partial }_t,\nabla )$ and define the commuted wave energyAll these L2 norms are finite on any smooth existence interval by finite propagation speed. At $t=0$, the polynomial coefficients of the vector fields are bounded on the fixed compact support of the Cauchy data. Whenever a higher time derivative occurs, use ${\varphi }_{tt}=\Delta \varphi -{\varphi }_t^2$ and its differentiated versions to express it in terms of initial spatial derivatives. Every term contains at least one factor of $\epsilon$; consequentlyfor a constant depending only on finitely many derivatives and the support radius of ${\varphi }_0,{\varphi }_1$.

The commutators $[Z,{\partial }_{\mu }]$ are constant linear combinations of translations. Together with $[\square ,S]=2\square$ and the Leibniz rule, this shows that each commuted source is a finite linear combination of productsThis statement includes the extra copies of the original source produced by the scaling vector field. In each product put the factor with fewer commutations in the Lp norm $L^{\infty }$ and the other in the L2 norm. The lower order is at most $\lfloor m/2\rfloor$. Applying the Klainerman-Sobolev inequality to ${\partial }_{\mu }{Z}^J\varphi$ costs at most two additional commutations; commuting those past ${\partial }_{\mu }$ introduces only lower-order translations. Since $\lfloor m/2\rfloor +2\le m$,The inhomogeneous wave energy estimate now givesLet $T_{\epsilon }={\epsilon }^{-N}$. Use a bootstrap argument with $A_m(t)\le 2C_0\epsilon$ up to the smaller of $T_{\epsilon }$ and the maximal existence time. The energy estimate improves this toFor each fixed $N$,Choose ${\epsilon }_N\le 1$ so that $4C_m{C}_0\epsilon \log (1+{\epsilon }^{-N})\le 1/2$ for every $0<\epsilon \le {\epsilon }_N$. Then $A_m(t)\le (3/2)C_0\epsilon$, a strict improvement. A continuity argument closes the bootstrap argument.

Finally, the translation terms in $A_m$ control ordinary spatial Sobolev norms of $\partial \varphi$. The missing L2 norm of $\varphi$ satisfiesThus the full local-existence Sobolev norms remain bounded on this finite interval. The smooth continuation criterion for semilinear wave equations extends the solution past any finite endpoint before $T_{\epsilon }$. To see smooth persistence explicitly, the tame Sobolev product estimate gives $\Vert {\varphi }_t^2{\Vert }_{H^k}\le C_k\Vert {\varphi }_t{\Vert }_{\infty }\Vert {\varphi }_t{\Vert }_{H^k}$. Ordinary differentiated wave energy estimates therefore bound each higher derivative energy by its initial value times $\exp (C_k{\int }_0^t\Vert {\varphi }_t(s){\Vert }_{\infty }ds)$. This is finite on the interval already controlled by the base commuted wave energy; no separate ${\epsilon }_N$ is needed for each derivative order. ThereforeFor $\epsilon =0$ the zero solution is global. The same energy estimate in fact permits an exponential lower bound for the lifespan, which is stronger than any fixed inverse power.

Smooth compatible wave map Cauchy data give a unique local smooth wave map. Relative to a constant map, Sobolev spaces $H^s\times H^{s-1}$ with $s>n/2+1$ provide a classical local theory. Iteration for the semilinear wave equation, Sobolev algebra and energy estimates give existence, uniqueness and continuous dependence. The smooth continuation criterion for semilinear wave equations extends the solution while these norms stay bounded. The sphere constraint and tangency constraint remain satisfied.

Global regularity for one-dimensional wave maps holds for arbitrary smooth localized compatible data. In retarded and advanced null coordinates $u=t-x$, $v=t+x$, the wave map equation says $D_v{\varphi }_u=D_u{\varphi }_v=0$, with $D$ the target covariant derivative. Hence $|{\varphi }_u{|}^2$ depends only on $u$, and $|{\varphi }_v{|}^2$ only on $v$. Initial derivative bounds persist. Compactness of the sphere and differentiated energy estimates then prevent finite-time loss of smoothness.

Small data global regularity for wave maps also holds in three spatial dimensions. For localized compatible data small in high weighted Sobolev norms, the key is the classical null condition for wave equations. Each derivative contraction is a null form for wave equations, vanishing for parallel null derivatives. The vector field method for wave equations exploits derivatives tangent to the light cone and weighted energy estimates to close the bootstrap argument; ordinary $(1+t{)}^{-1}$ decay alone is insufficient.

The original PDF assumes $u\ge 0$ at the beginning of this question; the TeX conversion omits that hypothesis. First replace a smooth nonnegative solution by $v=u+\epsilon >0$, which solves the same homogeneous divergence-form elliptic equation. For $w=v^{q/2}$, the chain rule gives the favorable termin the distributional sense. Thus $w$ is a weak subsolution. Testing its subsolution inequality with ${\zeta }^{2}w$ yields the same Caccioppoli inequality as before. More precisely, testing the equation for $v$ with ${\zeta }^2{v}^{q-1}$ givesIndeed the main term is at least $(q-1)\lambda \int {\zeta }^2{v}^{q-2}|\nabla v{|}^2$, and the cross term is bounded by twice $\Lambda$ times the square roots of this weighted gradient integral and $\int v^q|\nabla \zeta {|}^2$. Let $\epsilon \downarrow 0$ to obtain the stated power energy estimate.

Apply the fixed-domain Sobolev embedding theorem to $\zeta u^{q/2}$. The crucial constant stays uniform before taking the $2/q$ power:where $D_{\ell }\ge 1$ is independent of $q$. This is uniform power gain for elliptic solutions. The exponent is $q\alpha$, not the TeX conversion's $q^{\alpha }$. The PDF's weaker prefactor $C_{\ell }$ outside the $2/q$ power follows from this stronger estimate, but that weaker form alone cannot be iterated infinitely.

For completeness, the estimate also extends to nonnegative locally $H^1$ weak solutions without assuming they are already bounded. For $s\ge 0$ define $F_M(s)=s^{q/2}$ up to $M$ and continue it linearly with slope $(q/2)M^{q/2-1}$ above $M$; set ${\Psi }_M(s)={\int }_0^s{F}_M^{\prime }(r{)}^{2}dr$. Both have bounded derivatives for fixed $M$. Direct calculation gives ${\Psi }_M/F_M^{\prime }\le F_M$ and $F_M(s)\le (q/2)s^{q/2}$. Test the weak equation with ${\zeta }^2{\Psi }_M(u)$, using smooth approximations to these Sobolev chain rule functions if needed. The resulting weighted energy bound is $\int {\zeta }^2|\nabla F_M(u){|}^2\le 4R^2\int F_M(u{)}^2|\nabla \zeta {|}^2$. If $u\in L^q(B_{r_2})$, dominated convergence on the right and the Fatou lemma on the Sobolev left give the boxed gain as $M\to \infty$, with the same uniform constant. Starting with $q=2$, this justifies every subsequent iteration step.